Statistics Module 2 & 3

MASTER OF ARTS IN NURSING

INTRODUCTIONINTRODUCTION

Undoubtedly, statistics is a very Undoubtedly, statistics is a very useful tool in the various in the various useful tool in the various in the various activities of man. During the primitive activities of man. During the primitive period, people were not conscious that period, people were not conscious that they were already using statistics in they were already using statistics in counting events, activities, things, etc. counting events, activities, things, etc. They were not also aware that they wereThey were not also aware that they were

STATISTICS AS A SCIENCE

using statistics in determining the birth rate, crop yield, occurrence of events at a certain

period of time, etc.

The use of statistics in modern time is of course different from those of primitive past.The

modern man utilizes statistics, as a science, in the various field of studies, professional endeavors,

and even for personal profit.

You will then understand the nature and meaning of statistics, its brief historical development, the difference between

sample and population, the meaning and kinds of variables, and the importance of

statistics especially in the field of research.

GENERAL OBJECTIVESGENERAL OBJECTIVESAt the end of this module, you are expected At the end of this module, you are expected to:to:

1.1. State the nature and scientific definition of State the nature and scientific definition of statistics;statistics;

2.2. Trace the brief historical development of Trace the brief historical development of statistics;statistics;

3.3. Distinguish sample from a population; Distinguish sample from a population;

4.4. Enumerate and differentiate the kinds of Enumerate and differentiate the kinds of variables; andvariables; and

5.5. Explain the uses of statistics.Explain the uses of statistics.

TIME FRAMETIME FRAME 3 Hours3 Hours

PRE-TEST 1PRE-TEST 1Test your knowledge if the basic ideas Test your knowledge if the basic ideas

in statistics. As much as possible, avoid in statistics. As much as possible, avoid guessing. At any rate, this test in not guessing. At any rate, this test in not graded. Choose the letter of the best graded. Choose the letter of the best answer from the given four choices. Write answer from the given four choices. Write your answer on the blank before the your answer on the blank before the number.number.

_____ 1. From the research point of view, _____ 1. From the research point of view, statistics as a statistics as a science deals with the science deals with the following activities:following activities:

A. collection and gathering of dataA. collection and gathering of data

B. presentation and analysis of dataB. presentation and analysis of data

C. interpretation of dataC. interpretation of data

D. All of the above D. All of the above

______ 2. In counting events, objects, people, ______ 2. In counting events, objects, people, etc., the etc., the measurements that are collected measurements that are collected from the original from the original information are called information are called _________._________.

A. dataA. data

B. scores B. scores

C. raw dataC. raw data

D. none of the aboveD. none of the above

_____ 3. In making generalizations about the _____ 3. In making generalizations about the population population from which the sample has been from which the sample has been drawn, the measure drawn, the measure to use is called to use is called __________.__________.

A. descriptive statisticsA. descriptive statistics

B. inferential statisticsB. inferential statistics

C. correlational statistics C. correlational statistics

D. statistics D. statistics

_____ 4. It refers to the aggregates of people, objects, _____ 4. It refers to the aggregates of people, objects, materials, etc. materials, etc. of any form.of any form.

A. population A. population

B. sample B. sample

C. estimate C. estimate

D. statistic D. statistic

_____ 5. If you are interested with just a few members _____ 5. If you are interested with just a few members of the of the population to represent their traits and population to represent their traits and properties, then these properties, then these selected few members selected few members constitute a/an __________.constitute a/an __________.

A. SampleA. Sample

B. AggregateB. Aggregate

C. EstimateC. Estimate

D. statisticD. statistic

_____ 6. This term refers to a property, trait or _____ 6. This term refers to a property, trait or characteristic whereby the members of the group vary characteristic whereby the members of the group vary or differ from one another.or differ from one another.

A. VariableA. Variable

B. ConstantB. Constant

C. MeasurementC. Measurement

D. None of the aboveD. None of the above

_____ 7. A variable which allows making of statements _____ 7. A variable which allows making of statements only of equality or difference among the members of a only of equality or difference among the members of a group.group.

A. Nominal variableA. Nominal variable

B. Ratio B. Ratio

C. Interval variableC. Interval variable

D. Ordinal variableD. Ordinal variable

_____ 8. If you judge individuals according to their _____ 8. If you judge individuals according to their level of job satisfaction by ranking them, the resulting level of job satisfaction by ranking them, the resulting variable is a/an ________.variable is a/an ________.

A. NominalA. Nominal

B. RatioB. Ratio

C. OrdinalC. Ordinal

D. IntervalD. Interval

_____ 9. The number of make students in a class is _____ 9. The number of make students in a class is referred to a/anreferred to a/an

__________ variable.__________ variable.

B. NominalB. Nominal

C. OrdinalC. Ordinal

D. RatioD. Ratio

______ 10. Which of the following statements is not ______ 10. Which of the following statements is not true about the uses of Statistics?true about the uses of Statistics?

A. IntervalA. Interval

A. It can predict the behavior of individuals like A. It can predict the behavior of individuals like students, workers, school administrations, etc.students, workers, school administrations, etc.

B. It can give precise description of data.B. It can give precise description of data.

C. It can be used to test a hypothesis in research.C. It can be used to test a hypothesis in research.

D. It can be used to solve emotional problems.D. It can be used to solve emotional problems.

LESSON 1.1LESSON 1.1 THE NATURE AND SCIENTIFIC THE NATURE AND SCIENTIFIC DEFINITION OF STATISTICSDEFINITION OF STATISTICS

The Nature of StatisticsThe Nature of Statistics

The employment of statistics in man’s various The employment of statistics in man’s various activities during the past several centuries is said to be activities during the past several centuries is said to be in a limited sense. Its usefulness was trapped basically in a limited sense. Its usefulness was trapped basically in counting or determining the number of events that in counting or determining the number of events that have occurred at a certain period of time, birth rate, have occurred at a certain period of time, birth rate, mortality rate, etc.mortality rate, etc.

In counting activities, events, things, etc., the In counting activities, events, things, etc., the measurements that are gathered are referred to measurements that are gathered are referred to raw raw data. data. These data may be treated These data may be treated

by statistical tools in order to relate, associate, or by statistical tools in order to relate, associate, or describe the data. In the method of description, the describe the data. In the method of description, the statistical tool to apply is called statistical tool to apply is called descriptive descriptive statistics. statistics. In the method or relation and correlation In the method or relation and correlation two variables, two variables, correlational statistics correlational statistics is utilized. is utilized. Finally, in drawing generalizations regarding the Finally, in drawing generalizations regarding the population from which the sample has been gathered, population from which the sample has been gathered, the tool to utilize is the tool to utilize is inferential statisticsinferential statistics..

Scientific Definition of StatisticsScientific Definition of Statistics

Statistics can be defined operationally. From Statistics can be defined operationally. From the point of view of a researcher, the point of view of a researcher, statistics is a statistics is a science which deals with the methods of science which deals with the methods of collecting, gathering, presenting, analyzing and collecting, gathering, presenting, analyzing and interpreting data. interpreting data. The data gathering includes the The data gathering includes the collection of information through questionnaires, collection of information through questionnaires, observations, interviews, experiments, test, etc. The observations, interviews, experiments, test, etc. The information are usually converted into numerical or information are usually converted into numerical or quantitative data. The data collected can be quantitative data. The data collected can be displayeddisplayed

through the use of graphs, tables, figures and other through the use of graphs, tables, figures and other ways of exhibiting the data. There are two ways of ways of exhibiting the data. There are two ways of presenting data in tabular form. The presenting data in tabular form. The text or text or summary table summary table is usually found in the body of the is usually found in the body of the research work. The research work. The reference tablereference table is usually found is usually found in the appendices of the research work. The data in the appendices of the research work. The data analysis is a procedure wherein the resolution of the analysis is a procedure wherein the resolution of the information takes place by application of statistical information takes place by application of statistical principles. It involves the employment of any principles. It involves the employment of any statistical method and the choice of which depends statistical method and the choice of which depends largely upon the objectives of the research problem. largely upon the objectives of the research problem. After the analysis of data has been undertaken, the After the analysis of data has been undertaken, the results can be explained and interpreted. The findings results can be explained and interpreted. The findings of the study will then be compared to the existing of the study will then be compared to the existing theories and earlier researches or studies in a theories and earlier researches or studies in a particular field.particular field.

Activity 1.1Activity 1.1

Consider the following research situations then Consider the following research situations then specify the appropriate or the best manner of specify the appropriate or the best manner of gathering data whether gathering data whether interview, questionnaire, interview, questionnaire, experiment, observation, test, etc. experiment, observation, test, etc.

________ 1. Job Satisfaction of Public School Teachers________ 1. Job Satisfaction of Public School Teachers

________ 2. Emotionally Disturbed Grade School ________ 2. Emotionally Disturbed Grade School ChildrenChildren

________ 3. Sexually Harassed Adolescents________ 3. Sexually Harassed Adolescents

________ 4. Effect of Modularized Instruction to ________ 4. Effect of Modularized Instruction to Graduate Graduate Students’ Academic Students’ Academic Performance.Performance.

________ 5. Profile of the Faculty in Catholic Schools________ 5. Profile of the Faculty in Catholic Schools

________ 6. Factors Affecting the Performance of Staff ________ 6. Factors Affecting the Performance of Staff Nurses in Nurses in the Rural Areas the Rural Areas

________ 7. Comparative Study on the Various ________ 7. Comparative Study on the Various Instructional Instructional Strategies Applied to Strategies Applied to Handicapped LearnersHandicapped Learners

_____ 8. Diagnosing the Needs of Adult Learners in _____ 8. Diagnosing the Needs of Adult Learners in Tertiary LevelTertiary Level

_____ 9. Development of Instructional Materials in _____ 9. Development of Instructional Materials in Hydraulics.Hydraulics.

_____ 10. The Management Practices of Private School _____ 10. The Management Practices of Private School Principals in Principals in Region XII Region XII

LESSON 1. 2 BRIEF HISTORICAL VELOPMENTS LESSON 1. 2 BRIEF HISTORICAL VELOPMENTS OF OF STATISTICS STATISTICS

In the ancient times, statistics was utilized to In the ancient times, statistics was utilized to provide information that pertains to activities that provide information that pertains to activities that include farming, collection include farming, collection

of taxes, number of soldiers in a particular nation, of taxes, number of soldiers in a particular nation, number of events that occurred in a particular period number of events that occurred in a particular period of time, agricultural crops and even in athletic of time, agricultural crops and even in athletic endeavors of man. The employment of statistics was endeavors of man. The employment of statistics was later developed into an inferential science sometimes later developed into an inferential science sometimes in the sixteenth century. As an inferential science, it in the sixteenth century. As an inferential science, it largely depended on the theory of probability. The largely depended on the theory of probability. The development continued through the researches made development continued through the researches made by the people in various fields during the past 400 by the people in various fields during the past 400 years/years/

The inclination of man into gambling led to the The inclination of man into gambling led to the development of the development of the probability theoryprobability theory. During . During those times, the gamblers asked help from the those times, the gamblers asked help from the mathematicians to teach them the techniques on how mathematicians to teach them the techniques on how to win the games. The requests for such techniques to win the games. The requests for such techniques were consideredwere considered

by some mathematics among them were Pascal, by some mathematics among them were Pascal, Leibnitz, and James Bernoulli. It is very interesting to Leibnitz, and James Bernoulli. It is very interesting to note along this line that according to some winners of note along this line that according to some winners of the Lotto game, the chances of winning is attributed to the Lotto game, the chances of winning is attributed to the application of their knowledge of the application of their knowledge of probability and probability and statisticsstatistics..

In relation the historical development of In relation the historical development of statistics, statistics, De Moivre (1773) De Moivre (1773) discovered the equation discovered the equation for the for the normal distributionnormal distribution. The discovery of the . The discovery of the said equation became the basis of the development in said equation became the basis of the development in many theories of inferential statistics. The normal many theories of inferential statistics. The normal distribution which is a bell-shaped distribution was distribution which is a bell-shaped distribution was also referred to as the also referred to as the Gaussian distributionGaussian distribution. It was . It was during this time that the work of during this time that the work of LaplaceLaplace became so became so popular because of the application of statistics to popular because of the application of statistics to astronomy.astronomy.

Another significant event in the development of Another significant event in the development of statistics occurred when a Belgian statistician named statistics occurred when a Belgian statistician named Adolph Quetelet (1796-1874) Adolph Quetelet (1796-1874) made an application made an application of statistics in the field of psychology and education. of statistics in the field of psychology and education. He was considered to be the first statistician to He was considered to be the first statistician to demonstrate the statistical techniques derived in one demonstrate the statistical techniques derived in one area of research and applied to other areas.area of research and applied to other areas.

Another statistician who contributed his Another statistician who contributed his knowledge of statistics in the field of social sciences knowledge of statistics in the field of social sciences was was Sir Francis Galton (1822-1911).Sir Francis Galton (1822-1911). The The application of statistics to heredity and eugenics was application of statistics to heredity and eugenics was probably the most notable contribution of Galton to probably the most notable contribution of Galton to the development of statistics. He also discovered the the development of statistics. He also discovered the computation of computation of percentilespercentiles. Along with Galton was . Along with Galton was Karl Pearson (1857-1936)Karl Pearson (1857-1936) who exerted efforts and who exerted efforts and cooperated with Galton in developing thecooperated with Galton in developing the theory of theory of correlation and regression. correlation and regression. While Pearson was While Pearson was probably responsible for evolving the probably responsible for evolving the theories of theories of sampling sampling at present.at present.

Finally, at the rise of the twentieth century, Finally, at the rise of the twentieth century, William S. Gosset developed method for William S. Gosset developed method for decision-making decision-making derived from smaller sets of data. derived from smaller sets of data. Gosset worked in a brewery. He made a study and Gosset worked in a brewery. He made a study and published its results under the name “student.” He published its results under the name “student.” He disguised his real name because the brewery company disguised his real name because the brewery company which is owned by an Irish prohibited research since which is owned by an Irish prohibited research since results of the study might prove useful to its results of the study might prove useful to its competitors. The idea of Gosset was continued by competitors. The idea of Gosset was continued by another statistician named another statistician named Sir Ronald Fisher (1890-Sir Ronald Fisher (1890-1962) 1962) who was responsible for developing science of who was responsible for developing science of statistics for experimental designs.statistics for experimental designs.


Fill in the blanks with the correct answer.Fill in the blanks with the correct answer.

1.1. The inclination of man to gabling led to the The inclination of man to gabling led to the early development of _______________.early development of _______________.

2.2. ________________ discovered the equation for ________________ discovered the equation for normal distribution upon which man of the normal distribution upon which man of the theories of inferential statistics have been theories of inferential statistics have been based.based.

3.3. The normal distribution or the bell-shaped The normal distribution or the bell-shaped distribution was referred to ______________.distribution was referred to ______________.

4.4. The work of Laplace gained popularity for it was The work of Laplace gained popularity for it was about the application of statistics to about the application of statistics to _______________._______________.

5.5. ______________ made an application of statistics ______________ made an application of statistics in the field of psychology and education.in the field of psychology and education.

6.6. The greatest contribution of Sir Francis Galton The greatest contribution of Sir Francis Galton to the development of statistics to to the development of statistics to _______________._______________.

7.7. Pearson was probably responsible for evolving the Pearson was probably responsible for evolving the present theories of ________________.present theories of ________________.

8.8. _____________ developed methods for decision-_____________ developed methods for decision-making derived from smaller sets.making derived from smaller sets.

9.9. _____________ developed statistics for experimental _____________ developed statistics for experimental designs.designs.

LESSON 1.3 SOME BASIC CONCEPTS USEDLESSON 1.3 SOME BASIC CONCEPTS USED

IN STATISTICSIN STATISTICSWhat is a sample?What is a sample?

Suppose you are interested to study the behavior Suppose you are interested to study the behavior of handicapped students in a classroom situation. It of handicapped students in a classroom situation. It will be very tedious if you will consider to select the will be very tedious if you will consider to select the thousands of this type of students in a semester. thousands of this type of students in a semester. Instead you will only consider some of them to be Instead you will only consider some of them to be selected using an appropriate sampling technique. selected using an appropriate sampling technique. The portion of the totality of the handicapped The portion of the totality of the handicapped student is referred to as sample.student is referred to as sample.

What is a population ?What is a population ?

The term population refers to the aggregates of The term population refers to the aggregates of things, objects, people, events, etc. This could be things, objects, people, events, etc. This could be population of students, engineers, accountants, school population of students, engineers, accountants, school administrators, etc. In the research, the concern is to administrators, etc. In the research, the concern is to look at the properties of the aggregate or group rather look at the properties of the aggregate or group rather than the characteristic of each member.than the characteristic of each member.

What is a constant ?What is a constant ?

The word constant refers to a property whereby The word constant refers to a property whereby the members of a particular sample or aggregate do the members of a particular sample or aggregate do not differ from one another, For instance, a particular not differ from one another, For instance, a particular sex, say male, is a constant because the members do sex, say male, is a constant because the members do not differ.not differ.

What is a variable?What is a variable?

The variable refers to a property whereby the The variable refers to a property whereby the members of an aggregate differ from one another. members of an aggregate differ from one another. Thus, members of the group may vary or differ in the Thus, members of the group may vary or differ in the color of eyes, height, weight, civil status, etc.color of eyes, height, weight, civil status, etc.

What are the levels of measurement of a What are the levels of measurement of a variable?variable?

There are four levels of measurement.There are four levels of measurement.

1.1. Nominal variable: Nominal variable: This variable refers to a This variable refers to a characteristic or property of the members of the characteristic or property of the members of the group or aggregate defined by an operation which group or aggregate defined by an operation which allows making of statements only of equality or allows making of statements only of equality or difference. We can say that a member is different difference. We can say that a member is different or the same compare to another member of the or the same compare to another member of the group. For instance two male students are the group. For instance two male students are the same in sex while another two males may be same in sex while another two males may be different in height and weight.different in height and weight.

2.2. Ordinal variable: Ordinal variable: This variable refers to a This variable refers to a property or characteristic wherein the members of property or characteristic wherein the members of a group are compared say, one is greater that the a group are compared say, one is greater that the other or one is less than the other member. other or one is less than the other member. Ranking students based on the results of the Ranking students based on the results of the midterm examinations, will always have the first, midterm examinations, will always have the first, second, third and so on. In this case, the first in second, third and so on. In this case, the first in rank is higher than those who obtained other rank is higher than those who obtained other ranks. ranks.

3. Interval variable:3. Interval variable: This refers to a property or This refers to a property or characteristic defined by an operation which allows characteristic defined by an operation which allows making of statements of equality rather than making of statements of equality rather than statements of greater than or less that and sameness statements of greater than or less that and sameness or difference. An interval variable does not have a or difference. An interval variable does not have a “true” zero point.“true” zero point.

4. Ratio variable:4. Ratio variable: This refers to a property defined This refers to a property defined by an operation which allows making of statement or by an operation which allows making of statement or equality and ratios. This means that one value may equality and ratios. This means that one value may be thought of as five times another, triple of a certain be thought of as five times another, triple of a certain number, and so on. The measurements in the ratio number, and so on. The measurements in the ratio variable are made from an arbitrary zero point.variable are made from an arbitrary zero point.


A. Identify the concept: write your answer A. Identify the concept: write your answer on the blank before the number.on the blank before the number.

______ 1. The aggregates of objects, events, people, ______ 1. The aggregates of objects, events, people, etc.etc.

______ 2. The representative of an aggregate of ______ 2. The representative of an aggregate of handicapped handicapped learners in the tertiary learners in the tertiary level.level.

______ 3. It refers to a property or trait whereby the ______ 3. It refers to a property or trait whereby the members of the members of the group do not differ from one group do not differ from one another.another.

______ 4. It refers to a characteristic or property ______ 4. It refers to a characteristic or property whereby the whereby the members of a group vary of members of a group vary of differ from one another.differ from one another.

______ 5. The level of measurement wherein the ______ 5. The level of measurement wherein the property of property of members in a group are members in a group are considered in terms of sameness considered in terms of sameness or difference. or difference.

______ 6. The scale of measurement of a variable ______ 6. The scale of measurement of a variable wherein the wherein the characteristics or property characteristics or property of members in a particular of members in a particular aggregate say aggregate say individuals are ranked.individuals are ranked.

B.B. Write the level of measurement that corresponds to Write the level of measurement that corresponds to the variable in each item. Write your answer on the the variable in each item. Write your answer on the blank before each number.blank before each number.

______ 7. Second born and fifth born child in a family.______ 7. Second born and fifth born child in a family.

______ 8. Frequencies pf passing and failing the course ______ 8. Frequencies pf passing and failing the course in research.in research.

______ 9. Performance of 50 students in Statistics test.______ 9. Performance of 50 students in Statistics test.

______ 10. Socio-economic status of 30 subjects in a ______ 10. Socio-economic status of 30 subjects in a class.class.

LESSON 1.4 THE USES OF STATISTICSLESSON 1.4 THE USES OF STATISTICS

Statistics has an indespensable role particularly Statistics has an indespensable role particularly in the field of research. It enables a researcher to in the field of research. It enables a researcher to make a flawless and accurate statement of judgment make a flawless and accurate statement of judgment about a relationship of two or more variables. For about a relationship of two or more variables. For instance, describing the academic performance of the instance, describing the academic performance of the students in terms of the computed mean, standard students in terms of the computed mean, standard deviation, correlation in relation with another factor of deviation, correlation in relation with another factor of academic performance results. Thus, statistics can be academic performance results. Thus, statistics can be utilized to give a precise description of data.utilized to give a precise description of data.

In an educational research, the academic In an educational research, the academic performance can be predicted through the result of an performance can be predicted through the result of an entrance tests such as aptitude test, personality test, entrance tests such as aptitude test, personality test, etc. An instructor’s work performance can also be etc. An instructor’s work performance can also be predicted through the results of teacher inventory test. predicted through the results of teacher inventory test. In this light, statistics is useful in predicting the In this light, statistics is useful in predicting the behavior of individuals.behavior of individuals.

In order to determine the relationship between In order to determine the relationship between two or more variables, an appropriate statistical two or more variables, an appropriate statistical measure must be utilized. For instance, a correlational measure must be utilized. For instance, a correlational study may employ statistical measures such as t-test, study may employ statistical measures such as t-test, chi-square test, F-test, and others. With this purpose, chi-square test, F-test, and others. With this purpose, statistics can be used to test a hypothesis.statistics can be used to test a hypothesis.


What do you think are other uses of statistics? What do you think are other uses of statistics? Enumerate at least 5. Explain your answer.Enumerate at least 5. Explain your answer.

1.1. __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

2.2. __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

3.3. __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

4.4. __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

5. 5. ______________________________________________________________________________________________________

______________________________________________________________________________________________________

______________________________________________________________________________________________________

POST TEST 1POST TEST 11. 1. Fill in the blanks with the correct answer:Fill in the blanks with the correct answer:

1.1. In the method to relate or associate two variables, In the method to relate or associate two variables, the measure to apply is known as the measure to apply is known as ___________________.___________________.

2.2. To make generalizations about the population from To make generalizations about the population from which the sample has been drawn, the measure to which the sample has been drawn, the measure to use is known as __________________.use is known as __________________.

3.3. _______________ involves getting information with _______________ involves getting information with the employment of interviews, questionnaires, the employment of interviews, questionnaires, observations, psychological tests, etc.observations, psychological tests, etc.

4. _________________ is the resolution of information 4. _________________ is the resolution of information into simpler into simpler elements by the application of elements by the application of statistical principles.statistical principles.

5. The most notable contribution of Sir Francis Galton 5. The most notable contribution of Sir Francis Galton to the to the development of statistics was the development of statistics was the application of statistics to application of statistics to heredity and eugenics heredity and eugenics and his discoveries by ________.and his discoveries by ________.

6. The equation for the normal distribution was 6. The equation for the normal distribution was discovered by _____discovered by _____

7. ____ refers to the groups or aggregates of people, 7. ____ refers to the groups or aggregates of people, events, events, materials, etc. of any form.materials, etc. of any form.

8. ________ refers to a property whereby the members 8. ________ refers to a property whereby the members of a group or of a group or aggregate do not differ from one aggregate do not differ from one another.another.

9. The measures of the population are called 9. The measures of the population are called ____________________________

10. ________ refers to the properties or characteristics 10. ________ refers to the properties or characteristics whereby the whereby the members of the group or aggregate members of the group or aggregate vary or differ from one vary or differ from one another.another.

II. Write the level of measurement that II. Write the level of measurement that corresponds to the variable in each item. Write corresponds to the variable in each item. Write your answer on the blank before the number.your answer on the blank before the number.

___________ 1. Third born and fifth born child.___________ 1. Third born and fifth born child.

___________ 2. High and low scores in Statistics test.___________ 2. High and low scores in Statistics test.

___________ 3. Performance of boys and girls in an ___________ 3. Performance of boys and girls in an aptitude aptitude test. test.

____________ 4. Color preference of adults in Cebu.____________ 4. Color preference of adults in Cebu.

____________ 5. Failing and passing in a qualifying ____________ 5. Failing and passing in a qualifying

examination.examination.

____________ 6. Frequencies of strongly agree and ____________ 6. Frequencies of strongly agree and strongly strongly disagree responses to the disagree responses to the creation of E-vat.creation of E-vat.

7. __________Performance of 100 examinees in 7. __________Performance of 100 examinees in PBET.PBET.

8. __________Positions or ranks of graduate 8. __________Positions or ranks of graduate students on a social adjustment scale.students on a social adjustment scale.

9. __________The valedictorian and salutatorian in 9. __________The valedictorian and salutatorian in a graduating class.a graduating class.

10. _________Number of students who are in favor 10. _________Number of students who are in favor of the creation of the law on sexual of the creation of the law on sexual harassment.harassment.

ORGANIZING THE DATAORGANIZING THE DATA

Introduction:Introduction:

The collection of data entails a serious effort The collection of data entails a serious effort on the researcher. In doing so, the researcher must on the researcher. In doing so, the researcher must have good foresight, careful planning, and have good foresight, careful planning, and systematic organization of activities.systematic organization of activities.

The completion of data collection is not the The completion of data collection is not the end of the researcher’s task. The data must be end of the researcher’s task. The data must be analyzed using appropriate statistical tool or analyzed using appropriate statistical tool or treatment. From the analysis of data, results will be treatment. From the analysis of data, results will be obtained and test of hypotheses will be done.obtained and test of hypotheses will be done.

This module introduces the concepts of This module introduces the concepts of frequency, frequency, distribution, midpoint, class frequency, frequency, distribution, midpoint, class interval, proportion and percentage, cumulative interval, proportion and percentage, cumulative percentage, raw data, etc. A clearer understanding percentage, raw data, etc. A clearer understanding of these concepts will enable researchers to of these concepts will enable researchers to construct frequency distribution for the different construct frequency distribution for the different levels of measurement.levels of measurement.

GENERAL OBJECTIVESGENERAL OBJECTIVESAt the end of this module, you are expected to:At the end of this module, you are expected to:

1.1. Define or describe the following terms:Define or describe the following terms:a. frequencya. frequencyb. frequency distributionb. frequency distributionc. cumulative frequencyc. cumulative frequencyd. cumulative percentaged. cumulative percentagee. midpointe. midpointf. class intervalf. class intervalg. proportion and percentageg. proportion and percentageh. raw datah. raw data

2. Construct at frequency distribution of the 2. Construct at frequency distribution of the different levels of measurement.different levels of measurement.

Pre-TestPre-Test

Direction:Direction: Choose the letter of the correct answer. Choose the letter of the correct answer. Write your answer on the blank before each number.Write your answer on the blank before each number.

____1. The variants which have not been organized or ____1. The variants which have not been organized or classified in any way, and which are often recorded in classified in any way, and which are often recorded in the order observed.the order observed.

a. raw dataa. raw data

b.transmuted datab.transmuted data

c. raw scoresc. raw scores

d. none of thesed. none of these

____2. The distribution obtained by a simple process of ____2. The distribution obtained by a simple process of successive addition of the entries in the frequency successive addition of the entries in the frequency column.column.

a. relative frequencya. relative frequency b. cumulative frequencyb. cumulative frequency

c. frequencyc. frequency d. cumulative d. cumulative percentagepercentage

____3. It is obtained by dividing each entry in the cumulative ____3. It is obtained by dividing each entry in the cumulative frequency column by N and multiplying by 100%frequency column by N and multiplying by 100%

a. cumulative frequencya. cumulative frequency c. relative frequencyc. relative frequencyb. cumulative percentageb. cumulative percentage d. frequencyd. frequency

____4. It is obtained by adding the upper class limit and the lower ____4. It is obtained by adding the upper class limit and the lower class limit and dividing the sum by 2.class limit and dividing the sum by 2.

a. class intervala. class interval c. class boundaryc. class boundaryb. midpointb. midpoint d. none of thesed. none of these

____5. A frequency distribution is defined as __________5. A frequency distribution is defined as ______a. an orderly arrangement of data using arbitrarily defined a. an orderly arrangement of data using arbitrarily defined

classes or groupings and associated frequencies. classes or groupings and associated frequencies.b. an arrangement of numbers using a standard set of b. an arrangement of numbers using a standard set of

groupings.groupings.c. a classification of data showing the frequency of c. a classification of data showing the frequency of

occurrence of the occurrence of the variants. variants.d. an arrangement of a set of raw data from highest to d. an arrangement of a set of raw data from highest to

lowest or vice lowest or vice versa. versa.

_____6. The number of values or cases which is found _____6. The number of values or cases which is found in a class interval or category is called ______.in a class interval or category is called ______.

a. class intervala. class interval c. probabilityc. probabilityb. frequencyb. frequency d. data/scoresd. data/scores

_____7. The midpoint of class interval 15-19 is __________7. The midpoint of class interval 15-19 is _____a. 16a. 16 c. 18c. 18b. 17b. 17 d. 19d. 19

_____8. What is the cumulative percentage of the _____8. What is the cumulative percentage of the frequency in the class interval 30-34 from the table frequency in the class interval 30-34 from the table below?below?

Class interval Class interval f f15-1915-19 3320-2420-24 151525-2925-29 2230-3430-34 1010

a. 33.3%a. 33.3% c. 23%c. 23%

b. 30%b. 30% d. 20%d. 20%

______9. The table below shows an example of the ______9. The table below shows an example of the frequency distribution under ______frequency distribution under ______

Religion Religion FrequencyFrequency

ProtestantProtestant 1515

INCINC 1212

CatholicCatholic 1010

a. nominala. nominal c. intervalc. interval

ratioratio d. ordinald. ordinal

______10. The table below shows an example of ______10. The table below shows an example of frequency distribution under ________data.frequency distribution under ________data.

Class intervalClass interval f.f. %%

15-1915-19 1010 22.2%22.2%

20-2420-24 1212 26.6%26.6%

25-2925-29 88 17.7%17.7%

30-3430-34 55 11.1%11.1%

35-3935-39 1010 22.2%22.2%

a. nominala. nominal c. intervalc. interval

b. ordinalb. ordinal d. ratiod. ratio

FREQUENCY DISTRIBUTION OF NOMINAL DATAFREQUENCY DISTRIBUTION OF NOMINAL DATA

A tailor transforms raw cloth material into suits; a A tailor transforms raw cloth material into suits; a chef converts raw food material into more palatable chef converts raw food material into more palatable versions, served at the dinner table. On the same versions, served at the dinner table. On the same manner, the researcher aided by formulas and manner, the researcher aided by formulas and techniques, can transform raw data into an organized techniques, can transform raw data into an organized set of measures which can be used to test hypotheses.set of measures which can be used to test hypotheses.

What can a researcher do to organize the What can a researcher do to organize the gathered data? How to transform the raw data into an gathered data? How to transform the raw data into an easy-to-understand summary form? Perhaps, the first easy-to-understand summary form? Perhaps, the first step will be the construction of a step will be the construction of a frequency frequency distributiondistribution in a form of table.in a form of table.

Frequency distribution of nominal data has the Frequency distribution of nominal data has the following characteristics or properties. These following characteristics or properties. These characteristics are also held true to other levels of characteristics are also held true to other levels of measurement of data.measurement of data.

1.1. Table numberTable number – – it facilitates the reader to locate the table it facilitates the reader to locate the table easilyeasily

2.2. TitleTitle – – it gives the reader an idea as to the nature of the it gives the reader an idea as to the nature of the data being presented.data being presented.

Example:Example: Frequency Distribution of Nominal DataFrequency Distribution of Nominal Data

Table I Sex of Students Majoring in PsychologyTable I Sex of Students Majoring in Psychology

Sex of StudentsSex of Students Frequency (f)Frequency (f)

MaleMale 7474

FemaleFemale 2626

TotalTotal 100100

Table 1,Table 1, the left-hand column indicates what characteristic the left-hand column indicates what characteristic is being presented (is being presented (sex of studentssex of students) and contains the ) and contains the categories of analysiscategories of analysis ( (male and femalemale and female). An adjacent ). An adjacent column headedcolumn headed

““Frequency”Frequency” or “f”or “f” and indicates the and indicates the number of cases in number of cases in eacheach category (74 and 26category (74 and 26, , respectively)respectively) as well as as well as the the total number oftotal number of cases (N=100).cases (N=100). A glance in the A glance in the frequency distribution in Table I clearly shows that more frequency distribution in Table I clearly shows that more males are majoring in Psychology.males are majoring in Psychology.

Comparing DistributionsComparing DistributionsComparing the number of students majoring in Comparing the number of students majoring in Psychology at PWU Taft and those who are majoring in Psychology at PWU Taft and those who are majoring in the same course at PWU Q.C. requires statistical the same course at PWU Q.C. requires statistical knowledge and comparison. Thus, making comparisons knowledge and comparison. Thus, making comparisons between frequency distributions is a procedure often between frequency distributions is a procedure often used to clarify results and add information.used to clarify results and add information.

Recalling the hypothetical example above Recalling the hypothetical example above mentioned, you might ask: Are majors in Psychology at mentioned, you might ask: Are majors in Psychology at PWU Taft more likely to be male than at PWU QC? To PWU Taft more likely to be male than at PWU QC? To provide an answer, you might compare the students provide an answer, you might compare the students majoring at PWU Taft and at PWU QC in the said major majoring at PWU Taft and at PWU QC in the said major course.course.

Take a look at this example:Take a look at this example:

Table 2Table 2 SEX OF STUDENTS MAJORING IN SEX OF STUDENTS MAJORING IN PSYCHOLOGY AT PWU TAFT AND PWU QCPSYCHOLOGY AT PWU TAFT AND PWU QC

Psychology Psychology MajorsMajors

PWU Taft PWU Taft PWU QCPWU QC

Sex of StudentsSex of Students ff ff

MaleMale 8080 7070

FemaleFemale 2020 3030

TotalTotal 100100 100100

As shown in the table, 80 out of 100 who are majoring As shown in the table, 80 out of 100 who are majoring in Psychology at PWU Taft, and 70 out of 100 who are in Psychology at PWU Taft, and 70 out of 100 who are also majoring in Psychology at PWU QC, are males. also majoring in Psychology at PWU QC, are males. Thus, males predominate among Psychology majors in Thus, males predominate among Psychology majors in both schools, this tendency is more pronounced at both schools, this tendency is more pronounced at PWU Taft.PWU Taft.

One must be careful in making comparisons One must be careful in making comparisons which require data that are not presented.which require data that are not presented.

Proportions and PercentagesProportions and PercentagesWhen the researcher studies distributions of equal When the researcher studies distributions of equal total size, the frequency data can be used to make total size, the frequency data can be used to make comparisons between the groups. Thus, the number comparisons between the groups. Thus, the number of males majoring in Psychology at PWU Taft and QC of males majoring in Psychology at PWU Taft and QC can be directly compared, because there are exactly can be directly compared, because there are exactly 100 students majoring on each campus. It is generally 100 students majoring on each campus. It is generally not possible, however, to study distributions having not possible, however, to study distributions having number of cases. For instance, how to make sure that number of cases. For instance, how to make sure that precisely 100 students at both universities will decide precisely 100 students at both universities will decide to major in Psychology?to major in Psychology?

For more general use, there is a method of For more general use, there is a method of standardizingstandardizing frequency distributionsfrequency distributions for size. A for size. A way to compare groups despite differences in total way to compare groups despite differences in total frequencies. There are two popular and useful methods frequencies. There are two popular and useful methods of standardizing for size and comparing distributions. of standardizing for size and comparing distributions. These are the proportion and the These are the proportion and the percentage.percentage.

The The proportionproportion compares the number of cases compares the number of cases in a given category with the total size of the in a given category with the total size of the distribution. Any frequency can be converted to a distribution. Any frequency can be converted to a proportion P proportion P by dividing the number of cases in any by dividing the number of cases in any category fcategory f by the by the total number of cases in thetotal number of cases in the distribution Ndistribution N. The formula is:. The formula is:

ff

P = N P = N

Despite the usefulness of the proportion, Despite the usefulness of the proportion, may people prefer to indicate the relative size of a may people prefer to indicate the relative size of a series of numbers in terms of the series of numbers in terms of the percentage,percentage, the the frequency of occurrence of a category per frequency of occurrence of a category per 100 100 casescases. To calculate a percentage, the formula is.. To calculate a percentage, the formula is.

ff

% = (100)% = (100)NN

Activity 2.1.Activity 2.1.Given the following data, construct a Given the following data, construct a

frequency distribution in a form of table. Always frequency distribution in a form of table. Always have a have a table numbertable number and and titletitle for your frequency for your frequency distribution. Use the space provided for your distribution. Use the space provided for your answer.answer.

1. In the School Year 1995-1996, there were 250 1. In the School Year 1995-1996, there were 250 Education students majoring in Mathematics at Education students majoring in Mathematics at Philippine Normal University. Out of that total Philippine Normal University. Out of that total number of Education students, 105 were males.number of Education students, 105 were males.

2. A researcher tried to compare the number of 2. A researcher tried to compare the number of enrollees in two universities for the School Year enrollees in two universities for the School Year 1993-1994. There were 350 male and 650 females 1993-1994. There were 350 male and 650 females who enrolled at University Y. At University A, there who enrolled at University Y. At University A, there were 250 male enrollees and 750 were females.were 250 male enrollees and 750 were females.

3. The proportions of males who were majors of 3. The proportions of males who were majors of Mathematics at PNU.Mathematics at PNU.

4. The proportion of females who were majors of 4. The proportion of females who were majors of Mathematics at PNU.Mathematics at PNU.

5. The percentage of male enrollees in University Y 5. The percentage of male enrollees in University Y for the School Year 1993-1994.for the School Year 1993-1994.

6. The percentage of females who were of majors of 6. The percentage of females who were of majors of Mathematics at PNU.Mathematics at PNU.

7. The percentage of female enrollees in University A 7. The percentage of female enrollees in University A for the School Year 1993-1994.for the School Year 1993-1994.

8. The proportion of male enrollees in University A 8. The proportion of male enrollees in University A for the School Year 1993-1994.for the School Year 1993-1994.

9. The proportion of female enrollees in University Y 9. The proportion of female enrollees in University Y for the School Year 1993-1994.for the School Year 1993-1994.

10. The percentage of males who were majors of 10. The percentage of males who were majors of Mathematics at PNU.Mathematics at PNU.

PREQUENCY DISTRIBUTIONS OF NOMINAL, ORDINAL PREQUENCY DISTRIBUTIONS OF NOMINAL, ORDINAL AND INTERVAL DATAAND INTERVAL DATA

Nominal data are labeled rather than scaled. Nominal data are labeled rather than scaled. The categories of nominal-level distributions do The categories of nominal-level distributions do not have to be listed in any particular order.not have to be listed in any particular order.

The following table shows an example of The following table shows an example of ordinal data presented in three different, but ordinal data presented in three different, but equally acceptable arrangements.equally acceptable arrangements.

Table 3 THE DISTRIBUTION OF RELIGIOUS Table 3 THE DISTRIBUTION OF RELIGIOUS PREFERENCESPREFERENCES

(SHOWN IN THREE WAYS)(SHOWN IN THREE WAYS)

Religion f ReligionReligion f Religion f f ReligionReligion ff

Protestant 35Protestant 35 Catholic Catholic 50 50 IglesiaIglesia2020

Catholic 50Catholic 50 Iglesia Iglesia 2020 Catholic Catholic 5050

Iglesia 20 Iglesia 20 Protestant Protestant 35 35 Protestant 35Protestant 35

TotalTotal 105105 Total 105 Total 105 Total Total 105105

In contrast, the score values in ordinal or In contrast, the score values in ordinal or interval distributions represent the degree to which interval distributions represent the degree to which a particular characteristic is present. The ordinal a particular characteristic is present. The ordinal and interval categories are always arranged in and interval categories are always arranged in order. The arrangement is usually from highest to order. The arrangement is usually from highest to lowest values.lowest values.

The table below shows an example of interval The table below shows an example of interval data arranged correctly and incorrectly. data arranged correctly and incorrectly.

Table 4 FREQUENCY DISTRIBUTION OF ATTITUDES TOWARD Table 4 FREQUENCY DISTRIBUTION OF ATTITUDES TOWARD E-VAT ON A COLLEGE CAMPUS: INCORRECT AND CORRECT E-VAT ON A COLLEGE CAMPUS: INCORRECT AND CORRECT PRESENTATIONSPRESENTATIONS

Attitude Toward E-VatAttitude Toward E-Vat ff Attitude Toward E-Vat fAttitude Toward E-Vat f

Slightly favorableSlightly favorable 22 Strongly favorable 0 Strongly favorable 0

Somewhat unfavorableSomewhat unfavorable 11 Somewhat favorable11 Somewhat favorable 1 1

Strongly favorableStrongly favorable 00 Slightly favorable Slightly favorable 2 2

Slightly unfavorableSlightly unfavorable 66 Slightly unfavorable 6 Slightly unfavorable 6

Strongly unfavorableStrongly unfavorable 2020 Somewhat Somewhat unfavorable 11unfavorable 11

Somewhat favorableSomewhat favorable 11 Strongly unfavorable Strongly unfavorable 20 20

Total 40Total 40 Total 40 Total 40

INCORRECTINCORRECT CORRECT CORRECT

Which version do you find easier to read?Which version do you find easier to read?

Grouped Frequency Distribution of Interval DataGrouped Frequency Distribution of Interval DataInterval-level scores are sometimes spread over a Interval-level scores are sometimes spread over a

wide range, making the resultant simple frequency wide range, making the resultant simple frequency distribution long and difficult to read, but it can be distribution long and difficult to read, but it can be presented clearly by considering the separate scores presented clearly by considering the separate scores into a number of smaller groups. Each group or category into a number of smaller groups. Each group or category in a grouped frequency distribution is known as a in a grouped frequency distribution is known as a class class intervalinterval, , whose whose size size is determined by the number of is determined by the number of score values it contains.score values it contains.

Class LimitsClass LimitsIn accordance with its size, each class interval has In accordance with its size, each class interval has

an upper limit and a lower limit. The highest and lowest an upper limit and a lower limit. The highest and lowest scores in any category seem to be the limits. Thus, we scores in any category seem to be the limits. Thus, we might say that the upper and lower limits of 50-54 are 54 might say that the upper and lower limits of 50-54 are 54 and 50, respectively. and 50, respectively. Class boundariesClass boundaries are located at are located at the point halfway between adjacent class intervals and, the point halfway between adjacent class intervals and, therefore, serve to close the gap between them. Thus, therefore, serve to close the gap between them. Thus, the upper boundary of the class interval 40-44 is 44.5 the upper boundary of the class interval 40-44 is 44.5 and the lower boundary of the class interval 45-49 is also and the lower boundary of the class interval 45-49 is also 44.5.44.5.

The Class MidpointThe Class Midpoint

Another characteristic of a class interval is its Another characteristic of a class interval is its midpoint (m). It is the middlemost score value in midpoint (m). It is the middlemost score value in the class interval. As an example, the midpoint of the class interval. As an example, the midpoint of the class interval 55-59 is 57 as illustrated below:the class interval 55-59 is 57 as illustrated below:

lowestlowest ++ highest scorehighest score

m =m =

22

55 + 5955 + 59

==

22

m = 57m = 57

Therefore the midpoint of 55-59 is 57.Therefore the midpoint of 55-59 is 57.

Determining the Number of IntervalsDetermining the Number of IntervalsIn presenting interval data in a grouped In presenting interval data in a grouped

frequency distribution, the researcher considers the frequency distribution, the researcher considers the number of categories to employ. Textbooks generally number of categories to employ. Textbooks generally use as few as 3 or 4 intervals to as many as 20 use as few as 3 or 4 intervals to as many as 20 intervals. In this sense, it would be wise to remember intervals. In this sense, it would be wise to remember that grouped frequency distributions are utilized to that grouped frequency distributions are utilized to emphasize a group pattern.emphasize a group pattern.

Cumulative DistributionsCumulative DistributionsIt is sometimes desirable to display frequencies It is sometimes desirable to display frequencies

in a cumulative manner especially in locating a in a cumulative manner especially in locating a position of one case relative to overall group position of one case relative to overall group performance.performance.

Cumulative frequencies (cf)Cumulative frequencies (cf) are defined as are defined as the totalthe total number of cases having any given score or a number of cases having any given score or a score that isscore that is lower. Thus, thelower. Thus, the cumulative frequencycumulative frequency (cf)(cf) of any group or category (or class interval) is of any group or category (or class interval) is obtained by adding the frequency in that category to obtained by adding the frequency in that category to the total frequency for all category below it.the total frequency for all category below it.

The table below shows an example of a frequency The table below shows an example of a frequency distribution of interval data.distribution of interval data.

Table 5 GROUPED FREQUENCY DISTRIBUTION OF Table 5 GROUPED FREQUENCY DISTRIBUTION OF MIDTERM EXAMINATION SCORES FOR 50 MIDTERM EXAMINATION SCORES FOR 50 STUDENTS IN STATISTICS WITH COMPUTER STUDENTS IN STATISTICS WITH COMPUTER APPLICATIONSAPPLICATIONS

Class IntervalClass Interval ff %% mm cfcf85-8985-89 55 1010 8787 505080-8480-84 55 1010 8282 454575-7975-79 1010 2020 7777 353570-7470-74 55 1010 7272 303065-6965-69 88 1616 6767 252560-6460-64 77 1414 6262 171755-5955-59 1010 2020 5757 1010

TotalTotal 5050 100%100%

Cumulative percentageCumulative percentageIn addition to cumulative frequency, you can In addition to cumulative frequency, you can

also construct a distribution which indicates also construct a distribution which indicates cumulativecumulative percentage (c%),percentage (c%), the percent of the percent of case having any score or a score that is lower. To case having any score or a score that is lower. To calculate cumulative percentage, the following calculate cumulative percentage, the following formula can be employed:formula can be employed:

cfcfc% = (100%) c% = (100%)

NNwherewhere

cf = the cumulative frequency in any categorycf = the cumulative frequency in any categoryN = the total number of cases in the distributionN = the total number of cases in the distribution

Consider the following table and study how the cumulative Consider the following table and study how the cumulative percentage are being obtained.percentage are being obtained.

Table 6 CUMULATIVE PERCENTAGE DISTRIBUTION OF Table 6 CUMULATIVE PERCENTAGE DISTRIBUTION OF TEST SCORES IN STATISTICS FOR 50 STUDENTSTEST SCORES IN STATISTICS FOR 50 STUDENTS

Class intervalClass interval ff cfcf c%c%

35-3935-39 88 8 8 66

40-4440-44 55 1313 1010

45-4945-49 77 2020 1414

50-5450-54 1212 3232 2424

55-5955-59 1010 4242 2020

60-6460-64 55 4747 1010

65-6965-69 33 5050 66

Total N = 50Total N = 50 100% 100%

To illustrate, the cumulative percentage for the To illustrate, the cumulative percentage for the class interval 65-69 is obtained as shown in the class interval 65-69 is obtained as shown in the following solution:following solution:

c% = (100%) 3 c% = (100%) 3

5050

= (100%) .60= (100%) .60

= 6%= 6%

Activity 2.2.Activity 2.2.

1. Construct a table for the following distribution of 1. Construct a table for the following distribution of club preferences by 100 students in tertiary level:club preferences by 100 students in tertiary level:

ClubClub Frequency Frequency

SportsSports 3030

MissionMission 1010

ReligiousReligious 55

Math & ScienceMath & Science 1515

DramaDrama 4040

TotalTotal 100100

2. Construct a comprehensive tabular presentation 2. Construct a comprehensive tabular presentation of the following nominal data taken from the of the following nominal data taken from the opinions of 200 students about the creation of opinions of 200 students about the creation of sexual harassment.sexual harassment.

Level of agreement and disagreementLevel of agreement and disagreementFrequencyFrequency

Strongly agreeStrongly agree 7575

AgreeAgree 47 47

UncertainUncertain 23 23

DisagreeDisagree 30 30

Strongly disagreeStrongly disagree 25 25

TotalTotal 200 200

3. Complete the table below to be able to construct 3. Complete the table below to be able to construct a frequency distribution based on the following a frequency distribution based on the following scores obtained in Statistics test by 50 students.scores obtained in Statistics test by 50 students.

SCORES:SCORES:

7575 70 6470 64 6060 5555 4848 5050 4141 55 55 4141

9090 6565 55 55 5050 5050 7575 8080 5959 58586060

9191 8585 50 50 6565 6060 4141 7070 5050 41415050

9292 6060 75 75 8080 5555 4949 8080 5252 68689090

5858 9090 70 70 6060 5959 5555 6565 5050 53537474

Class intervalClass interval TallyTally ff %% MM cfcfc%c%

41-4541-45

46-5046-50

51-5551-55

56-6056-60

61-6561-65

66-7066-70

71-7571-75

76-8076-80

81-8581-85

86-9086-90

POST TEST 2POST TEST 2

1.1. From the table below representing the From the table below representing the academic performance of students from a rural academic performance of students from a rural area and an urban area, find the area and an urban area, find the

1.1. Percent of students from a rural area whose 1.1. Percent of students from a rural area whose level of level of achievement is high achievement is high

1.2. Percent of students from an urban area 1.2. Percent of students from an urban area whose level of whose level of achievement is high achievement is high

1.3. Proportion of students from a rural area 1.3. Proportion of students from a rural area whose level of whose level of achievement is high achievement is high

1.4. Proportion of students from an urban area 1.4. Proportion of students from an urban area whose level whose level of performance is low of performance is low

ACADEMIC PERFORMANCE OF STUDENTS FROM A ACADEMIC PERFORMANCE OF STUDENTS FROM A RURAL AREA AND AN URBAN AREARURAL AREA AND AN URBAN AREA

AREA AREA

Achievement LevelAchievement Level Rural Rural UrbanUrban

ff f f

HighHigh 83 83 146 146

LowLow 140 140 227 227

Total 223Total 223 373 373

2. Convert the following distribution of scores into a 2. Convert the following distribution of scores into a grouped frequency distribution containing five grouped frequency distribution containing five class intervals, andclass intervals, and

2.1. Determine the size of class intervals2.1. Determine the size of class intervals

2.2. Indicate the upper and lower limits of each 2.2. Indicate the upper and lower limits of each class intervalclass interval

2.3. Identify the midpoint of each class interval2.3. Identify the midpoint of each class interval

2.4. Find the percentage for each class interval2.4. Find the percentage for each class interval

2.5. Find the cumulative frequency for each class 2.5. Find the cumulative frequency for each class intervalinterval

2.6. Find the cumulative percentage for each 2.6. Find the cumulative percentage for each class intervalclass interval

SCORE VALUESCORE VALUE frequencyfrequency

2121 332020 441919 221818 221717 111616 551515 331414 221313 111212 111111 331010 2299 33

88 44

77 33

66 22

55 44

44 55

33 22

22 33

N = 55N = 55

GRAPHIC PRESENTATION OF DATAGRAPHIC PRESENTATION OF DATA(MODULE THREE)(MODULE THREE)

Introduction:Introduction:The previous module dealt with the organization of data The previous module dealt with the organization of data employing the tabular form. With this of approach, employing the tabular form. With this of approach, columns of numbers can evoke fear, boredom, a partly, columns of numbers can evoke fear, boredom, a partly, and misunderstanding on the part of the reader. While and misunderstanding on the part of the reader. While some people seem to some people seem to “tune out”“tune out” statistical information statistical information presented in tabular form, they may pay close attention presented in tabular form, they may pay close attention to the same data presented in graphic or picture form. to the same data presented in graphic or picture form. In fact, with the access of people to computer, the In fact, with the access of people to computer, the graphic or picture form of organizing and presenting graphic or picture form of organizing and presenting data can easily be prepared. As a result, many data can easily be prepared. As a result, many researchers and authors prefer to use graphs as researchers and authors prefer to use graphs as opposed to tables. For similar reasons, researchers opposed to tables. For similar reasons, researchers often use visual aids as often use visual aids as pie charts, bar graphs, pie charts, bar graphs, frequency polygons, and line graphsfrequency polygons, and line graphs in an effort to in an effort to increase and ensure the readability of findings.increase and ensure the readability of findings.

Objective:Objective:

At the end of this module, you are At the end of this module, you are expected to:expected to:

1.1. Organize and present data in graphic Organize and present data in graphic presentation such as:presentation such as:

1.1. Pie charts1.1. Pie charts

1.2. Bar graphs1.2. Bar graphs

1.3. Frequency polygons1.3. Frequency polygons

1.4. Line graphs1.4. Line graphs

2. Operationally define the following concepts used in 2. Operationally define the following concepts used in graphic presentation of data:graphic presentation of data:

2.1. Circular chart or pie chart2.1. Circular chart or pie chart

2.2. Histogram or bar graph2.2. Histogram or bar graph

2.3. Line graph2.3. Line graph

2.4. Frequency polygon2.4. Frequency polygon

PRE-TESTPRE-TESTChoose the letter of the best answer and write it on the Choose the letter of the best answer and write it on the

blank before the number.blank before the number._____1. A graph whose pieces add up to 100 percent is _____1. A graph whose pieces add up to 100 percent is

called ___called ___a. pie charta. pie chart c. line graphc. line graphb. map b. map d. bar graphd. bar graph

_____ 2. A graphic illustration in which rectangular bars _____ 2. A graphic illustration in which rectangular bars indicate indicate the frequencies for the range of score the frequencies for the range of score values or values or categories. categories.

a. pie charta. pie chart c. line graphc. line graphb. bar graphb. bar graph d. none of thesed. none of these

_____ 3. A graph in which frequencies are indicated by a _____ 3. A graph in which frequencies are indicated by a series of series of points placed over the score values of points placed over the score values of each class interval each class interval and connected with a and connected with a straight line dropped to the base straight line dropped to the base line line at at either end.either end.

a. picture grapha. picture graph c. bar grahp c. bar grahp

b. histogramb. histogram d. frequency d. frequency polygonpolygon

Study the figure below and answer the Study the figure below and answer the questions that follow:questions that follow:

Above AverageAbove Average

AverageAverage

Below Below AverageAverage

___ 4. If there are 20 students representing their IQ, ___ 4. If there are 20 students representing their IQ, then how then how many belong to average IQ? many belong to average IQ?

a. 50a. 50 c. 100c. 100

b. 75b. 75 d. 150d. 150

___ 5. How many belong to below average?___ 5. How many belong to below average?

a. 50a. 50 c. 90c. 90

b. 75b. 75 d. 125d. 125

___ 6. What level of measurement must a ___ 6. What level of measurement must a researcher have in researcher have in order to make use of pie order to make use of pie chart?chart?

a. ordinala. ordinal c. ratioc. ratio

b. nominalb. nominal d. intervald. interval

Study the illustration below and answer the Study the illustration below and answer the questions that follow:questions that follow:

600600

500500

400400

300300

200200

100100

00

Catholic Protestant Iglesia MethodistCatholic Protestant Iglesia Methodist

Bar graph of student’s religionsBar graph of student’s religions

____ 7. What is the frequency of students belonging to Catholic?____ 7. What is the frequency of students belonging to Catholic?

a. 200a. 200 c. 400c. 400

b. 300b. 300 d. 500d. 500

____ 8. What is the total population represented by the graph?____ 8. What is the total population represented by the graph?

a. 1000a. 1000 c. 1200c. 1200

b. 1100b. 1100 d. 1300d. 1300

Study the bar below and answer the questions that follow:Study the bar below and answer the questions that follow:

500500

Legend:Legend: 400400

malemale 300300

femalefemale 200200

100100

00

Never seldom Sometimes Never seldom Sometimes Frequent AlwaysFrequent Always

Bar graph of students using library facilitiesBar graph of students using library facilities

____ 9. How many students particularly males are ____ 9. How many students particularly males are frequently using frequently using the library facilities?the library facilities?

a. 100a. 100 c. 300c. 300

b. 200b. 200 d. 400d. 400

____ 10. What is the population of female students ____ 10. What is the population of female students represented by represented by the graph?the graph?

____ 11.What does the figure below show?____ 11.What does the figure below show?

freq.freq.

midpointmidpoint

a. frequency polygona. frequency polygon c. mapc. mapb. histogramb. histogram d. none of thesed. none of these

____ 12. What level of measurement must a researcher ____ 12. What level of measurement must a researcher have in have in order to make use of histogram of order to make use of histogram of frequency polygon?frequency polygon?

a. ordinala. ordinal c. intervalc. intervalb. nominalb. nominal d. all of thesed. all of these

PRESENTING DATA IN A PIE CHART & BAR GRAPHPRESENTING DATA IN A PIE CHART & BAR GRAPHWhat is a pie chart?What is a pie chart?When do you use a pie chart?When do you use a pie chart?

The The pie chartpie chart, a circular graph whose pieces sum up to , a circular graph whose pieces sum up to 100%, is one of the simplest methods of presenting 100%, is one of the simplest methods of presenting data in graphical form. Pie chart is p[particularly useful data in graphical form. Pie chart is p[particularly useful for depicting the differences in frequencies or percents for depicting the differences in frequencies or percents among categories of a nominal-level variable. The pie among categories of a nominal-level variable. The pie chart depicts the most noteworthy part. For instance, a chart depicts the most noteworthy part. For instance, a researcher would want you toresearcher would want you to

identify the course where most students are identify the course where most students are enrolled in a particular university. Looking at the enrolled in a particular university. Looking at the illustration below. Which section of the pie chart is illustration below. Which section of the pie chart is most noteworthy? Isn’t it the most students most noteworthy? Isn’t it the most students enrolled in a particular course? If the researcher enrolled in a particular course? If the researcher would want the reader to note the least number of would want the reader to note the least number of enrollees in a particular course, the section of the enrollees in a particular course, the section of the chart will be noted by the reader immediately.chart will be noted by the reader immediately.

Engr.Engr. NursingNursing

Educ.Educ.

Crim.Crim.CommerceCommerce

Figure 3.1 Pie Chart of Course of Students in a Figure 3.1 Pie Chart of Course of Students in a Particular UniversityParticular University

What is a bar graph?What is a bar graph?

When do you use graph?When do you use graph?

The pie provides a quick and easy The pie provides a quick and easy illustration of data. By comparison, the illustration of data. By comparison, the bar graph bar graph or histogramor histogram can accommodate any number of can accommodate any number of categories at any level of measurement.categories at any level of measurement.

The bar graph is constructed following the The bar graph is constructed following the standard arrangement: a horizontal base line (or x-standard arrangement: a horizontal base line (or x-axis) along which the score values or categories axis) along which the score values or categories are marked off, a vertical line (or y-axis) along the are marked off, a vertical line (or y-axis) along the left side displays the frequencies for each score left side displays the frequencies for each score value or category. For grouped data, the midpoints value or category. For grouped data, the midpoints of the class intervals are arranged along the base of the class intervals are arranged along the base line. Study the following illustration: Notice in the line. Study the following illustration: Notice in the illustration that the taller bar, the greater the illustration that the taller bar, the greater the frequency of the category. frequency of the category.

600600

500500

400400

300300

200200

100100

00

BSE BSN AB Commerce SecretariatBSE BSN AB Commerce Secretariat

Figure 3.2 Bar graph of students enrolled in different Figure 3.2 Bar graph of students enrolled in different coursescourses

In the illustration above, which course has the In the illustration above, which course has the most number of enrollees? Least number of enrollees? most number of enrollees? Least number of enrollees? Is it possible for you to arrange the data in an ordinal Is it possible for you to arrange the data in an ordinal manner? What level of measurement do you have in manner? What level of measurement do you have in the illustration?the illustration?

The course AB has the most number of enrollees The course AB has the most number of enrollees while Commerce has the least, I.e., 600 and 200, while Commerce has the least, I.e., 600 and 200, respectively. It is also possible to arrange the data in an respectively. It is also possible to arrange the data in an ordinal manner. In this case, the level of measurement is ordinal manner. In this case, the level of measurement is ordinal. However, the data presented is actually in the ordinal. However, the data presented is actually in the nominal-level variable.nominal-level variable.


1.1. Given the following distribution, construct a pie chart.Given the following distribution, construct a pie chart.

Club Preference of 300 Students in a Secondary SchoolClub Preference of 300 Students in a Secondary School

ClubClub Frequency (f)Frequency (f)

ArtArt 100100

DanceDance 5050

ScienceScience 100100

SpeechSpeech 5050

TotalTotal 300 300

2. Depict the following data in a pie chart2. Depict the following data in a pie chart

Religious Preference of StudentsReligious Preference of Students

ReligionReligion Frequency (f)Frequency (f)

CatholicCatholic 500500

ProtestantProtestant 400400

JewishJewish 100100

MethodistMethodist 800800

Iglesia ni ChristoIglesia ni Christo 200200

Total 2000Total 2000

3. On a graphing paper, draw a bar graph to 3. On a graphing paper, draw a bar graph to illustrate the following distribution of IQ scores:illustrate the following distribution of IQ scores:

Class intervalClass interval ff

151-155151-155 99

146-150146-150 77

141-145141-145 44

136-140136-140 33

131-135131-135 88

126-130126-130 99

N = 40N = 40

4. Depict the following in a bar graph.4. Depict the following in a bar graph.

Socio-economic status of employees in a particular Socio-economic status of employees in a particular companycompany

StatusStatus frequency (f)frequency (f)

Above averageAbove average 800800

AverageAverage 15001500

Below AverageBelow Average 700700

N = 3000N = 3000

PRESENTING DATA IN A FREQUENCY POLYGONPRESENTING DATA IN A FREQUENCY POLYGON

What is a frequency polygon?What is a frequency polygon?

Frequency PolygonFrequency Polygon is another graphic is another graphic method employed in the presentation of data. method employed in the presentation of data. Although the frequency polygon can accommodate a Although the frequency polygon can accommodate a wide variety of categories, it tends to emphasize wide variety of categories, it tends to emphasize ““continuity”continuity” along a scale rather than differences. along a scale rather than differences. It is however, advantageous or useful for depicting It is however, advantageous or useful for depicting ordinal and interval data. This is the reason why ordinal and interval data. This is the reason why frequencies are indicated by a series of points placed frequencies are indicated by a series of points placed over the score values or midpoints of each class over the score values or midpoints of each class interval. Adjacent points, however, are connected interval. Adjacent points, however, are connected with a straight line, which is dropped to the base line with a straight line, which is dropped to the base line at either end. In the figure below, the height of each at either end. In the figure below, the height of each point or dot indicates frequency of occurrence. point or dot indicates frequency of occurrence.

Class intervalClass interval f f

136-140136-140 10 10 ff 4040141-145141-145 15 15 rr 3535146-150146-150 30 30 ee 3030151-155151-155 35 35 qq 2525156-160156-160 30 30 uu 2020161-165161-165 20 20 ee 1515166-170166-170 15 15 nn 1010

cc 55yy 00

138138 143 148 153 158 143 148 153 158 163163

midpoints (IQ)midpoints (IQ)

Figure 3.3 Frequency polygon of a distribution of IQ scoreFigure 3.3 Frequency polygon of a distribution of IQ score

Looking at the frequency polygon (figure 3.3), what class Looking at the frequency polygon (figure 3.3), what class interval has the highest number of frequency? Is it not interval has the highest number of frequency? Is it not 136-140?136-140?

What is cumulative frequency polygon?What is cumulative frequency polygon?To construct a cumulative frequency polygon, To construct a cumulative frequency polygon,

cumulative frequencies (or cumulative percentages) are cumulative frequencies (or cumulative percentages) are needed. Recall your previous module wherein you needed. Recall your previous module wherein you learned the meaning of cumulative frequency and how it learned the meaning of cumulative frequency and how it is being presented in a frequency distribution. As shown is being presented in a frequency distribution. As shown in figure 3.4, cumulative frequencies are arranged along in figure 3.4, cumulative frequencies are arranged along the vertical line of the graph and are indicated by the the vertical line of the graph and are indicated by the height of points above the horizontal base line. height of points above the horizontal base line. Cumulative frequencies are a product of successive Cumulative frequencies are a product of successive additions. Any cumulative frequency is never less and is additions. Any cumulative frequency is never less and is usually more than the preceding cumulative frequency. usually more than the preceding cumulative frequency. The points in a cumulative graph are plotted above the The points in a cumulative graph are plotted above the upper limits of class intervals rather at their midpoints. upper limits of class intervals rather at their midpoints. This is the reason why cumulative frequency represents This is the reason why cumulative frequency represents the total number of cases both within and below a the total number of cases both within and below a particular class interval.particular class interval.

Using the data in figure 3, study how cumulative frequency Using the data in figure 3, study how cumulative frequency polygon is being constructed.polygon is being constructed.

cfcf ff 175175 rr ee 150150 qq uu 125125 ee nn 100100 cc yy 7575

5050252500

140.5 145.5 150.5 155.5 160.5 140.5 145.5 150.5 155.5 160.5 165.5 170.5165.5 170.5

Upper Class BoundariesUpper Class BoundariesFigure 3.4Figure 3.4 Cumulative Frequency Polygon Cumulative Frequency Polygon


1.1. On a graphing paper, draw a cumulative On a graphing paper, draw a cumulative frequency polygon to represent the following frequency polygon to represent the following grades on a final examination in Research with grades on a final examination in Research with Statistics:Statistics:

Class intervalClass interval ff cfcf

91-10091-100 55 3636

81-9081-90 88 3131

71-8071-80 1010 2323

61-7061-70 88 1313

51-6051-60 55 55

N = 36N = 36

2. Construct a cumulative frequency polygon based 2. Construct a cumulative frequency polygon based on the following frequency distributionon the following frequency distribution


11-2011-20 33

21-3021-30 66

31-4031-40 77

41-5041-50 99

51-6051-60 88

61-7061-70 55

71-8071-80 22

N = 40N = 40

Hint:Hint: You cannot construct the cumulative frequency You cannot construct the cumulative frequency polygon not unless you have the cumulative polygon not unless you have the cumulative frequencies. Therefore, you have to fill up first the frequencies. Therefore, you have to fill up first the column for cumulative frequencies. If you can notice column for cumulative frequencies. If you can notice the class interval in item # 1 are arranged from the class interval in item # 1 are arranged from highest to lowest. In contrast, the class intervals in highest to lowest. In contrast, the class intervals in item #3 are arranged from lowest to highest. So you item #3 are arranged from lowest to highest. So you have to very careful in filling up the column for have to very careful in filling up the column for cumulative frequencies in this item.cumulative frequencies in this item.

3. Draw a frequency polygon for these data:3. Draw a frequency polygon for these data:Class intervalClass interval ff 11-1511-15 33 16-2016-20 55

21-2521-25 88 26-3026-30 1010 31-3531-35 77 36-4036-40 55 41-4541-45 2 2


You can use extra sheet/s of paper preferably graphing paper for You can use extra sheet/s of paper preferably graphing paper for your answer in this posttest.your answer in this posttest.

1.1. Display the following unemployment rates both in a bar Display the following unemployment rates both in a bar graph and as a line chart. Draw the two graphs separately.graph and as a line chart. Draw the two graphs separately.

AGEAGE MALE UNEMPLOYMENT MALE UNEMPLOYMENT RATERATE

21-2521-25 20.520.5

26-3026-30 15.2515.25

31-3531-35 10.5010.50

36-4036-40 10.0010.00

41-4541-45 8.758.75

46-5046-50 6.506.50

51-5551-55 5.255.25

56-6056-60 3.503.50

2. depict the following data in a bar graph2. depict the following data in a bar graph

Employment StatusEmployment Status

ProbationaryProbationary PermanentPermanent

AgeAge ff ff

21-2521-25 2020 1515

26-3026-30 1010 2020

31-3531-35 2020 1010

36-4036-40 55 55

41-4541-45 2525 3030

46-5046-50 4040 5050

51-5551-55 3030 4545

3. Construct a cumulative frequency polygon for the 3. Construct a cumulative frequency polygon for the following data:following data:

CICI ff cfcf

52-5552-55 1010

46-5046-50 1818

41-4541-45 2020

36-4036-40 3030

31-3531-35 1515

26-3026-30 1010

21-2521-25 55

4. Construct a pie chart for the following data;4. Construct a pie chart for the following data;

Percentage of PBET Board passersPercentage of PBET Board passers

YearYear %%

19891989 1010

19901990 2020

19911991 2525

19921992 3030

19931993 1010

19941994 55

COPUTER APPLICATIONCOPUTER APPLICATIONMICROSTATMICROSTAT is a program used to prepare is a program used to prepare

statistical computations. With the use of computers statistical computations. With the use of computers and this program the statistical information can be and this program the statistical information can be prepared at once with accuracy and speed.prepared at once with accuracy and speed.

MICROSTATMICROSTAT has a command to prepare or has a command to prepare or construct frequency polygon and bar graph or construct frequency polygon and bar graph or histogram. In the discussion portion of the module, histogram. In the discussion portion of the module, specifically on presenting data into graphic form, the specifically on presenting data into graphic form, the histogram or the bar graph is constructed vertically. histogram or the bar graph is constructed vertically. The construction of histogram in this sense is done The construction of histogram in this sense is done manually, I.e., using pencil and paper. On the other manually, I.e., using pencil and paper. On the other hand, the construction and presentation of histogram hand, the construction and presentation of histogram employing computer and the employing computer and the MICROSTAT MICROSTAT program, program, the manner in done horizontally.the manner in done horizontally.

NOTE:NOTE: Where can you secure a copy of this software? Where can you secure a copy of this software? You can ask your professor or the school to provide You can ask your professor or the school to provide you a copy so that you will be able tp prepare all the you a copy so that you will be able tp prepare all the necessary statistical computations. necessary statistical computations.

APPLICATIONAPPLICATION

Encode the following set of score in the Encode the following set of score in the computer using the computer using the MICROSTAT program and MICROSTAT program and prepare a histogram.prepare a histogram.

2525 5555 5454 7878 4545 8888 7878 5555 5151

6565 4545 2525 8989 4747 7474 8282 7777 7070

3232 8585 4444 9393 2828 6565 4646 9999 6161

3636 3535 2222 1212 1010 2222 5858 2222 5353

6666 4848 5656 7575 2020 3737 6666 5858 1111

IS THIS YOUR FIRST TIME TO USE COMPUTER?IS THIS YOUR FIRST TIME TO USE COMPUTER?

If this is your time to make use of computer, I If this is your time to make use of computer, I suggest if you can have a little practice of typing. You suggest if you can have a little practice of typing. You don’t have to worry regarding the use of this program. don’t have to worry regarding the use of this program. All you have to do is to the understand what the All you have to do is to the understand what the computer tells and commands you to do. The first things computer tells and commands you to do. The first things you have to learn in this program is the you have to learn in this program is the DATA DATA MANAGEMENT SYSTEMMANAGEMENT SYSTEM which will enable you to which will enable you to encode, edit, list, etc. the data. The instructions in this encode, edit, list, etc. the data. The instructions in this program are simple and easy to grasp and follow. In fact, program are simple and easy to grasp and follow. In fact, high school and college students can learn the high school and college students can learn the commands in this program even without the assistance commands in this program even without the assistance or guidance of their teacher. You have more intelligence or guidance of their teacher. You have more intelligence than these high school and college students. So, you can than these high school and college students. So, you can do it!do it!

During your free time, you are advised to think or During your free time, you are advised to think or create your own sets of numbers or scores. Then encode create your own sets of numbers or scores. Then encode these in the computer. Prepare the computer print outs.these in the computer. Prepare the computer print outs.

However, if there is really a need for you to However, if there is really a need for you to consult your professor regarding the use of his consult your professor regarding the use of his program, please don’t hesitate to ask questions program, please don’t hesitate to ask questions from your professor. At any rate, you have to from your professor. At any rate, you have to submit the computer print outs to your professor submit the computer print outs to your professor for checking and recording the results of your for checking and recording the results of your computer task/activities.computer task/activities.

Module fourModule four

MEASUREMENT OF CENTRAL TENDENCYMEASUREMENT OF CENTRAL TENDENCY

IntroductionIntroduction

Researchers in many fields have used the term Researchers in many fields have used the term ““average”average” in asking questions such as What is the in asking questions such as What is the average average grade of students in fourth year? How many grade of students in fourth year? How many bottles of beer are consumed by the average teenager? bottles of beer are consumed by the average teenager? What is the What is the grade-point averagegrade-point average of a doctoral students? of a doctoral students? On the On the average,average, how many crimes are committed every how many crimes are committed every quarter in the Philippines?quarter in the Philippines?

A useful way to describe a group as a whole is to A useful way to describe a group as a whole is to find a single number that represents what is find a single number that represents what is “average”“average” of that set of data. In research, such a value is known of that set of data. In research, such a value is known as a measure of as a measure of central tendency.central tendency. Measures of central Measures of central tendency are generally located toward the center of tendency are generally located toward the center of distribution where most of the data tend to be distribution where most of the data tend to be concentrated.concentrated.

The three best known measures of central The three best known measures of central tendency are discussed in this module: the mean, tendency are discussed in this module: the mean, the median, and the mode.the median, and the mode.

GENERAL OBJECTIVESGENERAL OBJECTIVES

At the end of this module, you are expected to:At the end of this module, you are expected to:

1.1. Calculate the arithmetic mean, median, and mode;Calculate the arithmetic mean, median, and mode;

2.2. Explain the characteristics, use, advantages, and Explain the characteristics, use, advantages, and disadvantages of using the mean, mode, and disadvantages of using the mean, mode, and median in researches;median in researches;

3.3. Identify the position of arithmetic mean, median, Identify the position of arithmetic mean, median, and mode for both a systematical and a skewed and mode for both a systematical and a skewed distribution.distribution.

TIME FRAMETIME FRAME

PRE-TEST 4PRE-TEST 4

Read each item carefully. Choose the best Read each item carefully. Choose the best answer. Write only the letter of the correct answer answer. Write only the letter of the correct answer on the blank before the number.on the blank before the number.

_____ 1. The sum of a set of scores divided by the _____ 1. The sum of a set of scores divided by the total number of total number of scores in the set scores in the set

A. modeA. mode C. meanC. mean

B. medianB. median d. standard deviationd. standard deviation

_____ 2. The middlemost point in a frequency _____ 2. The middlemost point in a frequency distribution is know distribution is know asas


B. medianB. median D. rangeD. range

_____ 3. The most frequent score in a distribution is known _____ 3. The most frequent score in a distribution is known as __.as __.


B. medianB. median D. intervalD. interval

_____ 4. What is the mode in this set of scores? _____ 4. What is the mode in this set of scores? 1,2,3,1,1,6,4,3,1,2,2,4,5, 1,2,3,1,1,6,4,3,1,2,2,4,5,

A. 1A. 1 B. 2B. 2 C. 3C. 3 D.4D.4

_____ 5. What is the average of this set of scores? _____ 5. What is the average of this set of scores? 50,45,60,55,50,50,45,60,55,50,

35,65,50,40,6035,65,50,40,60

A. 35A. 35 B. 40B. 40 C. 45C. 45 D. 50D. 50

_____ 6. What is the median in this set of scores? _____ 6. What is the median in this set of scores? 3,5,7,9,11,15,20,3,5,7,9,11,15,20,

30,5030,50

A. 9A. 9 B. 11B. 11 C. 20C. 20 D. 21D. 21

_____ 7. Given the following data, compute the mean:_____ 7. Given the following data, compute the mean:RespondentRespondentX (IQ)X (IQ)

AllanAllan 9090BuddyBuddy 110110CathyCathy 100100DanielDaniel 112112EfrenEfren 7575FelipeFelipe 130130GarryGarry 8585Harold Harold 9393

A. 99.375A. 99.375 C. 132.49C. 132.49B. 113.57B. 113.57 D. 158.99D. 158.99

____ 8. Using the formula X = fX/N, compute the ____ 8. Using the formula X = fX/N, compute the mean from the mean from the following distribution:following distribution:

XX ff fXfX

55 1010 5050

44 88 3232

33 77 2121

22 99 1818

11 55 55

A. 3.23A. 3.23 B. 7.8B. 7.8 C. 8.4.C. 8.4.D. 25.20D. 25.20

____ 9. Find the mode of the data below:____ 9. Find the mode of the data below:

Class intervalClass interval ff

25-2925-29 33

30-2430-24 22

35-3935-39 55

40-4440-44 88

45-4945-49 88

50-5450-54 99

55-5955-59 88

60-6460-64 66

65-6965-69 44

70-7470-74 33

N = 56N = 56

A. 55A. 55 B. 56B. 56 C. 57C. 57 D. 58D. 58

____ 10. Find the median of the following distribution:____ 10. Find the median of the following distribution:

Class intervalClass interval ff cfcf25-2925-29 33 3330-3430-34 22 5535-3935-39 55 101040-4440-44 88 181845-4945-49 88 262650-5450-54 88 343455-5955-59 99 434360-6460-64 66 494965-6965-69 66 555570-7470-74 33 585875-7975-79 33 616180-8480-84 33 6464

A. 53.25A. 53.25 B. 52.25B. 52.25 C. 51.5C. 51.5D. 50.25D. 50.25

THE MEAN FOR THE UNGROUPED DATATHE MEAN FOR THE UNGROUPED DATA

What is an averageWhat is an averageYou often need a certain number to represent a You often need a certain number to represent a

set of data. This one number can be thought of as set of data. This one number can be thought of as being being “typical”“typical” of all the data. If the annual salary of a of all the data. If the annual salary of a school supervisor is P150,000.00, then P200,000.00 school supervisor is P150,000.00, then P200,000.00 would be would be above averageabove average. Likewise, if the annual salary . Likewise, if the annual salary of a classroom teacher is P75,000.00, a salary of of a classroom teacher is P75,000.00, a salary of 74,500.00 is 74,500.00 is about averageabout average. . What is an average?What is an average?

Average Average is a single value or number that is a single value or number that represents a set of data. It pinpoints a center of the represents a set of data. It pinpoints a center of the values.values.

The Sample MeanThe Sample MeanThe measure of central tendency (average) most The measure of central tendency (average) most

widely used is the arithmetic mean, usually shortened widely used is the arithmetic mean, usually shortened to the to the mean.mean. For raw data, I.e., the ungrouped data, For raw data, I.e., the ungrouped data, the mean is the sum of all the values divided by the the mean is the sum of all the values divided by the total number of values. To find the total number of values. To find the

Mean for a sample, you can use the following formula:Mean for a sample, you can use the following formula:

Sum of all the Values in the SampleSum of all the Values in the Sample Sample Mean = Sample Mean =

Number of Values in the Sample Number of Values in the Sample

Instead of writing the formula in words, it is Instead of writing the formula in words, it is convenient to use the shorthand notation of algebra. convenient to use the shorthand notation of algebra. Thus, in symbol the formula is:Thus, in symbol the formula is:

X X X X NN

Where: Where: X X stands for the sample mean – it is read as “X stands for the sample mean – it is read as “X bar”bar”

XX stands for a particular value stands for a particular value

is the Greek capital sigma and indicates the is the Greek capital sigma and indicates the operation of operation of

addingadding

So:So:

XX stands for the sum of all the Xs stands for the sum of all the Xs

NN is the total number of values in the sample is the total number of values in the sample

Try this example:Try this example:

What is the sample mean of the following What is the sample mean of the following ungrouped data?ungrouped data?

25,15,35,24,26,12,38,30,20,11,39.25,15,35,24,26,12,38,30,20,11,39.

What is your sample mean? Did you get 25? What is your sample mean? Did you get 25? The correct sample mean for these data is 25. The correct sample mean for these data is 25. What is a statistical term appropriately applied to What is a statistical term appropriately applied to the sample mean of 25? The mean of a sample, or the sample mean of 25? The mean of a sample, or any other measure based on a data, is called a any other measure based on a data, is called a statistic. Thus, statistic is a measurable statistic. Thus, statistic is a measurable characteristic of a sample.characteristic of a sample.

Lets have another exampleLets have another example

The results of the test of five students were The results of the test of five students were taken randomly. The scores are as follows: 55, taken randomly. The scores are as follows: 55, 76.5,55.25,75, and 88. What is the arithmetic 76.5,55.25,75, and 88. What is the arithmetic mean of these five scores of the students?mean of these five scores of the students?

Solution:Solution:

X = 55 + 76.5 + 55.25 + 75 +88X = 55 + 76.5 + 55.25 + 75 +88

55

X =X = 349.75 349.75

55

X = 69.95X = 69.95

Therefore, the arithmetic mean is 69.95Therefore, the arithmetic mean is 69.95

The Population MeanThe Population Mean

The The population meanpopulation mean is computed in the same is computed in the same manner as to how the sample mean is computed. manner as to how the sample mean is computed. The measurable characteristic of the mean is known The measurable characteristic of the mean is known as statistic. In contrast measurable characteristic of as statistic. In contrast measurable characteristic of a population is called a population is called parameterparameter. The formula used . The formula used for computing the population parameter is as for computing the population parameter is as follows: follows: µµ = = X X

NN

Where:Where: µµ stands for the population mean stands for the population mean N stands for the total number of observation in the N stands for the total number of observation in the

populationpopulation X stands for the sum of all the XsX stands for the sum of all the Xs

The Properties of the Arithmetic MeanThe Properties of the Arithmetic Mean1. Every set of interval-level and ratio-level data has a 1. Every set of interval-level and ratio-level data has a mean.mean.2. All the values are included in computing the mean.2. All the values are included in computing the mean.3. A set of data has only one mean.3. A set of data has only one mean.4. The mean is a very useful measure for comparing two or 4. The mean is a very useful measure for comparing two or more more

populations.populations.5. The arithmetic mean is the only measure of central 5. The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value tendency where the sum of the deviations of each value from the mean will always be zero. In symbols, it is from the mean will always be zero. In symbols, it is expressed as follows:expressed as follows:

(X – X) = 0(X – X) = 0

Let’s take an example:Let’s take an example:The mean of 3, 7, and 5 is 5. The:The mean of 3, 7, and 5 is 5. The:

(X- X) (X- X) = (3 – 5) + (7-5) + (5 – 5)= (3 – 5) + (7-5) + (5 – 5)= -2 + 2 + 0= -2 + 2 + 0= 0= 0

In algebra, this is called additive inverse. This In algebra, this is called additive inverse. This means that the sum of two values having opposite signs means that the sum of two values having opposite signs will always be equal to zero. Like (-4) + (+4) = 0, (-25) + will always be equal to zero. Like (-4) + (+4) = 0, (-25) + (25) = 0 etc.(25) = 0 etc.

Now, do you understand this property of the mean? Now, do you understand this property of the mean? Do you still need some example? You can do the following Do you still need some example? You can do the following examples on your own: Find the mean of the following examples on your own: Find the mean of the following sets of scores and show that the sum of the deviations sets of scores and show that the sum of the deviations from the mean is equal to zero:from the mean is equal to zero:

1.1. 33,27,18,32,50,0,35, and 1033,27,18,32,50,0,35, and 10

2.2. 12,8,10,5,15,10,9,and 1112,8,10,5,15,10,9,and 11

3.3. 6,3,7,8,2,7,3,and 56,3,7,8,2,7,3,and 5

After answering these drill, check your own After answering these drill, check your own work. The key to correction is as follows:work. The key to correction is as follows:

1. 251. 25 2. 102. 10 3. 53. 5

Of course the sum of the deviations from Of course the sum of the deviations from the mean in all items is zero. Now, you are ready the mean in all items is zero. Now, you are ready to answer the following more challenging to answer the following more challenging exercises if you got all the answers correctly.exercises if you got all the answers correctly.


1.1. Five scores from a Statistics test were taken Five scores from a Statistics test were taken randomly. The test results of the five students are randomly. The test results of the five students are as follows: 85,93,78,65,and 93. Find the average or as follows: 85,93,78,65,and 93. Find the average or the arithmetic mean and show that the sum of the arithmetic mean and show that the sum of deviations from the mean is zero.deviations from the mean is zero.

2.2. What would be the grade-point average of a What would be the grade-point average of a doctoral student after a semester if he had the doctoral student after a semester if he had the following grades:following grades:

Philosophy of EducationPhilosophy of Education 1.751.75

ResearchResearch 1.251.25

Foundations of EducationFoundations of Education 2.502.50

Educational StatisticsEducational Statistics 3.003.00

3. The average number of crimes committed every quarter in 3. The average number of crimes committed every quarter in a particular country is 15. For the first quarter there were a particular country is 15. For the first quarter there were 13 crimes committed; second quarter 17; and third 13 crimes committed; second quarter 17; and third quarter 8. Find the number of crimes committed for the quarter 8. Find the number of crimes committed for the last quarter.last quarter.

4. For the past five years, the enrolment in University XYZ 4. For the past five years, the enrolment in University XYZ were as follows:were as follows:

School Year:School Year: EnrolmentEnrolment::

1991-19921991-1992 2,0502,0501992-19931992-1993 2,5002,5001993-19941993-1994 2,4502,4501994-19951994-1995 2,7802,7801995-19961995-1996 3,0503,050

Find the mean annual number of enrolment in XYZ Find the mean annual number of enrolment in XYZ university.university.

5. The U.S. Education Department reported that for the 5. The U.S. Education Department reported that for the past several years 5,033; 5,652; 6, 407; 7,201; 8,719; past several years 5,033; 5,652; 6, 407; 7,201; 8,719; 11,154; and 15,121 people received bachelor’s degree 11,154; and 15,121 people received bachelor’s degree in computer and information sciences. What is the in computer and information sciences. What is the mean annual number receiving this degree?mean annual number receiving this degree?

is it a sample mean or a popular mean? (Mason, Robert is it a sample mean or a popular mean? (Mason, Robert et al. 1990, p.79).et al. 1990, p.79).

THE MEAN OF THE GROUPED DATATHE MEAN OF THE GROUPED DATAThe computation of mean for the grouped data is The computation of mean for the grouped data is

being done when there is a large number of scores to be being done when there is a large number of scores to be treated in statistics. For instance, you want to obtain the treated in statistics. For instance, you want to obtain the arithmetic mean or average performance of the 10,000 arithmetic mean or average performance of the 10,000 examinees in Professional Board Examination for examinees in Professional Board Examination for Teachers.Of course, it is very tedious to add all the Teachers.Of course, it is very tedious to add all the scores and then divide by the total number of scores and then divide by the total number of examinees. Also, the speed and accuracy will of great examinees. Also, the speed and accuracy will of great problem on the part of the researcher. Instead, the problem on the part of the researcher. Instead, the scores or raw data are grouped in order to make the scores or raw data are grouped in order to make the computation convenient.computation convenient.

Recall the procedure on how to prepare the Recall the procedure on how to prepare the frequency distribution. In this case, you need to obtain frequency distribution. In this case, you need to obtain the highest and lowest scores in order to get the range; the highest and lowest scores in order to get the range; the desired number of steps; the class intervals; and the the desired number of steps; the class intervals; and the frequency for each class interval.frequency for each class interval.

Consider the following example:Consider the following example:

Fifty students took an entrance test in University Fifty students took an entrance test in University ABC. The scores obtained by the students from a 100-ABC. The scores obtained by the students from a 100-item test are as follows:item test are as follows:

88,75,77,90,56,66,28,84,39,4088,75,77,90,56,66,28,84,39,4088,30,50,66,30,39,56,76,44,8888,30,50,66,30,39,56,76,44,8871,73,92,99,55,87,38,40,20,1071,73,92,99,55,87,38,40,20,1090,22,55,76,20,55,44,10,29,090,22,55,76,20,55,44,10,29,057,39,74,39,67,55,39,82,40,23.57,39,74,39,67,55,39,82,40,23.

What is the highest score? Lowest score?They are What is the highest score? Lowest score?They are 99 and 0, respectively. How about the range? 99 and 0, respectively. How about the range? Obviously, it is 99 because 99-0 = 99. How many steps Obviously, it is 99 because 99-0 = 99. How many steps do you want to apply? Suppose you decide to have 10, do you want to apply? Suppose you decide to have 10, what will be the size of the class interval? You simply what will be the size of the class interval? You simply divide 99 by 10 and the size of the class interval divide 99 by 10 and the size of the class interval becomes 9.9 or 10 as the rounded value. Therefore, becomes 9.9 or 10 as the rounded value. Therefore, what will be what will be

the first class interval? The first class interval will be 0-the first class interval? The first class interval will be 0-9; second class interval will be 10-19; and so on. Thus, you are 9; second class interval will be 10-19; and so on. Thus, you are now ready to tally the scores and prepare the frequency now ready to tally the scores and prepare the frequency distribution.distribution.

Class intervalClass interval TallyTally frequency (f)frequency (f) MMfMfM

0-90-9

10-1910-19

20-2920-29

30-3930-39

40-4940-49

50-5950-59

60-6960-69

70-7970-79

80-8980-89

90-9990-99

N =N =

The formula for computing the mean of the grouped data is as The formula for computing the mean of the grouped data is as follows:follows:

fmfm NN

Where: X stands for the meanWhere: X stands for the meanfM stands for the product of the frequency and midpointfM stands for the product of the frequency and midpointfM stands for the sum of the product of all fs and MsfM stands for the sum of the product of all fs and MsN stands for the total number of scores or casesN stands for the total number of scores or cases

You will notice that in the table of the frequency distribution, You will notice that in the table of the frequency distribution, there is a column for M (or midpoint) and another column for there is a column for M (or midpoint) and another column for fM (or product of frequency and midpoint).fM (or product of frequency and midpoint). These two These two columns are left for your own computation. This is to apply columns are left for your own computation. This is to apply what you have learned in the previous modules. Do you still what you have learned in the previous modules. Do you still know how to get the midpoint? After you have all the know how to get the midpoint? After you have all the midpoints midpoints

for all the class intervals, you multiply them for all the class intervals, you multiply them with their corresponding frequency then get the with their corresponding frequency then get the summation of the column for fM. summation of the column for fM.

YourYour summation summation for the column of for the column of fMfM should should be be 27152715. If you divide this by 50 which is the total . If you divide this by 50 which is the total number of scores, you will obtain a mean of number of scores, you will obtain a mean of 54.3.54.3. Therefore, the mean performance of the fifty Therefore, the mean performance of the fifty students who took the entrance test of University students who took the entrance test of University ABC is 54.3. Now, are you ready for the activity?ABC is 54.3. Now, are you ready for the activity?


Given the set of scores below, find the following:Given the set of scores below, find the following:

1.1. Highest and lowest scores Highest and lowest scores

2.2. RangeRange

3.3. Size of the class interval (use 10 steps)Size of the class interval (use 10 steps)

4.4. Class intervalsClass intervals

5. Determine the frequency in each class interval5. Determine the frequency in each class interval

6. Identify the midpoints6. Identify the midpoints

7. Determine the summation of the product of all fs and 7. Determine the summation of the product of all fs and MsMs

8. Solve for the mean8. Solve for the mean

Scores:Scores: 10, 9, 20, 7, 18, 40, 25, 18, 20, 2310, 9, 20, 7, 18, 40, 25, 18, 20, 23

4, 1, 49, 25, 33, 23, 19, 23, 47, 334, 1, 49, 25, 33, 23, 19, 23, 47, 33

48, 44, 25, 37, 13, 34, 30, 20, 10, 27.48, 44, 25, 37, 13, 34, 30, 20, 10, 27.

The next lesson for you to learn is median The next lesson for you to learn is median which is another measure of central tendency . Since which is another measure of central tendency . Since median is another measure of centrality, you will median is another measure of centrality, you will notice that value of it is close to the value of the notice that value of it is close to the value of the mean. However, you have to learn first how to obtain mean. However, you have to learn first how to obtain the median of the ungrouped data then the median of the median of the ungrouped data then the median of the grouped data.the grouped data.

THE MEDIAN OF UNGROUPED AND GROUPED DATATHE MEDIAN OF UNGROUPED AND GROUPED DATA

What is median?What is median?

Median is know as the middlemost point in a Median is know as the middlemost point in a frequency distribution. In an ungrouped data, frequency distribution. In an ungrouped data, median can be defined as the centermost score in a median can be defined as the centermost score in a distribution. For instance, you want to obtain the distribution. For instance, you want to obtain the median from a 10-item test taken by 5 students. median from a 10-item test taken by 5 students. The scores obtained are as follows: The scores obtained are as follows: 7, 10, 5,3,7, 10, 5,3, and and 1.1. What is the median for this ungrouped data. To What is the median for this ungrouped data. To get the median, arrange the scores from lowest to get the median, arrange the scores from lowest to highest or vice-versa. Thus, the order of the scores highest or vice-versa. Thus, the order of the scores will be will be 1,3,5,7,1,3,5,7, and and 10 10 or or 10,7,5,3,10,7,5,3, and 1. Looking and 1. Looking at the order of the scores, the centermost score is 5 at the order of the scores, the centermost score is 5 either from the other of lowest to highest or vice-either from the other of lowest to highest or vice-versa.versa.

Try these examples: Find the median:Try these examples: Find the median:

1.1. 73,54,33,12,0,23, and 4173,54,33,12,0,23, and 412.2. 6,8,29,44,19,23,and 56,8,29,44,19,23,and 53.3. 14,34,97,33,and 6714,34,97,33,and 674.4. 10,45,66,78,and 4410,45,66,78,and 445.5. 27,87,and 23027,87,and 230

What are your answer? Did you get What are your answer? Did you get 33,19,34,4533,19,34,45, , and and 8787 for item 1,2,3,4, and 5, respectively? If you for item 1,2,3,4, and 5, respectively? If you have noticed the number of scores in every item is have noticed the number of scores in every item is odd. For items 1 and 2, there are 7 scores; in items 3 odd. For items 1 and 2, there are 7 scores; in items 3 and 4,5 and so on. How about if the scores are even-and 4,5 and so on. How about if the scores are even-numbered. For instance, you have 4 scores, 10 scores, numbered. For instance, you have 4 scores, 10 scores, etc. How will you obtain the median? To find the etc. How will you obtain the median? To find the median of even-numbered scores, simply add the two median of even-numbered scores, simply add the two centermost scores then divide by 2. For example: centermost scores then divide by 2. For example: What is the median of these scores: 5,24,12, and 40. What is the median of these scores: 5,24,12, and 40. There are four scores There are four scores

And if you arrange them, say from lowest to And if you arrange them, say from lowest to highest, you will have highest, you will have 5,12,245,12,24, and , and 40.40. Obviously the Obviously the two centermost scores are two centermost scores are 1212 and and 2424. The sum of . The sum of these two centermost scores is 36. If you divide 36 by these two centermost scores is 36. If you divide 36 by 2, then you will have a quotient of 18. 2, then you will have a quotient of 18. Therefore, the Therefore, the median is 18.median is 18.

Try solving the median of these sets of scores:Try solving the median of these sets of scores:

1. 24,90,156,761. 24,90,156,762. 25, 78, 12, 56, 77, 902. 25, 78, 12, 56, 77, 903. 45, 98, 12, 89, 34, 333. 45, 98, 12, 89, 34, 334. 256, 345, 897, 2344. 256, 345, 897, 2345. 45, 88, 72, 135. 45, 88, 72, 13

Let checked if you got the correct answers. You Let checked if you got the correct answers. You should have the following median: should have the following median: 50,66.5,39.5,400.5, and 58.5 for item 1,2,3,4, and 5, 50,66.5,39.5,400.5, and 58.5 for item 1,2,3,4, and 5, respectively.respectively.

The Median of Grouped DataThe Median of Grouped DataAs mentioned earlier in this module, the mean for As mentioned earlier in this module, the mean for

a large group of numbers is computed by organizing the a large group of numbers is computed by organizing the data in a way that these data are grouped accordingly. data in a way that these data are grouped accordingly. In like manner, the median for a large number of scores In like manner, the median for a large number of scores is also computed by organizing and grouping the data. is also computed by organizing and grouping the data. Also, as mentioned earlier, the purpose of grouping Also, as mentioned earlier, the purpose of grouping large number of scores is to make the computation easy large number of scores is to make the computation easy and to ensure accuracy and speed. It is accurate as far and to ensure accuracy and speed. It is accurate as far as the employment of formula and mathematical as the employment of formula and mathematical operations are concerned. However, with the use of operations are concerned. However, with the use of grouped data for the computation of the median and grouped data for the computation of the median and even the mean, you should bear in mind that the even the mean, you should bear in mind that the obtained mean and median from grouped frequency obtained mean and median from grouped frequency distributions are only approximations of what you would distributions are only approximations of what you would get if they were calculated from the raw scores.get if they were calculated from the raw scores.

Suppose a set of 100 scores were grouped and Suppose a set of 100 scores were grouped and translated into frequency distribution. How will you translated into frequency distribution. How will you obtain the median? So, take a look at this example and obtain the median? So, take a look at this example and examine how the median is obtained?examine how the median is obtained?


20-2420-24 33 3325-2925-29 1010 131330-3430-34 1212 252535-3935-39 1515 404040-4440-44 1818 585845-4945-49 1414 727250-5450-54 1010 828255-5955-59 99 919160-6460-64 66 979765-6965-69 33 100100

100100

The formula for finding the median of grouped data is The formula for finding the median of grouped data is given as follows:given as follows:

N/2 – FN/2 – FMdnMdn = L + = L + (i) (i)

fmfmWhere:Where:

MdnMdn = median= medianLL = lower boundary of interval containing = lower boundary of interval containing

the median the median class classFF = the sum of all the frequencies below L= the sum of all the frequencies below Lfmfm = frequency of interval containing the = frequency of interval containing the

median classmedian classNN = the total number of cases= the total number of casesii = class interval= class interval

To solve for the median of the data above, To solve for the median of the data above, the following values must be determined first then the following values must be determined first then substitution follows:substitution follows:

1.1. The lower boundary of the interval containing the The lower boundary of the interval containing the median class or the L.median class or the L.

2.2. The sum of all the frequencies below the exact The sum of all the frequencies below the exact lower limit or the F.lower limit or the F.

3.3. The frequency of the interval containing the median The frequency of the interval containing the median class or the fmclass or the fm

4.4. The total number of cases or the N.The total number of cases or the N.5.5. The class interval or the i.The class interval or the i.6.6. The 50% or half of N or th N/2.The 50% or half of N or th N/2.

Based on the data presented, what isBased on the data presented, what isL? L is equal to ________________L? L is equal to ________________F? F is equal to ________________F? F is equal to ________________

Fm? Fm? fm is equal to ____________fm is equal to ____________

N?N? N is equal to _____________N is equal to _____________

i?i? i is equal to ______________i is equal to ______________

N/2? N/2 is equal to _____________N/2? N/2 is equal to _____________

Checked if you got the answers correctly. The Checked if you got the answers correctly. The answers are as follows: explain how to set this L = answers are as follows: explain how to set this L = 39.5; F = 40; fm = 18; N = 100; i = 5; and N/2 = 39.5; F = 40; fm = 18; N = 100; i = 5; and N/2 = 50. If you got them correctly, you are ready to 50. If you got them correctly, you are ready to substitute the values to the formula.substitute the values to the formula.

To substitute, you will have the following:To substitute, you will have the following:

100 - 40100 - 40

MdnMdn = 39.5 + 2 = 39.5 + 2

.5.5

1818

= 39.5 + 50= 39.5 + 50

1818

= 39.5 + 2.9= 39.5 + 2.9

= 42.4= 42.4

Now, do you understand the steps on how to solve for Now, do you understand the steps on how to solve for the median of grouped data. Try this.the median of grouped data. Try this.


20-2420-24 77 7725-2925-29 1010 171730-3430-34 1515 323235-3935-39 99 414140-4440-44 77 484845-4945-49 55 535350-5450-54 33 565655-5955-59 44 6060

Based on the data presented above, what is:Based on the data presented above, what is:

L? L? L is equal to ________________L is equal to ________________

F? F? F is equal to ________________F is equal to ________________

fm? fm? fm is equal to ____________fm is equal to ____________

N?N? N is equal to _____________N is equal to _____________

i?i? i is equal to ______________i is equal to ______________

N/2? N/2 is equal to _____________N/2? N/2 is equal to _____________

This time you have to discover the answer to This time you have to discover the answer to this exercise. Anyway,for purposes of checking this exercise. Anyway,for purposes of checking your answer, the median or the final answer is your answer, the median or the final answer is 34.0.34.0.

Activity 4.3.Activity 4.3.A.A. Find the median of the following sets of scores:Find the median of the following sets of scores:

1. 34,98,56,99,23,and 121. 34,98,56,99,23,and 122. 55,90,23,45, and 772. 55,90,23,45, and 773. 40,34,78,22,12,and 663. 40,34,78,22,12,and 664. 45,8,35,18,39,55,and 894. 45,8,35,18,39,55,and 895. 9,0,3,12,5,4,67,71, and 355. 9,0,3,12,5,4,67,71, and 35

B.B. Compute the median of the following frequency distribution:Compute the median of the following frequency distribution:


45-4945-49 1010 101050-5450-54 2929 393955-5955-59 3232 717160-6460-64 4040 11111165-6965-69 3535 146146

70-7470-74 3030 17617675-7975-79 2525 20120180-8480-84 1515 21621685-8985-89 99 225225

THE MODE OF THE UNGROUPED AND GROUPED DATATHE MODE OF THE UNGROUPED AND GROUPED DATA

What is mode?What is mode?What is unimodal? Bimodal? Trimodal?What is unimodal? Bimodal? Trimodal?

How will you find for the mode of ungrouped data? How will you find for the mode of ungrouped data? Grouped data?Grouped data?

For ungrouped data, the mode is defined as that For ungrouped data, the mode is defined as that datum value or specific score which the highest datum value or specific score which the highest frequency. In other words, the frequency. In other words, the most frequency most frequency occurring score is the modeoccurring score is the mode. If there is one. If there is one

frequently occurring score in a set of values, the mode frequently occurring score in a set of values, the mode is classified as is classified as unimodal.unimodal. If the are two, the If the are two, the classification is known as bimodal and if there are classification is known as bimodal and if there are three, the classification is known as three, the classification is known as trimodal,trimodal, and so and so on.on.

Example:Example: Find the mode of the following set of score: Find the mode of the following set of score: 1,2,2,2,6,6,8,8,8,8,8,9,1,2,2,2,6,6,8,8,8,8,8,9,

By inspection, the mode (Mo0 is 8 because it appears By inspection, the mode (Mo0 is 8 because it appears the most number of times.the most number of times.

How about this example?How about this example? 5,5,5,5,5,6,6,6,6,6,7,7,9 5,5,5,5,5,6,6,6,6,6,7,7,9

What value or values appear the most number of times? What value or values appear the most number of times? Are there more one values? What are they? What is Are there more one values? What are they? What is the classification of this mode? The values that appear the classification of this mode? The values that appear most number of times are 5 and 6. There are two most number of times are 5 and 6. There are two values, therefore the classification of mode in this values, therefore the classification of mode in this case is bimodal.case is bimodal.

The mode for the grouped data is defined as the The mode for the grouped data is defined as the midpoint of the interval containing largest number of midpoint of the interval containing largest number of cases.cases.

Example:Example: Find the mode of the data below: Find the mode of the data below:

Class intervalClass interval ff34-3934-39 1515

40-4440-44 111145-4945-49 202050-5450-54 343455-5955-59 505060-6460-64 393965-6965-69 171770-7470-74 101074-7974-79 99

The mode (Mo) is 57 because the class interval 55-57 has The mode (Mo) is 57 because the class interval 55-57 has the largest frequency and 57 is the midpoint of this class intervalthe largest frequency and 57 is the midpoint of this class interval


A. Find the mode of the following sets of scores:A. Find the mode of the following sets of scores:

1. 23,34,34,55,55,55,55,60,90,1001. 23,34,34,55,55,55,55,60,90,100

2. 12,34,12,55,12,67,23,11,10,2002. 12,34,12,55,12,67,23,11,10,200

3. 25,66,66,75,66,66,13,13,10,443. 25,66,66,75,66,66,13,13,10,44

4. 4,5,7,2,5,5,5,5,8,94. 4,5,7,2,5,5,5,5,8,9

5. 7,34,34,3,12,34,8,9,10,405. 7,34,34,3,12,34,8,9,10,40

B.Find the mode of the following:B.Find the mode of the following:

1. Class interval1. Class interval ff 2. Class interval2. Class intervalff

10-1410-14 1414 30-3430-34 99

15-1915-19 1010 35-3935-39 1515

20-2420-24 1717 40-4440-44 6060

25-2925-29 4040 45-4945-49 2323

30-3430-34 5555 50-5450-54 77

3. Class interval3. Class interval ff 4. Class interval4. Class interval ff45-4945-49 11 55-5955-59 3350-5450-54 88 60-6460-64 101055-5955-59 2525 65-6965-69 454560-6460-64 1717 70-7470-74 9965-6965-69 99 75-7975-79 5570-7470-74 55 80-8480-84 22

5. Class interval5. Class interval ff34-3934-39 171740-4440-44 232345-4945-49 787850-5450-54 545455-5955-59 353560-6460-64 2222


Test I.Test I. Write Mn if the statement tells something about the Write Mn if the statement tells something about the mean, Mo if it is about the mode and Md if it is about the mean, Mo if it is about the mode and Md if it is about the median. Write your answer on the blank before the median. Write your answer on the blank before the number.number.

___ 1. It is used for interval data.___ 1. It is used for interval data.___ 2. It is used for ordinal or interval data.___ 2. It is used for ordinal or interval data.___ 3. It is used for nominal, ordinal or interval data.___ 3. It is used for nominal, ordinal or interval data.___ 4. It is most appropriate for unimodal systematical ___ 4. It is most appropriate for unimodal systematical

distributiondistribution___ 5. It is most appropriate for bimodal distribution___ 5. It is most appropriate for bimodal distribution___ 6. It is most appropriate for highly skewed distribution___ 6. It is most appropriate for highly skewed distribution___ 7. It is used when the purpose is to consider the value of ___ 7. It is used when the purpose is to consider the value of

each each scorescore___ 8. It is used when the center score is needed to avoid the ___ 8. It is used when the center score is needed to avoid the

influence of influence of extreme valuesextreme values___ 9. It is used when the researcher wants to identify the ___ 9. It is used when the researcher wants to identify the

most frequency most frequency occurring scoreoccurring score

___ 10. This value is referred as to the result of driving the sum of ___ 10. This value is referred as to the result of driving the sum of scores by the total number of cases.scores by the total number of cases.

Test II.Test II. Find the mean for the following grouped and ungrouped Find the mean for the following grouped and ungrouped data.data.

1.1. 89 45 57 34 23 90 56 44 34 7889 45 57 34 23 90 56 44 34 78

2.2. 134 345 879 345 223 590 276134 345 879 345 223 590 276

3.3. 87 34 77 40 122 45 66 88 3987 34 77 40 122 45 66 88 39

4.4. Class interval Class interval ff MM fMfM

20-2220-22 3333

23-2523-25 2525

26-2826-28 4040

29-3129-31 3737

32-3432-34 2020

35-3735-37 1414

N=N=

5. Class Interval5. Class Interval ff MM fMfM

15-1715-17 4545

18-2018-20 7878

21-2321-23 9090

24-2624-26 8080

27-2927-29 6565

30-3230-32 3030

N=N=

6. Class interval6. Class interval ff MM fMfM

35-3935-39 88

40-4440-44 1010

50-5450-54 1515

55-5955-59 99

60-6460-64 77

65-6965-69 44

70-7470-74 11

N=N=

Test III.Test III. Find the mode of the following grouped and Find the mode of the following grouped and ungrouped data. Write your answer on the blank ungrouped data. Write your answer on the blank before the number.before the number.

___ 1. 34,56,56,56,78,89,90,and 45___ 1. 34,56,56,56,78,89,90,and 45

___ 2. 45,12,10,10,34,10,78,23,and 10___ 2. 45,12,10,10,34,10,78,23,and 10

___ 3. 16,39,345,100,39,39,23,and 39___ 3. 16,39,345,100,39,39,23,and 39

___ 4. 23,90,23,88,23,67,23,23,12,and 23___ 4. 23,90,23,88,23,67,23,23,12,and 23

___ 5. 10,20,30,20,20,50,60,70,80,and 40___ 5. 10,20,30,20,20,50,60,70,80,and 40

___ 6. Class interval___ 6. Class interval ff

44-4644-46 1212

47-4947-49 99

50-5250-52 3434

53-5553-55 1818

56-5856-58 1010

___ 7. Class interval___ 7. Class interval ff

1-21-2 88

3-43-4 1010

5-65-6 1515

7-87-8 99

9-109-10 33

Test IIITest III. Find the median of the following grouped and . Find the median of the following grouped and ungrouped and ungrouped data. Write your ungrouped and ungrouped data. Write your answer on the blank before the number.answer on the blank before the number.

1.1. 34,78,90,12,45,76,34, and 5634,78,90,12,45,76,34, and 56

2.2. 10,45,20,55,30,67,90,and 4010,45,20,55,30,67,90,and 40

3.3. 16,34,88,77,23,76,and 4516,34,88,77,23,76,and 45

4.4. 66,90,34,22,19,30,and 7966,90,34,22,19,30,and 79

5.5. 44,90,56,34,67,21,43,and 4544,90,56,34,67,21,43,and 45

6. Class interval6. Class interval ff cfcf

10-1410-14 99 99

15-1915-19 1212 1212

20-2420-24 1515 3636

25-2925-29 3030 6666

30-3430-34 1414 8080

35-3935-39 1010 9090

40-4440-44 1010 100100

7. Class interval 7. Class interval ff cfcf

33-3633-36 99 99

37-4037-40 1313 2222

41-4441-44 1818 4040

45-4845-48 3030 7070

49-5249-52 1717 8787

53-5653-56 99 9696

57-6057-60 44 100100

COMPUTER APPLICATIONSCOMPUTER APPLICATIONS

Using the MICROSTAT software, encode the following Using the MICROSTAT software, encode the following data:data:

123123 564564 879879 998998 243243 456456 789789 345345 101101

456456 238238 875875 345345 897897 345345 234234 675675 234234

543543 689689 409409 497497 100100 346346 368368 203203 234234

544544 409409 567567 290290 567567 298298 456456 109109 309309

545545 208208 390390 476476 190190 267267 457457 333333 289289

785785 390390 386386 509509 390390 520520 345345 987987 289289

With the data that you encoded, compute the With the data that you encoded, compute the mean and prepare a computer print out.mean and prepare a computer print out.

________ DESCRIPTIVE ________ DESCRIPTIVE STATISTICS_________STATISTICS_________

HEADER DATA FORHEADER DATA FOR: A: PHILIP: A: PHILIP LABELLABEL: mean: mean

NUMBER OF CASESNUMBER OF CASES: 54: 54 NUMBER OF VARIABLESNUMBER OF VARIABLES: 1: 1

NO. NAME N MEAN STD.DEV. MINIMUM MAXIMUMNO. NAME N MEAN STD.DEV. MINIMUM MAXIMUM

1 SCORES 54 428.7073 224.3080 100.000 1 SCORES 54 428.7073 224.3080 100.000 998.0000998.0000

PRESS ANY KEY TO CONTINUE.PRESS ANY KEY TO CONTINUE.

MODULE FIVEMODULE FIVE

MEASURES OF VARIATIONMEASURES OF VARIATION

IntroductionIntroduction

In module number 4, you learned that the In module number 4, you learned that the mode, median, and mean could be utilized to mode, median, and mean could be utilized to summarize in a single number what is “typical” or summarize in a single number what is “typical” or “average” of a particular distribution. However, “average” of a particular distribution. However, when these measures of central tendency are when these measures of central tendency are employed alone, they can be misleading because employed alone, they can be misleading because they do not depict the complete picture of a set of they do not depict the complete picture of a set of data.data.

In addition to the measures of central In addition to the measures of central tendency, you need an index of how the scores are tendency, you need an index of how the scores are scattered the distribution. Thus, this will introduce scattered the distribution. Thus, this will introduce you to the concepts on variety.you to the concepts on variety.

OBJECTIVESOBJECTIVES

At the end of this module, you are expected to:At the end of this module, you are expected to:

1.1. Explain the uses of measures of variationExplain the uses of measures of variation

2.2. Compute the range, mean deviation, variance Compute the range, mean deviation, variance and standard deviation from a grouped data and and standard deviation from a grouped data and ungrouped data; andungrouped data; and

3.3. Compare the range, mean deviation, variance Compare the range, mean deviation, variance and standard deviation.and standard deviation.

PRE-TEST 5PRE-TEST 5Read each item carefully. Choose the best answer from Read each item carefully. Choose the best answer from the given four options. Write only the letter of your the given four options. Write only the letter of your answer on the blank before the number.answer on the blank before the number.

____ 1. The manner in which the score are scattered around ____ 1. The manner in which the score are scattered around the center of the distribution. Also known as dispersion the center of the distribution. Also known as dispersion or spread.or spread.

A. variabilityA. variability C. standard C. standard deviationdeviation

B. rangeB. range D. mean deviationD. mean deviation____ 2. The mean of the squared deviation from the mean of ____ 2. The mean of the squared deviation from the mean of

a distribution. A measure of variability in a distribution.a distribution. A measure of variability in a distribution.A. deviationA. deviation C. varianceC. varianceB. rangeB. range D. standard deviationD. standard deviation

____3. The difference between the highest and lowest scores ____3. The difference between the highest and lowest scores in distribution in distribution

A. deviation A. deviation C. range C. range B. mean deviation B. mean deviation D. varianceD. variance

____ 4. The sum of the absolute deviations from the mean ____ 4. The sum of the absolute deviations from the mean divided divided by the number of scores in a distribution.by the number of scores in a distribution.

A. A. mean deviationmean deviation C. standardC. standard

B. varianceB. variance D. none of theseD. none of these

____ 5. The square root of the mean of the squared ____ 5. The square root of the mean of the squared deviations from the mean of a distribution.deviations from the mean of a distribution.

A. standard deviationA. standard deviation C. mean C. mean deviationdeviation

B. varianceB. variance D. none of theseD. none of these

____ 6. Suppose the highest grade of a student is ____ 6. Suppose the highest grade of a student is 98 and 98 and his lowest grade is 94. Find the range his lowest grade is 94. Find the range between the between the two grade values.two grade values.

A. 4A. 4 C. 6C. 6

B. 5B. 5 D.7D.7

____ 7. The average of 5,8,9, and 6 is 7. Find the mean ____ 7. The average of 5,8,9, and 6 is 7. Find the mean deviation.deviation.

A. 1.5.A. 1.5. C. 2.0C. 2.0

B. 1.75B. 1.75 D. none of theseD. none of these

____ 8. The standard deviation of a certain distribution is ____ 8. The standard deviation of a certain distribution is 5.6. What 5.6. What is its variance?is its variance?

A. 29.36A. 29.36 C. 31.36C. 31.36

B. 30.36B. 30.36 D. 32.36D. 32.36

____ 9. The variance of a distribution is 50.41. What is its ____ 9. The variance of a distribution is 50.41. What is its standard standard deviation?deviation?

A. 7.30A. 7.30 C. 7.10C. 7.10

B. 7.20B. 7.20 D. 7.00D. 7.00

____ 10. The following are measures of variability except ____ 10. The following are measures of variability except one.one.

A. varianceA. variance C. rangeC. range

B. mean deviationB. mean deviation D. meanD. mean

THE RANGE AND THE MEAN DEVIATIONTHE RANGE AND THE MEAN DEVIATIONWhat is rangeWhat is rangeWhat is mean deviation?What is mean deviation?

To get quick but rough measure of variability, you To get quick but rough measure of variability, you might find what is known as the range. The range is might find what is known as the range. The range is the difference between the highest and lowest score. the difference between the highest and lowest score. Some teachers usually find the difference between Some teachers usually find the difference between these two scores, say from a mid term test results, to these two scores, say from a mid term test results, to set the passing score.set the passing score.

Study these examplesStudy these examples::1.1. Five students in Statistics class took a special Five students in Statistics class took a special

midterm test. The scores obtained by the students midterm test. The scores obtained by the students were as follows: 76,45,90,55, and 85. Find the range.were as follows: 76,45,90,55, and 85. Find the range.

2.2. The temperature in a certain place was recorded for The temperature in a certain place was recorded for one week. The record shows the following one week. The record shows the following temperature expressed in degree Celsius: temperature expressed in degree Celsius: 45,55,39,40,44,49, and 26. Find the range.45,55,39,40,44,49, and 26. Find the range.

Obviously, the highest and lowest scores in example Obviously, the highest and lowest scores in example 1 are 90 and 45, respectively.Applying the equation 1 are 90 and 45, respectively.Applying the equation R=HS-LS, the range will be 45. In example 2, the highest R=HS-LS, the range will be 45. In example 2, the highest and lowest scores and 55 and 26, respectively. Thus, the and lowest scores and 55 and 26, respectively. Thus, the range for this is 29 because 55-26 = 29. Suppose you range for this is 29 because 55-26 = 29. Suppose you have the following grades during the first semester: have the following grades during the first semester: 1.00,2.00, 3.00,2.75, and 5.00. Find the range. The range 1.00,2.00, 3.00,2.75, and 5.00. Find the range. The range for this example is 4.00 because the two extreme grades for this example is 4.00 because the two extreme grades are 5.00 and 1.00 and finding the difference between are 5.00 and 1.00 and finding the difference between these two, the range becomes 4.00these two, the range becomes 4.00

What is mean deviation?What is mean deviation?Mean deviation is defined as the distance of any Mean deviation is defined as the distance of any

given raw score from the mean. To find for such deviation, given raw score from the mean. To find for such deviation, subtract the mean from any raw score. For instance, the subtract the mean from any raw score. For instance, the mean of 5 scores is 7 Find the deviation from the mean of mean of 5 scores is 7 Find the deviation from the mean of score whose magnitude is 10. Obviously, the answer is 2. score whose magnitude is 10. Obviously, the answer is 2. How about a score whose magnitude is 3? What is the How about a score whose magnitude is 3? What is the absolute value of its deviation from the mean? The absolute value of its deviation from the mean? The absolute value of any raw score is simply referred to itsabsolute value of any raw score is simply referred to its

Magnitude, I. E., the sign whether positive or Magnitude, I. E., the sign whether positive or negative, is disregarded. Thus, the absolute value of –10 negative, is disregarded. Thus, the absolute value of –10 and + 10 is 10.and + 10 is 10.

How will you compute for the mean deviation of a set How will you compute for the mean deviation of a set of score? For instance, you have these scores: 9,8,6,4,2, of score? For instance, you have these scores: 9,8,6,4,2, and 7.and 7.

1.1. What will be the formula to solve for the mean deviation of What will be the formula to solve for the mean deviation of these scores? The formula is as follows: these scores? The formula is as follows:

/X – X//X – X/MD MD

NNWhere:Where:

MD = the mean deviationMD = the mean deviation /X – X/ = the sum of the absolute deviation (disregarding /X – X/ = the sum of the absolute deviation (disregarding

plus the minus signs)plus the minus signs)NN = total number of scores or cases= total number of scores or cases

Activity 5.1.Activity 5.1.1.1. Find the range of the following sets of scores. Write the Find the range of the following sets of scores. Write the

answer on the blank before the the number.answer on the blank before the the number.____ 1. 36,75,16,95,80____ 1. 36,75,16,95,80____ 2. 39,40,78,10,36____ 2. 39,40,78,10,36____ 3. 84,85,79,64,78,85.5____ 3. 84,85,79,64,78,85.5____ 4. 33,44,55,66,77,88____ 4. 33,44,55,66,77,88____ 5. 2,5,9,18,1,3,7,90____ 5. 2,5,9,18,1,3,7,90II. Compute the mean deviation or MD for the following II. Compute the mean deviation or MD for the following

distribution.distribution.1)1) XX X-XX-X /X-X//X-X/

77 ________ __________55 ________ __________99 ________ __________1212 ________ __________99 ________ __________66 ________ __________

X =X =

/X-X//X-X/

Find the mean:Find the mean:

XX

X = = X = =

NN

Find the mean deviation:Find the mean deviation:

MD = MD = /X –X /X –X = ______= ______

NN

2)2) XX X-XX-X /X-X/X-X

1010 _10-10______10-10_____ ____________

99 ____________ ____________

1111 ____________ ____________

55 ____________ ____________

1515 ____________ ____________

77 ____________ ____________

1313 ____________ ____________

XX /X-X//X-X/

Find the mean:Find the mean:

X X XX

== = _______ == _______ =

NN

Find the deviation:Find the deviation:

MDMD /X-X//X-X/

== = _______ == _______ =

NN

THE VARIANCE AND STANDARD DEVIATIOTHE VARIANCE AND STANDARD DEVIATIONNWhat is variance?What is variance?

In the previous learning activity, you learned that In the previous learning activity, you learned that the mean deviation avoid the problem of negative the mean deviation avoid the problem of negative values that cancel out positive values by simply values that cancel out positive values by simply ignoring positive (+) and negative (-) signs and getting ignoring positive (+) and negative (-) signs and getting the summation of absolute deviation from the mean. the summation of absolute deviation from the mean. This procedure, however, has disadvantage in creating This procedure, however, has disadvantage in creating a measure of variability, I.e, such absolute values are a measure of variability, I.e, such absolute values are not always useful in more advanced statistical analysis not always useful in more advanced statistical analysis because they cannot easily be manipulated because they cannot easily be manipulated algebraically.algebraically.

Thus, to overcome this kind of problem and Thus, to overcome this kind of problem and obtain a measure of variability which is more accepted obtain a measure of variability which is more accepted to advanced statistical procedure, you might square to advanced statistical procedure, you might square each of the actual deviations from the mean and add each of the actual deviations from the mean and add the squared values together. In symbols, it becomes:the squared values together. In symbols, it becomes:

(X-X)(X-X)²²

Table 5.2.1Table 5.2.1 SQUARING Deviations to Eliminate SQUARING Deviations to Eliminate Negative Negative Values: An IllustrationValues: An Illustration

XX X-XX-X (X-X) (X-X) ²²

55 -3-3 99

88 00 00

1010 22 44

1212 44 1616

55 -3-3 99

X = 40X = 40 0 0 (X-X) (X-X) ² = 38² = 38

Under the column of X-X, how will you show the Under the column of X-X, how will you show the solution for the first value which is –3? Since the solution for the first value which is –3? Since the mean is 8, subtract it from the first value of X which mean is 8, subtract it from the first value of X which is 5. Therefore, 5-8= -3. So, squaring –3 or is 5. Therefore, 5-8= -3. So, squaring –3 or multiplying by itself twice, it yields 9.multiplying by itself twice, it yields 9.

Now, what is the summation of all the squared Now, what is the summation of all the squared deviations from the mean? If you will divide this by N, deviations from the mean? If you will divide this by N, you will arrive at a measure of variability which is you will arrive at a measure of variability which is known as variance. Variance is actually the mean of known as variance. Variance is actually the mean of the squared deviations. To illustrate the solution for the squared deviations. To illustrate the solution for finding the variance, you have to employ the finding the variance, you have to employ the following formula:following formula:

(X-X) (X-X) ²²

ss² =² =

NN

Where:Where:

ss² is the variance² is the variance

(X-X)² is the summation of all squared (X-X)² is the summation of all squared deviationsdeviations

N is the total number of casesN is the total number of cases

To substitute the values you will have To substitute the values you will have the following equation:the following equation:

3838

ss² =² = = 7.6= 7.6

55

Therefore, the variance is 7.6.Therefore, the variance is 7.6.Can you tell now the advantage of variance over the Can you tell now the advantage of variance over the

mean deviation? Variance given appropriately greater mean deviation? Variance given appropriately greater emphasis to extreme values. In other words, it is more emphasis to extreme values. In other words, it is more sensitive to the degree of deviation in the distribution.sensitive to the degree of deviation in the distribution.

With the use of variance, however, another problem With the use of variance, however, another problem arises. As a direct result of having squared the deviations, arises. As a direct result of having squared the deviations, the unit of measurement is altered. Thus, it makes the the unit of measurement is altered. Thus, it makes the variance rather difficult to interpret. For instance, the variance rather difficult to interpret. For instance, the variance is 7.6, but 7.6 of what?variance is 7.6, but 7.6 of what?

What will you do with the variance in order to put the What will you do with the variance in order to put the measure of variability in the right perspective? Isn’t it that measure of variability in the right perspective? Isn’t it that you will find the square root of the variance?you will find the square root of the variance?

What will you obtain when extracting the root is What will you obtain when extracting the root is done? You will obtain another measure of variability which done? You will obtain another measure of variability which is called standard deviation. Standard deviations is the is called standard deviation. Standard deviations is the result of summing the squared deviations from the mean, result of summing the squared deviations from the mean, dividing it by N, and finally, taking the square root.dividing it by N, and finally, taking the square root.

To translate this definition into symbols, you will have the To translate this definition into symbols, you will have the formula:formula:

s = s = (X-X)(X-X)²²

NN

Where:Where:

s s is the standard deviationis the standard deviation

(X-X)(X-X)² ² is the sum of squared deviations from is the sum of squared deviations from the meanthe mean

NN is the total number of scores/casesis the total number of scores/cases

Now, how will you summarize the procedure for Now, how will you summarize the procedure for computing the standard deviation? The procedure does computing the standard deviation? The procedure does not differ much from the method you learned earlier to not differ much from the method you learned earlier to obtain the mean deviation with reference to the obtain the mean deviation with reference to the example on page 6, table 7, the following steps are example on page 6, table 7, the following steps are carried out. carried out.

Step IStep I Find the mean for the distribution. Find the mean for the distribution.XX55 XX88 X =X =1010 N N121255 40 40

N= 40N= 40 = 5 = 5

= 8= 8

Step 2 Subtract the mean from each raw score to obtain deviationStep 2 Subtract the mean from each raw score to obtain deviationXX X – XX – X55 -3 -388 0 01010 2 21212 4 455 -3 -3

Step 3Step 3 Square each deviation, then add the squared Square each deviation, then add the squared deviations.deviations.

XX X-XX-X (X-X) (X-X) ²²55 -3-3 9988 00 001010 22 441212 44 161655 -3-3 99

(X-X) (X-X) ² = 38² = 38

Step 4Step 4 Divide the summation of squared deviations by N Divide the summation of squared deviations by N and get the and get the square root of the results.square root of the results.

s = s = (X-X) (X-X) ²² NN

= 38= 38 55

= 7.6= 7.6

= 2.76= 2.76You can now conclude that the standard deviation You can now conclude that the standard deviation

for the set of scores 5, & 10,12 and 5 is 2.76.for the set of scores 5, & 10,12 and 5 is 2.76.

The Raw score Formula for Variance and Standard The Raw score Formula for Variance and Standard DeviationDeviation

The computation of variance and standard The computation of variance and standard deviation employing the raw score formula is very deviation employing the raw score formula is very simple and easy.simple and easy.

The formula for variance and standard deviation are as The formula for variance and standard deviation are as follows:follows:

XX²²

ss² = - X² ² = - X²

NN

s = X²s = X²

- X²- X²

NN

Where Where

X² is the sum of the squared scoresX² is the sum of the squared scores

N is the total number of scoresN is the total number of scores

X² is the mean squaredX² is the mean squared

The step-by-step procedure for computing the The step-by-step procedure for computing the ss² and s by the raw score method can be illustrated ² and s by the raw score method can be illustrated by returning to the data in Table 7.by returning to the data in Table 7.

Step IStep I. Square each raw score then add the squares . Square each raw score then add the squares raw scores.raw scores.

XX X²X²

55 2525

88 6464

1010 100100

1212 144144

55 2525

X² = 358X² = 358

Step 2Step 2 Obtain the mean and square it. Obtain the mean and square it.

4040

X =X =

55

= 8= 8 XX² ² = 64 = 64

Step 3Step 3. Plug the results from steps I and 2 into formula. Plug the results from steps I and 2 into formula

X² - X²X² - X² s = s = X² - X²X² - X²

ss² =² =

NN N N

358358

= = - 64 = 358 - 64 - 64 = 358 - 64

5 55 5

= 71.6 – 64 = 71.6 - 64= 71.6 – 64 = 71.6 - 64

= 7.6= 7.6 = 7.6 = 7.6

= 2.76= 2.76

Obtaining the Variance and Standard deviation from a Obtaining the Variance and Standard deviation from a frequency Distributionfrequency Distribution

To obtain the variance and standard deviation To obtain the variance and standard deviation from a frequency distribution, you will apply the from a frequency distribution, you will apply the following formulafollowing formula

X² X²

ss² =² = - X² - X²

NN

s = s = X² X²

- X² - X²

NN

Study the following example:Study the following example:

Score valueScore value frequencyfrequency77 1166 2255 3344 7733 4422 2211 11

N= 20N= 20

Step 1 Step 1 Multiply each score value (X) by its f to obtain fx.Multiply each score value (X) by its f to obtain fx.

XX ff fXfX

77 11 7766 22 1212

55 22 121244 77 282833 44 121222 22 4411 11 11

Step 2Step 2 Multiply each by Multiply each by X² to obtain f X².X² to obtain f X².

XX ff X² X² f X²f X²77 11 4949 494966 22 3636 727255 33 2525 757544 77 1616 11211233 44 99 363622 33 44 8811 11 11 11

f f X² = 353X² = 353

Step 3Step 3 Obtain the mean and square it. Obtain the mean and square it.

fXfX fXfX

77 X = X = N N X² = (3.95)²X² = (3.95)²

1212

1515 = 79 = 79 X² = 15.6025X² = 15.6025

28 2028 20

1212

44 = 3.95= 3.95

11

fX = 79fX = 79

Step 4.Step 4. Plug the results from steps 1, 2, and 3 into the Plug the results from steps 1, 2, and 3 into the formulas.formulas.

fXfX²² fXfX²²

s² = -s² = -XX²² s = s = --XX²²

NN N N

353353 = 353 = 353

= -15.6025= -15.6025 - - 15.602515.6025

2020 20 20

= 17.65= 17.65 = 17.65 – 15.6025 = 17.65 – 15.6025

= 2.0475= 2.0475 = 2.0475 = 2.0475

= 1.4309= 1.4309


1.1. Compute the variance and standard deviation for the Compute the variance and standard deviation for the following set of of scores. Use deviations from the mean and following set of of scores. Use deviations from the mean and squaring.squaring.

XX (X – X)(X – X) (X-X)(X-X)²²2244556688

2. Compute for the variance and standard deviation using raw 2. Compute for the variance and standard deviation using raw score formula. Use the data in question # 1.score formula. Use the data in question # 1.

XX X²X²2244556688

3. Compute the variance and standard deviation for the following 3. Compute the variance and standard deviation for the following frequency distributionfrequency distribution

XX ff fXfX fXfX²²

1212 33

1010 66

88 1010

99 55

55 22


Solve the following problems:Solve the following problems:

1.1. The examination scores of a group of five students are The examination scores of a group of five students are 8,5,4,3, and 0 on a 10-point scale. For this set of scores, 8,5,4,3, and 0 on a 10-point scale. For this set of scores, find:find:

1.1 range1.1 range

1.2. Mean deviation1.2. Mean deviation

1.3 variance1.3 variance

1.4 standard deviation1.4 standard deviation

2. For the following frequency distribution of scores, 2. For the following frequency distribution of scores, find:find:

2.1 variance2.1 variance

2.2 standard deviation2.2 standard deviation

Note: Note: This time you are provided with the answer in This time you are provided with the answer in the posttest. The key to correction might give you the posttest. The key to correction might give you some hint to answer the problems. However, this some hint to answer the problems. However, this does not mean that you will be the one to check does not mean that you will be the one to check your post test. The professor’s task in this case is your post test. The professor’s task in this case is to check your computations/solutions.to check your computations/solutions.

So, after answering the problems, you may So, after answering the problems, you may submit your answer to your professor. After the submit your answer to your professor. After the answers have been checked, he will tell you answers have been checked, he will tell you whether to proceed to the next module or not.whether to proceed to the next module or not.

Good luck!Good luck!

MODULE SIXMODULE SIX

NORMAL PROBABILITY DISTRIBUTIONNORMAL PROBABILITY DISTRIBUTION

Introduction: Introduction:

One of the most commonly used theoretical One of the most commonly used theoretical distributions in statistical inference is the normal distributions in statistical inference is the normal probability curve. The distribution is sometimes known probability curve. The distribution is sometimes known as Gaussian distribution in honor of Gauss, who derived as Gaussian distribution in honor of Gauss, who derived the mathematical equation for the normal curve. the mathematical equation for the normal curve. Frequency distributions can take a variety of shapes Frequency distributions can take a variety of shapes and and forms. Some are perfectly symmetric or free of and and forms. Some are perfectly symmetric or free of skew ness; others are skewed, either positively, skew ness; others are skewed, either positively, negatively skewed; still others have more than one negatively skewed; still others have more than one “hump”, and so on. Within this diversity, there is one “hump”, and so on. Within this diversity, there is one distribution as mentioned earlier which is commonly distribution as mentioned earlier which is commonly used in statistical inferences, I,e., the normal curve. used in statistical inferences, I,e., the normal curve. This module will discuss the meaning uses, This module will discuss the meaning uses, characteristics and applications of the normal curve in characteristics and applications of the normal curve in statistical inferences. statistical inferences.

GENERAL OBJECTIVESGENERAL OBJECTIVES

After you have completed this module, you are After you have completed this module, you are expected to:expected to:

1.1. Enumerate and explain the characteristics of a Enumerate and explain the characteristics of a normal probability distribution;normal probability distribution;

2.2. Define and calculate Z values;Define and calculate Z values;

3.3. Determine the probability that an observation will Determine the probability that an observation will lie between two points using the standard normal lie between two points using the standard normal distribution;distribution;

4.4. Determine the probability that an observation will Determine the probability that an observation will lie above or below, a value using standard normal lie above or below, a value using standard normal distribution; anddistribution; and

5.5. Compare two or more observations that are in Compare two or more observations that are in different probability.different probability.

PRE-TEST 6PRE-TEST 6Directions:Directions: Read each item very carefully. Choose Read each item very carefully. Choose

the letter of the nest answer. Write your answer on the letter of the nest answer. Write your answer on the blank before the number.the blank before the number.

___ 1. What characteristics of the normal curve is ___ 1. What characteristics of the normal curve is described in the described in the following statements? The normal following statements? The normal curve has a single peak at curve has a single peak at the exact center of the the exact center of the distribution. The arithmetic mean, distribution. The arithmetic mean, mode and mode and median of the distribution are equal and located at median of the distribution are equal and located at

the peak.the peak.a. bell-shapeda. bell-shaped c. symmetricc. symmetricb. asymptoticb. asymptotic d. none of thesed. none of these

___ 2. What characteristic of the normal curve is ___ 2. What characteristic of the normal curve is described in this described in this statement? If you cut the normal statement? If you cut the normal curve vertically at the curve vertically at the central value, two halves central value, two halves will coincide or will be mirror will coincide or will be mirror images.images.

a. bell-shapeda. bell-shaped c. symmetricc. symmetricb. asymptoticb. asymptotic d. none of thesed. none of these

____ 3. What characteristic of the normal curve is ____ 3. What characteristic of the normal curve is described in the described in the following statement? The normal curve following statement? The normal curve falls off smoothly in falls off smoothly in either direction from the central either direction from the central value. The curve gets closer value. The curve gets closer and closer to the X-axis and closer to the X-axis but never touches it. The tails of the but never touches it. The tails of the curve extend curve extend indefinitely in both directions.indefinitely in both directions.

a. bell-shapeda. bell-shaped c. symmetricc. symmetricb. asymptoticb. asymptotic d. none of thesed. none of these

____ 4. The percent of the area under the normal curve ____ 4. The percent of the area under the normal curve which is which is within plus one (or positive one) and minus within plus one (or positive one) and minus one (or positive one (or positive one) and minus one (or negative one) one) and minus one (or negative one) standard deviation of standard deviation of the mean.the mean.

a. 68%a. 68% b. 80%b. 80% c. 95%c. 95%d. 99%d. 99%

____ 5. The percent of the area under the normal curve ____ 5. The percent of the area under the normal curve which is which is within plus two (or positive two) and minus within plus two (or positive two) and minus two (or negative two (or negative two) standard deviations of the mean.two) standard deviations of the mean.

a. 68%a. 68% b. 80%b. 80% c. 95%c. 95%d. 99.73%d. 99.73%

____ 6. The percent of the area under the normal curve ____ 6. The percent of the area under the normal curve which is which is within three standard deviations of the mean.within three standard deviations of the mean.

a. 68%a. 68% b. 80%b. 80% c. 95%c. 95%d. 99.73%d. 99.73%

____ 7. The mean of a normally distributed group of weekly ____ 7. The mean of a normally distributed group of weekly income income of a large group of mile administrators in a of a large group of mile administrators in a certain school is certain school is P1,000.00; the standard deviation is P1,000.00; the standard deviation is 100.00. What is the Z 100.00. What is the Z value for an income X of value for an income X of P1,000.00?P1,000.00?

a. + 1.00a. + 1.00 b. +1.50b. +1.50 c. –1.00c. –1.00 d. –1-50d. –1-50____ 8. The mean of a normally distributed test scores in ____ 8. The mean of a normally distributed test scores in

statistics of statistics of 20 students is 100; the standard 20 students is 100; the standard deviation is 5.0. What is the deviation is 5.0. What is the Z value for a score X of 90?Z value for a score X of 90?

a. –1.00a. –1.00 b. –2.00b. –2.00 c. +1.00c. +1.00 d. +2.00d. +2.00____ 9. The distance between a selected value, designated ____ 9. The distance between a selected value, designated

by X, and by X, and the population mean, u, divided by the the population mean, u, divided by the population standard population standard deviation, 0.deviation, 0.

a. sstandard score b. t value c. z valuea. sstandard score b. t value c. z value d. d. raw scoreraw score

____ 10. The normal distribution is a ____ distribution.____ 10. The normal distribution is a ____ distribution.

a. continuousa. continuous c. either negativelyc. either negatively

b. discontinuousb. discontinuous d. none of thesed. none of these

CHARACTERISTICS OF A NORMAL PROBABILITY CHARACTERISTICS OF A NORMAL PROBABILITY DISTRIBUTIONDISTRIBUTION

What is a normal probability distribution?What is a normal probability distribution?

What are its main characteristics?What are its main characteristics?

What do you mean by bell-shaped? Symmetrical? What do you mean by bell-shaped? Symmetrical? Asymptotic?Asymptotic?

Study how a normal curve is constructed in the Study how a normal curve is constructed in the example below. Also, try to recall how a histogram is example below. Also, try to recall how a histogram is constructed. Consider the following frequency constructed. Consider the following frequency distribution as a basis for constructing a normal curve distribution as a basis for constructing a normal curve . But, first you will illustrate or draw the histogram for . But, first you will illustrate or draw the histogram for the distribution.the distribution.

Frequency Distribution Depicting a Normal CurveFrequency Distribution Depicting a Normal Curve

Class intervalClass interval Frequency (f)Frequency (f)

15-1915-19 55

20-2420-24 1010

25-2925-29 1515

30-3430-34 2020

35-3935-39 2525

40-4440-44 2020

45-4945-49 1515

50-5450-54 1010

55-5955-59 55

Table 6.1.1Table 6.1.1

Examine the distribution of frequency. What Examine the distribution of frequency. What is the frequency on both ends of the distribution? Is is the frequency on both ends of the distribution? Is is is 55? Yes, it is 5. How about the class intervals ? Yes, it is 5. How about the class intervals 20-20-2424 and and 50-5450-54 which are just near the two ends? which are just near the two ends? Both of them have a frequency of Both of them have a frequency of 10.10. The peak of The peak of the distribution is that class interval with the the distribution is that class interval with the highest frequency of highest frequency of 25.25.

Now, try to illustrate or depict the distribution Now, try to illustrate or depict the distribution into a histogram.into a histogram.

25252020151510105500

1717 2222 2727 3232 3737 4242 4747 5252 5757

Will your histogram look like this? If not, recall Will your histogram look like this? If not, recall what you have learned in the first three modules what you have learned in the first three modules particularly the module that discusses the topic on particularly the module that discusses the topic on organizing data.organizing data.

After you have drawn the histogram, try to After you have drawn the histogram, try to draw a curve out of it. Will your curve look like this?draw a curve out of it. Will your curve look like this?

2525

2020

1515

1010

55

00

1717 22 22 2727 3232 3737 4242 4747 5252 5757

Now, you describe the curve. Why do you call it Now, you describe the curve. Why do you call it a normal curve? If you cut the curve vertically at the a normal curve? If you cut the curve vertically at the central value of 25, what will happen to the two central value of 25, what will happen to the two halve? Will they coincide? Yes, the two halves will halve? Will they coincide? Yes, the two halves will coincide or sometimes they are describe as mirror coincide or sometimes they are describe as mirror images. What is the shape of the curve? Is it bell-images. What is the shape of the curve? Is it bell-shaped? How about the tails of the curve? Do they shaped? How about the tails of the curve? Do they actually touch the baseline? With all these questions, actually touch the baseline? With all these questions, you must be able to identify the following main you must be able to identify the following main characteristics of a normal curve such as:characteristics of a normal curve such as:

1.1. The normal curve is bell-shapedThe normal curve is bell-shaped. It has a single peak . It has a single peak

at the exact center of the distribution. The example at the exact center of the distribution. The example above has a frequency of 25 as its peak.above has a frequency of 25 as its peak.

2.2. The normal probability distribution is The normal probability distribution is symmetricalsymmetrical about itsabout its meanmean. The term symmetrical here means . The term symmetrical here means that if you cut the normal curve vertically at the that if you cut the normal curve vertically at the central value or at the peak value, the two halves of central value or at the peak value, the two halves of the normal curve will the normal curve will coincidecoincide or or willwill be mirror be mirror imagesimages..

3. The normal curve is 3. The normal curve is asymptoticasymptotic which means that the which means that the curve gets closer and closer to the X-axis but never curve gets closer and closer to the X-axis but never actually touches it.actually touches it.

The characteristic of a normal curve can be summarized The characteristic of a normal curve can be summarized in the following figure:in the following figure:

Figure 1Figure 1 Characteristics of a normal distribution Characteristics of a normal distribution

Normal curve is symmetricalNormal curve is symmetrical

The two halves are identicalThe two halves are identical

tailtail tailtail

Theoretically, curve extends to (-) Theoretically, curve extends to (-) Theoretical, curve Theoretical, curve extends to (+)extends to (+)


Draw a normal curve for each of the following Draw a normal curve for each of the following frequency distributions:frequency distributions:

1.1. C.1 10-14C.1 10-14 15-19 20-2415-19 20-24 25-29 25-29 30- 30-3434

ff 55 15 20 15 20 151555

2.2. C.1. 3-5C.1. 3-5 6-8 6-8 9-11 9-11 12-1412-1415-17 18-20 21-2315-17 18-20 21-23

ff 5 15 25 30 25 15 5 15 25 30 25 15 5 5

3.3. C.I. 1-2 3-4 5-6 7-8 9-10 11-12 C.I. 1-2 3-4 5-6 7-8 9-10 11-12 13-1413-14

f 5 10 15 25 15 10 f 5 10 15 25 15 10 5 5

AREAS UNDER THE NORMAL CURVEAREAS UNDER THE NORMAL CURVE

What is standard deviation?What is standard deviation?

What is + 1.00 standard deviation?What is + 1.00 standard deviation?

What is – 1.00 standard deviation?, etc?What is – 1.00 standard deviation?, etc?

What is the total area under the normal curve?What is the total area under the normal curve?

What is the difference between proportion and percent?What is the difference between proportion and percent?

If you can still recall, you learned about If you can still recall, you learned about standard deviation in the first few modules that it is standard deviation in the first few modules that it is the square root of the mean of the squared the square root of the mean of the squared deviations from the mean of a distribution. With this, deviations from the mean of a distribution. With this, module 6 will expand your knowledge about standard module 6 will expand your knowledge about standard deviation.deviation.

Study the area under the normal curve in the following Study the area under the normal curve in the following illustration:illustration:

-3s-3s -2s-2s -1s-1s XX +1s+1s +2s+2s +3s+3sAs you can see in the illustration, the percent of the As you can see in the illustration, the percent of the

area within area within _1.00_1.00 and and + 1.00+ 1.00 standard deviation of the standard deviation of the mean is about mean is about 68%.68%. The percent of the are within The percent of the are within –2.00–2.00 and and 2.002.00 standard deviation of the mean is about standard deviation of the mean is about 95%.95%. Lastly, the percent of the area within three standard Lastly, the percent of the area within three standard deviations of the mean is deviations of the mean is 99.73%.99.73%.

The area under the normal curve which is expressed The area under the normal curve which is expressed in percent can be translated in terms of proportion, I.e., in in percent can be translated in terms of proportion, I.e., in decimals. Thus, the area under the normal curve within decimals. Thus, the area under the normal curve within ––1.001.00 and and +1.00+1.00 is about is about .6827..6827. The area under the normal The area under the normal curve within curve within –2.00–2.00 and + and + 2.002.00 is about is about .9545.9545. Finally, the . Finally, the area under the normal curve within area under the normal curve within three standard three standard deviationsdeviations is about is about .9973.9973. However, the . However, the

Total area under the normal curve is Total area under the normal curve is 1.001.00. . The mathematical equation for the normal curve The mathematical equation for the normal curve is very useful in finding the percent or proportion is very useful in finding the percent or proportion of a particular population or sample without of a particular population or sample without having a tedious computation. You just simply having a tedious computation. You just simply employ the table of the are under the normal employ the table of the are under the normal curve.curve.

For instance, in a Statistics test, with a For instance, in a Statistics test, with a sample of 150 cases, the mean score is 60 and sample of 150 cases, the mean score is 60 and standard deviation is 10.0. Assuming normally, standard deviation is 10.0. Assuming normally, what percentage of the cases falls between the what percentage of the cases falls between the mean and a score of 80?mean and a score of 80?

The solution for the example mentioned is as follows:The solution for the example mentioned is as follows:

1.1. Convert the raw score of 80 into sigma score:Convert the raw score of 80 into sigma score:

Note:Note: sigma score is score sometimes known as the Z value sigma score is score sometimes known as the Z value

X-XX-X

Z = Z =

ss

80-6080-60

= =

1010

Z = 2.0 sZ = 2.0 s

2. Refer to the value of 2.0 s in the Table under the normal 2. Refer to the value of 2.0 s in the Table under the normal curve. curve. (The table is(The table is found on the next page).found on the next page). The value of The value of 2.0 s is .4772.2.0 s is .4772.

3. Convert .4772 into percent: .4772 = 47.72%3. Convert .4772 into percent: .4772 = 47.72%

To illustrate the normal curve, you should have the following:To illustrate the normal curve, you should have the following:

-3-3 -2-2 -1-1 XX +1+1 +2+2 +3+3

In the illustration, the shaded are is what actually In the illustration, the shaded are is what actually asked in the problem, I.e., the area under the normal asked in the problem, I.e., the area under the normal curve between the mean and two standard deviations.curve between the mean and two standard deviations.

Conclusion:Conclusion:The percent of students who got a score of 60 is The percent of students who got a score of 60 is

47.72% or about 48%. In this case, you can not say 47.72% or about 48%. In this case, you can not say 47.72% because you can not have head counts of students 47.72% because you can not have head counts of students with a decimal percent of 47.72%. Instead, 47.72 is with a decimal percent of 47.72%. Instead, 47.72 is rounded off. So, if you will be asked to get the number of rounded off. So, if you will be asked to get the number of students who got a score of 60, you will simply find for the students who got a score of 60, you will simply find for the 48% of the 150 cases. Thus, 48% of 150 is 72. Therefore, 48% of the 150 cases. Thus, 48% of 150 is 72. Therefore, the number of students who got a score of 60 is 72.the number of students who got a score of 60 is 72.


A.A. Given the mean, raw score, and standard Given the mean, raw score, and standard deviation, convert the following into Z values.deviation, convert the following into Z values.

Mean (X)Mean (X) Raw Score (X)Raw Score (X) Standard Standard deviation (s)deviation (s)

1. 1101. 110 120120 55

2. 802. 80 7070 44

3. 2353. 235 250250 1010

4. 1054. 105 100100 66

5. 305. 30 4040 2.52.5

B. Find the area under the normal curve:B. Find the area under the normal curve:1. Between the mean and + 1.50 standard deviation1. Between the mean and + 1.50 standard deviation2. To the right of –2.50 standard deviation2. To the right of –2.50 standard deviation3. Above +1.75 standard deviation3. Above +1.75 standard deviation4. To the left of +1.11 standard deviation4. To the left of +1.11 standard deviation5. Between –1.55 and +2.33 standard deviations5. Between –1.55 and +2.33 standard deviations

THE APPLICATIONS OF THE STANDARD NORMAL THE APPLICATIONS OF THE STANDARD NORMAL DISTRIBUTIONDISTRIBUTION

As mentioned earlier, with the employment of As mentioned earlier, with the employment of the Table of Areas Under the Normal Curve, the the Table of Areas Under the Normal Curve, the computations for finding a percent or proportion of the computations for finding a percent or proportion of the sample or the population based on a defined sample or the population based on a defined characteristic will not be tedious as the manual characteristic will not be tedious as the manual computations permit. The following examples illustrate computations permit. The following examples illustrate the applications of the normal curve or normal the applications of the normal curve or normal distribution.distribution.

The First ApplicationThe First Application

The first application of the standard normal The first application of the standard normal distribution is determining the area under the normal distribution is determining the area under the normal curve between the mean and a selected value. Using curve between the mean and a selected value. Using the same example on page 74, i,e., in a Statistical the same example on page 74, i,e., in a Statistical test with a mean of 60 and a standard deviation of test with a mean of 60 and a standard deviation of 10, find the area under the normal curve of the cases 10, find the area under the normal curve of the cases that falls between the mean and a score of 80. The Z that falls between the mean and a score of 80. The Z value has been computed and a value of 2.0 was value has been computed and a value of 2.0 was obtained. To repeat:obtained. To repeat:

X-XX-X Z = 2.0Z = 2.0Z =Z =

ss 80-6080-60==

1010

Using the Table of Areas Under the Normal Using the Table of Areas Under the Normal Curve, a Z = 2.0 has an area of .4772. A small Curve, a Z = 2.0 has an area of .4772. A small portion of the table is shown below. To locate the portion of the table is shown below. To locate the area, go down the left column to 2.00. Then move area, go down the left column to 2.00. Then move horizontally to the right, and read the area under horizontally to the right, and read the area under the curve in the column marked “area”.the curve in the column marked “area”.

Z valueZ value AreaArea1.961.96 .4750.47501.971.97 .4756.47561.981.98 .4761.47611.991.99 .4767.47672.002.00 .4772.47722.012.01 .4778.47782.022.02 .4783.47832.032.03 .4788.4788

As shown m the table above, the area under As shown m the table above, the area under the normal curve for a Z = 2.00 is .4772.the normal curve for a Z = 2.00 is .4772.

Here is another example:Here is another example:

In a Mathematics achievement test, with a In a Mathematics achievement test, with a sample of 100 cases, the mean score is 30 and sample of 100 cases, the mean score is 30 and standard deviation is 4.0. Assuming normality, what standard deviation is 4.0. Assuming normality, what percentage of the cases falls between the mean percentage of the cases falls between the mean and a score of 36?and a score of 36?

Solution:Solution:

1. Convert the raw score of 36 into sigma score1. Convert the raw score of 36 into sigma score

36-3036-30

Z =Z =

44

66

==

44

Z = 1.50 sZ = 1.50 s

2. Refer to the Table of Area Under the Normal Curve.2. Refer to the Table of Area Under the Normal Curve.The table shows that 1.5 corresponds to .4332.The table shows that 1.5 corresponds to .4332.

3. Convert .4332 to percent: .4332 = 43.32%3. Convert .4332 to percent: .4332 = 43.32%

Conclusion:Conclusion:Therefore, the percentage of the cases that Therefore, the percentage of the cases that

falls between the mean and a score of 36 is 43.32%.falls between the mean and a score of 36 is 43.32%.

The Second ApplicationThe Second ApplicationThe second application of the standard normal The second application of the standard normal

distribution involves finding the areas to the right or distribution involves finding the areas to the right or above and to the left or below, a specified value.above and to the left or below, a specified value.

Example:Example:In a Reading test of 100 cases, the mean is 85 In a Reading test of 100 cases, the mean is 85

and standard deviation is 5.0. What is the probability and standard deviation is 5.0. What is the probability above the score of 90?above the score of 90?

Solution:Solution:1.1. Convert the raw score of 90 to Z value or sigma score.Convert the raw score of 90 to Z value or sigma score.

90-8590-85Z = Z =

55

55 = =

55

Z = 1.0Z = 1.0

2. Locate the value of Z = 1.00 in the table.2. Locate the value of Z = 1.00 in the table.Z = 1.00 has a value of .3413.Z = 1.00 has a value of .3413.

3. Draw the normal curve and label the area that is asked.3. Draw the normal curve and label the area that is asked.

4. Since half of the area under the normal curve 4. Since half of the area under the normal curve equals .5000, subtract .3413 from that value. equals .5000, subtract .3413 from that value. Thus, the difference is .1587.Thus, the difference is .1587.

5. 5. Conclusion:Conclusion: Therefore, the probability or the area Therefore, the probability or the area under the normal curve of score above 90 under the normal curve of score above 90 is .1587.is .1587.

Note:Note: the term above in the problem means to the the term above in the problem means to the right as you have learned in the previous right as you have learned in the previous examples.examples.


The average performance of 5,000 cases in The average performance of 5,000 cases in the board examination is 75 and standard the board examination is 75 and standard deviation is 5.0 What is the probability of those deviation is 5.0 What is the probability of those who scored below 87?who scored below 87?

Solution:Solution:

1. Convert the raw score of 87 to Z value.1. Convert the raw score of 87 to Z value.

87-7587-75

Z =Z =

55

1212

==

55

Z = 2.4Z = 2.4

2. Locate the value of Z = 2.4 in the table.2. Locate the value of Z = 2.4 in the table.

Z = 2.4 corresponds to .4918.Z = 2.4 corresponds to .4918.

3. Draw the normal curve and label the area that is asked.3. Draw the normal curve and label the area that is asked.

4. Since half of the area under the normal curve is equal 4. Since half of the area under the normal curve is equal to .5000, subtract .4918 from that value. Thus, the difference to .5000, subtract .4918 from that value. Thus, the difference is .0082. is .0082.

5. 5. Conclusion:Conclusion: Therefore, the probability or the area Therefore, the probability or the area under the normal curve of those who scored below under the normal curve of those who scored below 87 is .0082.87 is .0082.

Note:Note: The term below in the problem means to the The term below in the problem means to the left.left.

The Third ApplicationThe Third Application

The third application of the standard normal The third application of the standard normal distribution is combining areas to the right and to distribution is combining areas to the right and to the left of the mean.the left of the mean.

Example: Example: In a Comprehensive Examination of 100 cases, the mean score In a Comprehensive Examination of 100 cases, the mean score is 50 and standard deviation is 5.0. Assuming normality, what is 50 and standard deviation is 5.0. Assuming normality, what is the probability between a score of 45 and a score of 60?is the probability between a score of 45 and a score of 60?

Solution:Solution:1.1. Convert the two raw scores into sigma scores or Z values.Convert the two raw scores into sigma scores or Z values.

45-5045-50Z =Z =

55

Z = -1.00Z = -1.00

60-5060-50Z = Z =

55

Z = 2.00Z = 2.00

2. Locate the values of Z = -1.00 and Z = 2.00 in the 2. Locate the values of Z = -1.00 and Z = 2.00 in the table. The table shows that Z = -1.00 corresponds table. The table shows that Z = -1.00 corresponds to .3413 and Z = 2.00 corresponds to .4772.to .3413 and Z = 2.00 corresponds to .4772.

Note: There is no negative Z values in the table. Thus, Note: There is no negative Z values in the table. Thus, when you locate certain value, you have to disregard when you locate certain value, you have to disregard the sign. The sign of negative here means below the the sign. The sign of negative here means below the mean or to the left of the mean. Likewise, the sign of mean or to the left of the mean. Likewise, the sign of positive means above or to the right of the mean.positive means above or to the right of the mean.

3. Draw the normal curve and label or shade the areas 3. Draw the normal curve and label or shade the areas that are asked.that are asked.

4. Combine the two values. In the previous application 4. Combine the two values. In the previous application of the standard normal distribution, you always of the standard normal distribution, you always subtract the computed value from .5000 or half the subtract the computed value from .5000 or half the value of the area under the normal curve. This time, value of the area under the normal curve. This time, instead of subtracting the two values, you are going instead of subtracting the two values, you are going to combine them. Thus, the sum of .3413 and .4772 to combine them. Thus, the sum of .3413 and .4772 is .8185.is .8185.

5. 5. Conclusion:Conclusion: Therefore, the probability between Therefore, the probability between scores 45 and 60 is .1885.scores 45 and 60 is .1885.


Using the same situation above, what is the Using the same situation above, what is the probability or area under the normal curve of the probability or area under the normal curve of the scores between 27 and 57?scores between 27 and 57?

Solution:Solution:

1.1. Convert the raw scores 37 and 57 into sigma Convert the raw scores 37 and 57 into sigma scores.scores.

37 – 5037 – 50

Z = 5Z = 5

Z = -2.60Z = -2.60

Z = 57 – 50Z = 57 – 50

55

Z = 1.4Z = 1.4

2. Locate the values of Z = -2.60 and Z = 1.4. The table 2. Locate the values of Z = -2.60 and Z = 1.4. The table shows that Z = 2.60 corresponds to .4953 and Z = shows that Z = 2.60 corresponds to .4953 and Z = 1.4 corresponds to .4192.1.4 corresponds to .4192.

3. Draw the normal curve and label or shade the areas 3. Draw the normal curve and label or shade the areas that are asked.that are asked.

4. Combine the two values: .4953 + .4192 = .9145.4. Combine the two values: .4953 + .4192 = .9145.5. 5. Conclusion:Conclusion: Therefore, the probability between scores Therefore, the probability between scores

37 and 57 is .9145.37 and 57 is .9145.

The Fourth ApplicationThe Fourth ApplicationThe fourth application of the standard normal The fourth application of the standard normal

distribution is finding the value of the observation X, distribution is finding the value of the observation X, when the percent below or above the mean is given.when the percent below or above the mean is given.

Example:Example:In a Statistics test, with a sample of 150 cases, In a Statistics test, with a sample of 150 cases,

the mean score is 30 and standard deviation is 2.5. the mean score is 30 and standard deviation is 2.5. Assuming normally, Assuming normally,

what is the score that a student should get to belong to what is the score that a student should get to belong to the upper 20%?the upper 20%?

Solution:Solution:1.1. 20% is at the end of the curve. So find 30% of the 20% is at the end of the curve. So find 30% of the

area from the mean.area from the mean.2.2. Refer to the table, column for area. There is no 30% Refer to the table, column for area. There is no 30%

or .30 but the value nearest to it is .2995.or .30 but the value nearest to it is .2995.3.3. .2995 correspond to a Z-score of .84..2995 correspond to a Z-score of .84.4.4. Convert Z = .84 to a raw score.Convert Z = .84 to a raw score.

X = ZX = Z ss ++ XX = (.84)= (.84) (2.5)(2.5) ++ 3030 = 2.1= 2.1 + 30+ 30X = 32.1X = 32.1

5. 5. ConclusionConclusion: Therefore, the score that a student should : Therefore, the score that a student should get in order for him to belong to the upper 20% is get in order for him to belong to the upper 20% is 32.1.32.1.


Solve the following problems:Solve the following problems:1.1. A teacher in Biology administered a 100-item to 50 A teacher in Biology administered a 100-item to 50

students. The mean performance of the students in the students. The mean performance of the students in the test is 76 and standard deviation is 8.4. Assuming test is 76 and standard deviation is 8.4. Assuming normality, what is the probability between the mean and a normality, what is the probability between the mean and a score of 97?score of 97?

2.2. The mean of a normal distribution is 500 pounds. The The mean of a normal distribution is 500 pounds. The standard deviation is 25. What is the area above 490 standard deviation is 25. What is the area above 490 pounds?pounds?

3.3. The mean age of 300 students in a certain school is 15 The mean age of 300 students in a certain school is 15 and standard deviation is 2.0. Assuming normality, what is and standard deviation is 2.0. Assuming normality, what is the area under the normal curve between ages 10 and the area under the normal curve between ages 10 and 17?17?

4.4. The mean performance of 200 students in the national The mean performance of 200 students in the national Elementary Achievement test is 80 and standard deviation Elementary Achievement test is 80 and standard deviation is 15.0. Assuming normality, what should be the score of a is 15.0. Assuming normality, what should be the score of a particular student in order for him or her to belong to particular student in order for him or her to belong to upper 15%?upper 15%?


Test 1. Draw a normal curve for each of the following Test 1. Draw a normal curve for each of the following distribution:distribution:

1. Class interval1. Class interval frequency frequency

5-95-9 1010

10-1410-14 3030

15-1915-19 4040

20-2420-24 3030

25-2925-29 1010

2. Class interval2. Class interval frequency frequency

2-42-4 55

5-75-7 1010

8-108-10 1515

11-1311-13 2020

14-1614-16 1515

17-1917-19 1010

20-2220-22 55

Note:Note: To make the drawing accurate, use graphing paper: To make the drawing accurate, use graphing paper:

Test II. Given the mean and standard deviation, convert Test II. Given the mean and standard deviation, convert the following raw scores into Z score.the following raw scores into Z score.

raw scoreraw score mean mean standard deviationstandard deviation1.1. 9090 70 70 10 102.2. 120120 200 200 50503.3. 1010 55 444.4. 1,0001,000 1,500 1,500 8008005.5. 300300 450450 6060

Test III. Find the area under the normal curve. Draw the Test III. Find the area under the normal curve. Draw the normal curve and label or shade the area that is normal curve and label or shade the area that is asked.asked.

1.1. To the right of Z = 1.8To the right of Z = 1.82.2. Below 2.75 standard deviationsBelow 2.75 standard deviations3.3. Between Z = 1.33 and Z = -2.33Between Z = 1.33 and Z = -2.334.4. To the left of Z = 0.99To the left of Z = 0.995.5. Above Z = 1.98Above Z = 1.98

Test IV. Solve the following problems:Test IV. Solve the following problems:

1.1. The mean of a normal distribution is The mean of a normal distribution is $500. The $500. The standard deviation is 80. What is the probability standard deviation is 80. What is the probability between the mean and $440?between the mean and $440?

2.2. The average enrollees every first semester in Philips The average enrollees every first semester in Philips College for the past 6 years is 2,000. The standard College for the past 6 years is 2,000. The standard deviation is 750. What is the probability for an deviation is 750. What is the probability for an enrollment of 3,000?enrollment of 3,000?

3.3. A salesman earns an average income of P15,000.00 A salesman earns an average income of P15,000.00 every month for the past two years. The standard every month for the past two years. The standard deviation is 1,800. What is the probability above an deviation is 1,800. What is the probability above an earning of 10,000.00?earning of 10,000.00?

4.4. The mean performance of 100 students in a 100-The mean performance of 100 students in a 100-item entrance test is 65. The standard deviation is item entrance test is 65. The standard deviation is 20. What must a student get in order to belong to 20. What must a student get in order to belong to upper 25%?upper 25%?

PAMPAINIT NG ULO

TESTING DIFFERENCES BETWEEN MEANSTESTING DIFFERENCES BETWEEN MEANSIntroductionIntroduction

In testing the differences between means, In testing the differences between means, first you must know the meaning of a first you must know the meaning of a hypothesis; the difference between null and hypothesis; the difference between null and alternative hypothesis; the difference between alternative hypothesis; the difference between directional and non-directional hypotheses, the directional and non-directional hypotheses, the steps in hypothesis testing, and finally, the t-steps in hypothesis testing, and finally, the t-tests of significance.tests of significance.

Thus, when testing the differences Thus, when testing the differences between means, researchers ask questions between means, researchers ask questions such as: such as: Do students exposed to traditional Do students exposed to traditional teaching methods differ from those exposed to teaching methods differ from those exposed to innovative teaching methods in terms of innovative teaching methods in terms of academic performance? Do professionals academic performance? Do professionals discipline their children morediscipline their children more severely than severely than those who have no education at allthose who have no education at all ? Note that ? Note that each research question involves making a each research question involves making a comparison between two groups. comparison between two groups.

OBJECTIVESOBJECTIVES

After you have completed this module, you are After you have completed this module, you are expected to:expected to:

1.1. Define what is meant by a hypothesis and Define what is meant by a hypothesis and hypothesis testing;hypothesis testing;

2.2. Distinguish between a null hypothesis and an Distinguish between a null hypothesis and an alternative or research hypothesis;alternative or research hypothesis;

3.3. Distinguish between a directional and a non-Distinguish between a directional and a non-directional hypothesis;directional hypothesis;

4.4. Describe the steps in hypothesis testing; andDescribe the steps in hypothesis testing; and

5.5. Conduct a test of hypothesis about the difference Conduct a test of hypothesis about the difference between two sample means.between two sample means.

TIME FRAMETIME FRAME

PRE-TESTPRE-TESTTest I. Fill in the blank with the correct answer:Test I. Fill in the blank with the correct answer:

Hint: This test is all about hypothesis.Hint: This test is all about hypothesis.1.1. _____ is an idea about the nature of reality which is _____ is an idea about the nature of reality which is

testable by systematic research.testable by systematic research.2.2. _____ States that there is no existence of relationship _____ States that there is no existence of relationship

between the variables under studybetween the variables under study3.3. _____ is sometimes called research hypothesis._____ is sometimes called research hypothesis.4.4. _____ makes use of one tail or one side of the _____ makes use of one tail or one side of the

statistical or distribution.statistical or distribution.5.5. _____ makes use of the tails or two sides of the _____ makes use of the tails or two sides of the

statistical model or distribution.statistical model or distribution.

Test II. Write N on the blank before the number if the Test II. Write N on the blank before the number if the statement is a null hypothesis and R if it is a research statement is a null hypothesis and R if it is a research hypothesis.hypothesis.

_____1. There is no significant difference in the _____1. There is no significant difference in the performance ofperformance of

students whether they are exposed to traditional or students whether they are exposed to traditional or contemporary teaching methods.contemporary teaching methods.

____ 2. There is difference in work performance between two ____ 2. There is difference in work performance between two groups of employees, one is given incentive, the other is groups of employees, one is given incentive, the other is not.not.

____ 3. There is no relationship between the I.Q. and sex of ____ 3. There is no relationship between the I.Q. and sex of students in terms of their academic performance.students in terms of their academic performance.

____ 4. Values integration will have significant effect on the ____ 4. Values integration will have significant effect on the behavior pf the students.behavior pf the students.

_____ 5. The discovery approach method of teaching will _____ 5. The discovery approach method of teaching will have no relation to the final year’s mark of the students.have no relation to the final year’s mark of the students.

Test III. Write D on the blank before the number if the Test III. Write D on the blank before the number if the statement is a directional hypothesis and ND if it is non-statement is a directional hypothesis and ND if it is non-directional.directional.

____ 1. The students exposed to values clarification lesson ____ 1. The students exposed to values clarification lesson will have better self-esteem than those who are not will have better self-esteem than those who are not exposed to such lessons.exposed to such lessons.

2. 2. The method of reaching is independent in terms of the The method of reaching is independent in terms of the achievement of the students.achievement of the students.

____ 3. The movies that have moral lessons will have better ____ 3. The movies that have moral lessons will have better effect to character development of students than those effect to character development of students than those action movies.action movies.

____ 4. The facial lotion used by most women is dependent to ____ 4. The facial lotion used by most women is dependent to their skin sensitivity.their skin sensitivity.

____ 5. The professional student will perform better in the ____ 5. The professional student will perform better in the comprehensive examination than a working student.comprehensive examination than a working student.

Test IV. Arrange the following procedure in hypothesis testing by Test IV. Arrange the following procedure in hypothesis testing by numbering them from 1 to 7.numbering them from 1 to 7.

_____ Determine the measurement level of the variable _____ Determine the measurement level of the variable understudyunderstudy

_____ State the null hypothesis._____ State the null hypothesis.

_____ Determine the significance of the computed value._____ Determine the significance of the computed value.

_____ Compute the value of the statistical test._____ Compute the value of the statistical test.

____Choose the statistical test appropriate to test the hypothesis.____Choose the statistical test appropriate to test the hypothesis.

____ Interpret and discuss the result.____ Interpret and discuss the result.

____ Specify a significance level and the sample size.____ Specify a significance level and the sample size.

The null Hypothesis: No Difference Between Means The The null Hypothesis: No Difference Between Means The Research Hypothesis: A Difference Between MeansResearch Hypothesis: A Difference Between Means

What is a null hypothesis?What is a null hypothesis?

What is a research or an alternative hypothesis?What is a research or an alternative hypothesis?

In statistical analysis, it has become conventional to In statistical analysis, it has become conventional to set out testing the null hypothesis – the hypothesis that says set out testing the null hypothesis – the hypothesis that says there is no difference between means of two samples drawn there is no difference between means of two samples drawn from the same population. According to the null hypothesis, from the same population. According to the null hypothesis, any observed difference between sample is regarded as a any observed difference between sample is regarded as a chance occurrence resulting from sampling error. Thus, it can chance occurrence resulting from sampling error. Thus, it can be concluded that an obtained difference between two be concluded that an obtained difference between two sample means does not actually represent a true difference sample means does not actually represent a true difference between their population means.between their population means.

The null hypothesis can be symbolized as:The null hypothesis can be symbolized as:

uu1 = 1 = uu22

Where:Where:

uu1 1 = means of the population= means of the population

uu2 2 = means of the second population= means of the second population

Examine the following null hypothesis:Examine the following null hypothesis:

1.1. Intelligent students are no more or less interested to Intelligent students are no more or less interested to listening to fairy tales.listening to fairy tales.

2.2. Nurses exposed to related learning experiences and those Nurses exposed to related learning experiences and those not exposed to such experiences perform to the same not exposed to such experiences perform to the same extent in their nursing care program.extent in their nursing care program.

3.3. Students who are given remedial class and those who Students who are given remedial class and those who attend the regular class do not differ in their academic attend the regular class do not differ in their academic performance.performance.

Note that each hypothesis states comparison between two Note that each hypothesis states comparison between two variables. The comparison states that there is no differencevariables. The comparison states that there is no difference

Between the two sample means. Thus, null hypotheses Between the two sample means. Thus, null hypotheses are stated in the manner that there will be no difference are stated in the manner that there will be no difference between means or no relationship exists between two between means or no relationship exists between two variables.variables.

Examine the following research or alternative hypotheses:Examine the following research or alternative hypotheses:

1. There is significant difference in the performance of 1. There is significant difference in the performance of students exposed to two different teaching students exposed to two different teaching methodologiesmethodologies

2. The performance of workers who are given incentives 2. The performance of workers who are given incentives differ from those who are not given incentives.differ from those who are not given incentives.

3. The perception of seminarians about sex differ 3. The perception of seminarians about sex differ significantly from laymensignificantly from laymen

4. The discipline of military people differ significantly from 4. The discipline of military people differ significantly from religious peoplereligious people

5. The academic performance of students attending regular 5. The academic performance of students attending regular class differ from those who study through distance class differ from those who study through distance education.education.

Note that each alternative hypothesis states comparison Note that each alternative hypothesis states comparison between two variables. The comparison states that there is between two variables. The comparison states that there is significant difference between two means or variables. Thus, significant difference between two means or variables. Thus, research or alternative hypothesis is stated in the manner research or alternative hypothesis is stated in the manner that there is an existence of relationship between means or that there is an existence of relationship between means or variables.variables.

Here are other examples of null hypotheses and Here are other examples of null hypotheses and research or alternative hypotheses. Be able to classify research or alternative hypotheses. Be able to classify them.them.

1.1. The 6-year old children do not differ from the 7-year old ones The 6-year old children do not differ from the 7-year old ones in terms of their readiness in schooling.in terms of their readiness in schooling.

2.2. The college professors compared to elementary teachers The college professors compared to elementary teachers differ significantly in terms of professional development.differ significantly in terms of professional development.

3.3. Government employees do not differ from private employees Government employees do not differ from private employees in terms of job satisfaction.in terms of job satisfaction.

4.4. The sense of responsibility of the first born children differ The sense of responsibility of the first born children differ significantly from those who are the last members in the significantly from those who are the last members in the family. family.

5. The morale of educated people does not differ 5. The morale of educated people does not differ from those who are illiterate.from those who are illiterate.

6. The performance of students exposed to modular 6. The performance of students exposed to modular instruction does not differ from those who are instruction does not differ from those who are exposed to the usual lecture method of instruction.exposed to the usual lecture method of instruction.

7. The number of achievers in terms of sex differ 7. The number of achievers in terms of sex differ significantly in the grade school level.significantly in the grade school level.

Which of the hypotheses above are stated in Which of the hypotheses above are stated in the null form? In research or alternative form? The the null form? In research or alternative form? The hypotheses stated in the null form are items 1,3,5, hypotheses stated in the null form are items 1,3,5, and 6. Item 2,4, and 7 are in the research or and 6. Item 2,4, and 7 are in the research or alternative form. Now, if you got these answers alternative form. Now, if you got these answers correctly, then you are ready to answer the correctly, then you are ready to answer the following exercises.following exercises.

ACTIVITY 7.1ACTIVITY 7.1

Write a null hypothesis and a research or alternative Write a null hypothesis and a research or alternative hypothesis for the following research objectives.hypothesis for the following research objectives.

A.A. To test the effect of modular instruction on the academic To test the effect of modular instruction on the academic performance of students with high intelligence quotient.performance of students with high intelligence quotient.

HHo o : ________________________________________: ________________________________________

HH11: ________________________________________: ________________________________________

B. To compare the temperament of respondents who are B. To compare the temperament of respondents who are classified as professionals and non-professionals.classified as professionals and non-professionals.

HHo o : ________________________________________: ________________________________________

HH11: ________________________________________: ________________________________________

C. To determine whether the Christian Formation Program in most C. To determine whether the Christian Formation Program in most Catholic schools develop the character among college Catholic schools develop the character among college students.students.

HHo o : ________________________________________: ________________________________________

HH11: ________________________________________: ________________________________________

D. To study how the mathematical abilities of D. To study how the mathematical abilities of students are developed under teachers who use students are developed under teachers who use contemporary approaches in teaching compared contemporary approaches in teaching compared with those of students under teachers who utilize with those of students under teachers who utilize the traditional lecture method.the traditional lecture method.

HHo o : ________________________________________: ________________________________________

HH11: ________________________________________: ________________________________________

E. To describe the male and female teachers in terms E. To describe the male and female teachers in terms of creativity in teaching.of creativity in teaching.

HHo o : ________________________________________: ________________________________________

HH11: ________________________________________: ________________________________________

DIRECTIONAL AND NON-DIRECTIONAL AND NON-DIRECTIONAL DIRECTIONAL HYPOTHESISHYPOTHESIS What is a directional hypothesis?What is a directional hypothesis? What is a non-directional hypothesis?What is a non-directional hypothesis? When the null hypothesis is rejected, the When the null hypothesis is rejected, the

alternative or research hypothesis is alternative or research hypothesis is accepted. accepted.

When the alternative or research hypothesis When the alternative or research hypothesis is accepted, it indicates existence of is accepted, it indicates existence of difference between the 2 means.difference between the 2 means.

When the direction or nature of the When the direction or nature of the diference between the 2 means or variable diference between the 2 means or variable is not stated, the test is considered, non is not stated, the test is considered, non directional.directional.

Examples of Examples of Hypothesis in Non Hypothesis in Non Directional FormDirectional Form1.1. Values clarification lessons will have Values clarification lessons will have

significant effect on the moral significant effect on the moral development of studentsdevelopment of students

2.2. There will be no significsnt effect on the There will be no significsnt effect on the academic performance of students under academic performance of students under teacher using contemporary and teacher using contemporary and traditional lecture method of instructiontraditional lecture method of instruction

3.3. There is significant diferrence in IQ There is significant diferrence in IQ between the group that will be exposed between the group that will be exposed to critical thinking lessons and the group to critical thinking lessons and the group that will be exposed to the same lessons.that will be exposed to the same lessons.

Do the hypothesis above use both Do the hypothesis above use both sides of statistical model or sides of statistical model or distribution?distribution?

Will the result of the study consider Will the result of the study consider those that belong below and above the those that belong below and above the mean?mean?

Nondirectional hypothesis or test make Nondirectional hypothesis or test make use of the two-tails or two-sides of the use of the two-tails or two-sides of the statistical model or distributionstatistical model or distribution

No assertion is made as to wether the No assertion is made as to wether the difference falls within the positive or difference falls within the positive or negative side or end of the distribution.negative side or end of the distribution.

Examples of Hypothesis Examples of Hypothesis Stated in Directional Stated in Directional FormForm1.1. The academic performance of students exposed to The academic performance of students exposed to

modular instruction is greater than those exposed to modular instruction is greater than those exposed to the traditional lecture method.the traditional lecture method.

2.2. The self-esteem of students exposed to values The self-esteem of students exposed to values clarification lessons is greater than those that are clarification lessons is greater than those that are not given the same lessonsnot given the same lessons

3.3. The nurses who are given incentives will have high The nurses who are given incentives will have high sense of responsibility than those who are not given sense of responsibility than those who are not given such incentives.such incentives.

IF THE DIRECTION OF THE DIFFERENCE IS STATED, IF THE DIRECTION OF THE DIFFERENCE IS STATED, THE TEST OF HYPOTHESIS BECOMES DIRECTIONALTHE TEST OF HYPOTHESIS BECOMES DIRECTIONAL

CAN YOU CONVERT A NON-DIRECTIONAL CAN YOU CONVERT A NON-DIRECTIONAL HYPOTHESIS INTO DIRECTIONAL FORM?HYPOTHESIS INTO DIRECTIONAL FORM?

Yes, you can convert a non-Yes, you can convert a non-directonal hypothesis ilnto directonal hypothesis ilnto directional formatdirectional format

Note: D = directional; N = non Note: D = directional; N = non directionaldirectionalN: There is no significant difference N: There is no significant difference in the behavior of nurses who are in the behavior of nurses who are given incentives and those who are given incentives and those who are not given the same.not given the same.D: the nurses who are given D: the nurses who are given incentives will perform better than incentives will perform better than those who are not given incentivesthose who are not given incentives

What have you What have you noticed?noticed? Directional hypothesis – pinpoints the Directional hypothesis – pinpoints the

group that will have significant effect, group that will have significant effect, or difference after a treatment is or difference after a treatment is givengiven

Non-directional hypothesis – the 2 Non-directional hypothesis – the 2 ends, or both sides of the statistical ends, or both sides of the statistical model or distribution are considered.model or distribution are considered.

Hypothesis testing – the process of Hypothesis testing – the process of determining the acceptability of a determining the acceptability of a hypothesis as derived from a theory hypothesis as derived from a theory through scientific data collection and through scientific data collection and through an application of appropriate through an application of appropriate statistical teststatistical testSTEPS:STEPS:

1.1. State the null hypothesis (Ho)State the null hypothesis (Ho)2.2. Determine the level of measurement of the Determine the level of measurement of the

variables under study and choose the statistical variables under study and choose the statistical test appropriate to test the hypothesistest appropriate to test the hypothesis

3.3. Specify the significance level and the sample Specify the significance level and the sample size.size.

4.4. Compute the value of the statistical test and Compute the value of the statistical test and determine the significance of the computed determine the significance of the computed value. Remember: a computed value that is value. Remember: a computed value that is equal or greater than the critical value is said to equal or greater than the critical value is said to be statistically significantbe statistically significant

5.5. Interpret and discuss the resultInterpret and discuss the result

The t-tests of The t-tests of significancesignificance What is a t- distribution?What is a t- distribution? What is the t-test of independent What is the t-test of independent

or uncorrelated means?or uncorrelated means? What is the t-test for dependent What is the t-test for dependent

or correlated means?or correlated means?

The t-ratioThe t-ratio

x-u

s 2n

t=

By W.S. Gosset