Meanings of Statistics

8/6/2019 Meanings of Statistics

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1.0 INTRODUCTION TO STATISTICS

Meanings of Statistics

The word statistics has three different meanings (sense) which are discussed below:

(1) Plural Sense (2) Singular Sense (3) Plural of the word Statistic

(1) Plural Sense: In plural sense, the word statistics refer to numerical facts and figurescollected in a systematic manner with a definite purpose in any field of study. In this sense,statistics are also aggregates of facts which are expressed in numerical form. For example,Statistics on industrial production, statistics or population growth of a country in different yearsetc.

(2) Singular Sense: In singular sense, it refers to the science comprising methods which are usedin collection, analysis, interpretation and presentation of numerical data. These methods are usedto draw conclusion about the population parameter.

For Example: If we want to have a study about the distribution of weights of students in acertain college. First of all, we will collect the information on the weights which may be obtainedfrom the records of the college or we may collect from the students directly. The large number ofweight figures will confuse the mind. In this situation we may arrange the weights in groups suchas: 50 Kg to 60 Kg 60 Kg to 70 Kg and so on and find the number of students fall in eachgroup. This step is called a presentation of data. We may still go further and compute theaverages and some other measures which may give us complete description of the original data.

(3) Plural of Word Statistic: The word statistics is used as the plural of the word Statisticwhich refers to a numerical quantity like mean, median, variance etc, calculated from samplevalue. For Example: If we select 15 student from a class of 80 students, measure their heights

and find the average height. This average would be a statistic.

Kinds or Branches Statistics

Statistics may be divided into two main branches:(1) Descriptive Statistics (2) Inferential Statistics

(1) Descriptive Statistics: In descriptive statistics, it deals with collection of data, itspresentation in various forms, such as tables, graphs and diagrams and findings averages andother measures which would describe the data.

For Example: Industrial statistics, population statistics, trade statistics etc Such asbusinessman make to use descriptive statistics in presenting their annual reports, final accounts,bank statements.

(2) Inferential Statistics: In inferential statistics, it deals with techniques used for analysis ofdata, making the estimates and drawing conclusions from limited information taken on sample

basis and testing the reliability of the estimates. ForExample: Suppose we want to have an idea about the percentage of illiterates in our country. Wetake a sample from the population and find the proportion of illiterates in the sample. Thissample proportion with the help of probability enables us to make some inferences about the

population proportion. This study belongs to inferential statistics.

Definition of StatisticsStatistics like many other sciences is a developing discipline. It is not nothing static. It has


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gradually developed during last few centuries. In different times, it has been defined in differentmanners. Some definitions of the past look very strange today but those definitions had their

place in their own time. Defining a subject has always been difficult task. A good definition oftoday may be discarded in future. It is difficult to define statistics. Some of the definitions arereproduced here:

(1) The kings and rulers in the ancient times were interested in their manpower. They conductedcensus of population to get information about their population. They used information tocalculate their strength and ability for wars. In those days statistics was defined as the scienceof kings, political and science of statecraft

(2) A.L. Bowley defined statistics as statistics is the science of counting This definition placesthe entries stress on counting only. A common man also thinks as if statistics is nothing butcounting. This used to be the situation but very long time ago. Statistics today is not merecounting of people, counting of animals, counting of trees and counting of fighting force. It hasnow grown to a rich methods of data analysis and interpretation.

(3) A.L. Bowley has also defined as science of averages This definition is verysimple but it covers only some area of statistics. Average is very simple important in statistics.Experts are interested in average deaths rates, average birth rates, average increase in population,and average increase in per capita income, average increase in standard of living and cost ofliving, average development rate, average inflation rate, average production of rice per acre,average literacy rate and many other averages of different fields of practical life. But statistics isnot limited to average only. There are many other statistical tools like measure of variation,measure of correlation, measures of independence etc Thus this definition is weak andincomplete and has been buried in the past.

(4) Prof: Boddington has defined statistics as science of estimate and probabilities Thisdefinition covers a major part of statistics. It is close to the modern statistics. But it is notcomplete because it stress only on probability. There are some areas of statistics in which

probability is not used.

(5) A definition due to W.I. King is the science of statistics is the method of judgingcollection, natural or social phenomena from the results obtained from the analysis or

enumeration or collection of estimates. This definition is close to the modern statistics. But itdoes not cover the entire scope of modern statistics. Secrist has given a detailed definition ofstatistics in plural sense. His definition is given on the previous. He has not given any importance

to statistics in singular sense. Statistics both in the singular and the plural sense has beencombined in the following definition which is accepted as the modern definition of statistics.

Statistics are the numerical statement of facts capable of analysis and interpretation and the

science of statistics is the study of the principles and the methods applied in collecting,

presenting, analysis and interpreting the numerical data in any field of inquiry.

Characteristics of Statistics


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Importance of Statistics in Different Fields

Statistics plays a vital role in every fields of human activity. Statistics has importantrole in determining the existing position of per capita income, unemployment,

population growth rate, housing, schooling medical facilities etcin a country. Nowstatistics holds a central position in almost every field like Industry, Commerce,Trade, Physics, Chemistry, Economics, Mathematics, Biology, Botany, Psychology,Astronomy etc, so application of statistics is very wide. Now we discuss someimportant fields in which statistics is commonly applied.

(1) Business: Statistics play an important role in business. A successfulbusinessman must be very quick and accurate in decision making. He knows thatwhat his customers wants, he should therefore, know what to produce and sell and inwhat quantities. Statistics helps businessman to plan production according to the taste

of the costumers, the quality of the products can also be checked more efficiently byusing statistical methods. So all the activities of the businessman based on statisticalinformation. He can make correct decision about the location of business, marketingof the products, financial resources etc

(2) In Economics: Statistics play an important role in economics. Economics largelydepends upon statistics. National income accounts are multipurpose indicators for theeconomists and administrators. Statistical methods are used for preparation of theseaccounts. In economics research statistical methods are used for collecting andanalysis the data and testing hypothesis. The relationship between supply anddemands is studies by statistical methods, the imports and exports, the inflation rate,

the per capita income are the problems which require good knowledge of statistics.

(3) In Mathematics: Statistical plays a central role in almost all natural and socialsciences. The methods of natural sciences are most reliable but conclusions drawfrom them are only probable, because they are based on incomplete evidence.Statistical helps in describing these measurements more precisely. Statistics is branchof applied mathematics. The large number of statistical methods like probabilityaverages, dispersions, estimation etc is used in mathematics and differenttechniques of pure mathematics like integration, differentiation and algebra are usedin statistics.

(4) In Banking: Statistics play an important role in banking. The banks make use ofstatistics for a number of purposes. The banks work on the principle that all the

people who deposit their money with the banks do not withdraw it at the same time.The bank earns profits out of these deposits by lending to others on interest. The

bankers use statistical approaches based on probability to estimate the numbers ofdepositors and their claims for a certain day.

(5) In State Management (Administration): Statistics is essential for a country.Different policies of the government are based on statistics. Statistical data are nowwidely used in taking all administrative decisions. Suppose if the government wantsto revise the pay scales of employees in view of an increase in the living cost,statistical methods will be used to determine the rise in the cost of living. Preparation


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of federal and provincial government budgets mainly depends upon statistics becauseit helps in estimating the expected expenditures and revenue from different sources.So statistics are the eyes of administration of the state.

(6) In Accounting and Auditing: Accounting is impossible without exactness. Butfor decision making purpose, so much precision is not essential the decision may betaken on the basis of approximation, know as statistics. The correction of the valuesof current asserts is made on the basis of the purchasing power of money or thecurrent value of it. In auditing sampling techniques are commonly used. An auditordetermines the sample size of the book to be audited on the basis of error.

(7) In Natural and Social Sciences: Statistics plays a vital role in almost all thenatural and social sciences. Statistical methods are commonly used for analyzing theexperiments results, testing their significance in Biology, Physics, Chemistry,Mathematics, Meteorology, Research chambers of commerce, Sociology, Business,

Public Administration, Communication and Information Technology etc

(8) In Astronomy: Astronomy is one of the oldest branches of statistical study; itdeals with the measurement of distance, sizes, masses and densities of heavenly

bodies by means of observations. During these measurements errors are unavoidableso most probable measurements are founded by using statistical methods.

Example: This distance of moon from the earth is measured. Since old days theastronomers have been statistical methods like method of least squares for finding the

movements of stars.

Some Basic Definitions in Statistics

Constant: A quantity which can be assuming only one value is called a constant. It is

usually denoted by the first letters of alphabets .

For Example: Value of and value of

Variable: A quantity which can vary from one individual or object to and other is

called a variable. It is usually denoted by the last letters of alphabets .

For Example: Heights and Weights of students, Income, Temperature, No. ofChildren in a family etc

Continuous Variable: A variable which can assume each and every valuewithin a given range is called a continuous variable. It can occur in decimals.For Example: Heights and Weights of students Speed of a bus, the age of aShopkeeper, the life time of T.V etc

Continuous Data: Data which can be described by a continuous variable is calledcontinuous data. For Example: Weights of 50 students in a class.

Discrete Variable: A variable which can assume only some specific values within agiven range is called discrete variable. It cannot occur in decimals. It can occur in


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whole numbers.For Example:Number of students in a class, number of flowers on the tree, numberof houses in a street, number of chairs in a room etc

Discrete Data: Data which can be described by a discrete variable is called discretedata. For Example:Number of students in a college.

Quantitative Variable: A characteristic which varies only in magnitude from onindividual to another is called quantitative variable. It can be measurable.For Example: Wages, Prices, Heights, Weights etc

Qualitative Variable: A characteristic which varies only in quality from oneindividual to another is called qualitative variable. It cannot be measured.For Example: Beauty, Marital Status, Rich, Poor, Smell

Collection of Statistical Data

Statistical Data: A sequence of observation, made on a set of objects included in thesample drawn from population is known as statistical data.

(1) Ungrouped Data: Data which have been arranged in a systematic order are calledraw data or ungrouped data.

(2) Grouped Data: Data presented in the form of frequency distribution is calledgrouped data.

Collection of Data: The first step in any enquiry (investigation) is collection of data.The data may be collected for the whole population or for a sample only. It is mostlycollected on sample basis. Collection of data is very difficult job. The enumerator orinvestigator is the well trained person who collects the statistical data. Therespondents (information) are the persons whom the information is collected.

Types of Data: There are two types (sources) for the collection of data. (1) Primary Data (2) Secondary Data

(1) Primary Data: The primary data are the first hand informationcollected, compiled and published by organization for some purpose. They are most

original data in character and have not undergone any sort of statistical treatment.Example: Population census reports are primary data because these are collected,complied and published by the population census organization.(2) Secondary Data: The secondary data are the second hand information which arealready collected by some one (organization) for some purpose and are available forthe present study. The secondary data are not pure in character and have undergonesome treatment at least once.Example: Economics survey of England is secondary data because these arecollected by more than one organization like Bureau of statistics, Board of Revenue,the Banks etc


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Methods of Collecting Primary Data: Primary data are collected by the followingmethods:

Personal Investigation: The researcher conducts the survey him/herself and

collects data from it. The data collected in this way is usually accurate andreliable. This method of collecting data is only applicable in case of smallresearch projects.

Through Investigation: Trained investigators are employed to collect thedata. These investigators contact the individuals and fill in questionnaire afterasking the required information. Most of the organizing implied this method.

Collection through Questionnaire: The researchers get the data from localrepresentation or agents that are based upon their own experience. Thismethod is quick but gives only rough estimate.

Through Telephone: The researchers get information through telephone thismethod is quick and give accurate information.

Methods of Collecting Secondary Data: The secondary data are collected by thefollowing sources:

Official: e.g. The publications of the Statistical Division, Ministry of Finance,the Federal Bureaus of Statistics, Ministries of Food, Agriculture, Industry,Labor etc

Semi-Official: e.g. State Bank, Railway Board, Central Cotton Committee,Boards of Economic Enquiry etc

Publication of Trade Associations, Chambers of Commerce etc

Technical and Trade Journals and Newspapers.

Research Organizations such as Universities and other institutions.

Difference between Primary and Secondary Data: The difference between primary

and secondary data is only a change of hand. The primary data are the first hand datainformation which is directly collected form one source. They are most original datain character and have not undergone any sort of statistical treatment while thesecondary data are obtained from some other sources or agencies. They are not purein character and have undergone some treatment at least once.For Example: Suppose we interested to find the average age of MS students. Wecollect the ages data by two methods; either by directly collecting from each studenthimself personally or getting their ages from the university record. The data collected

by the direct personal investigation is called primary data and the data obtained fromthe university record is called secondary data.

Editing of Data: After collecting the data either from primary or secondary source,the next step is its editing. Editing means the examination of collected data to


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discover any error and mistake before presenting it. It has to be decided before handwhat degree of accuracy is wanted and what extent of errors can be tolerated in theinquiry. The editing of secondary data is simpler than that of primary data.


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PRESENTATION OF DATAClassification of Data

The process of arranging data into homogenous group or classes according tosome common characteristics present in the data is called classification.For Example: The process of sorting letters in a post office, the letters are classified

according to the cities and further arranged according to streets.

Bases of Classification:

There are four important bases of classification:(1) Qualitative Base (2) Quantitative Base (3) Geographical Base (4) Chronologicalor Temporal Base

(1) Qualitative Base: When the data are classified according to some quality orattributes such as sex, religion, literacy, intelligence etc

(2) Quantitative Base: When the data are classified by quantitative characteristics likeheights, weights, ages, income etc

(3) Geographical Base: When the data are classified by geographical regions orlocation, like states, provinces, cities, countries etc(4) Chronological or Temporal Base: When the data are classified or arranged bytheir time of occurrence, such as years, months, weeks, days etc For Example: Timeseries data.

Types of Classification:

(1) One -way Classification: If we classify observed data keeping in view singlecharacteristic, this type of classification is known as one-way classification. ForExample: The population of world may be classified by religion as Muslim, Christiansetc

(2) Two -way Classification: If we consider two characteristics at a time in order toclassify the observed data then we are doing two way classifications. For Example:The population of world may be classified by Religion and Sex.

(3) Multi -way Classification: We may consider more than two characteristics at a

time to classify given data or observed data. In this way we deal in multi-wayclassification.For Example: The population of world may be classified by Religion, Sex andLiteracy.

Tabulation of Data

The process of placing classified data into tabular form is known as tabulation. A tableis a symmetric arrangement of statistical data in rows and columns. Rows are horizontalarrangements whereas columns are vertical arrangements. It may be simple, double orcomplex depending upon the type of classification.


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Types of Tabulation

(1) Simple Tabulation or One-way Tabulation: When the data are tabulated to onecharacteristic, it is said to be simple tabulation or one-way tabulation.

For Example: Tabulation of data on population of world classified by onecharacteristic like Religion is example of simple tabulation.

(2) Double Tabulation or Two-way Tabulation: When the data are tabulatedaccording to two characteristics at a time. It is said to be double tabulation or two-waytabulation. For Example: Tabulation of data on population of world classified by twocharacteristics like Religion and Sex is example of double tabulation.

(3) Complex Tabulation: When the data are tabulated according to manycharacteristics, it is said to be complex tabulation. For Example: Tabulation of data on

population of world classified by two characteristics like Religion, Sex and Literacyetcis example of complex tabulation.

Construction of Statistical Table

A statistical table has at least four major parts and some other minor parts.(1) The Title (2) The Box Head (column captions) (3) The Stub (row captions)(4) The Body (5) Prefatory Notes (6) Foots Notes (7) Source NotesThe general sketch of table indicating its necessary parts is shown below:

----THE TITLE----

----Prefatory Notes----

----Box Head----

----Row Captions---- ----Column Captions----

----Stub Entries---- ----The Body----

Foot Notes

Source Notes

(1) The Title: A title is the main heading written in capital shown at the top of thetable. It must explain the contents of the table and throw light on the table as wholedifferent parts of the heading can be separated by commas there are no full stop be used

in the little.


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(2) The Box Head (column captions): The vertical heading and subheading of thecolumn are called columns captions. The spaces were these column headings arewritten is called box head. Only the first letter of the box head is in capital letters andthe remaining words must be written in small letters.

(3) The Stub (row captions): The horizontal headings and sub heading of the row arecalled row captions and the space where these rows headings are written is called stub.

(4) The Body: It is the main part of the table which contains the numerical informationclassified with respect to row and column captions.

(5) Prefatory Notes: A statement given below the title and enclosed in brackets usuallydescribes the units of measurement is called prefatory notes.

(6) Foot Notes: It appears immediately below the body of the table providing thefurther additional explanation.

(7) Source Notes: The source notes is given at the end of the table indicating the sourcefrom when information has been taken. It includes the information about compilingagency, publication etc

General Rules of Tabulation:

A table should be simple and attractive. There should be no need of furtherexplanations (details).

Proper and clear headings for columns and rows should be need. Suitable approximation may be adopted and figures may be rounded off. The unit of measurement should be well defined. If the observations are large in number they can be broken into two or three

tables.

Thick lines should be used to separate the data under big classes and thin linesto separate the sub classes of data.

Difference Between Classification and Tabulation(1) First the data are classified and then they are presented in tables, the classificationand tabulation in fact goes together. So classification is the basis for tabulation.

(2) Tabulation is a mechanical function of classification because in tabulationclassified data are placed in row and columns.

(3) Classification is a process of statistical analysis where as tabulation is a process ofpresenting the data in suitable form.

Frequency Distribution

A frequency distribution is a tabular arrangement of data into classes according to thesize or magnitude along with corresponding class frequencies (the number of values fall


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in each class).

Ungrouped Data or Raw Data: Data which have not been arranged in a systemicorder is called ungrouped or raw data.

Grouped Data: Data presented in the form of frequency distribution is called groupeddata.

Array: The numerical raw data is arranged in ascending or descending order is calledan array.

Example: Array the following data in ascending or descending order 6, 4, 13, 7, 10, 16,19.Solution:

Array in ascending order is 4, 6, 7, 10, 13, 16, and 19Array in descending order id 19, 16, 13, 10, 7, 6, and 4

Class Limits:

The variant values of the classes or groups are called the class limits. The smaller valueof the class is called lower class limit and larger value of the class is called upper classlimit. Class limits are also called inclusive classes.For Example: Let us take the class 10 19, the smaller value 10 is lower class limitand larger value 19 is called upper class limit.

Class Boundaries:

The true values, which describe the actual class limits of a class, are called class

boundaries. The smaller true value is called the lower class boundary and the larger truevalue is called the upper class boundary of the class. It is important to note that theupper class boundary of a class coincides with the lower class boundary of the nextclass. Class boundaries are also known as exclusive classes.For Example:

Weights in Kg No of Students

60 65 8

65 70 12

70 75 5

25

A student whose weights are between 60kg and 64.5kg would be included in the 60 65 class. A student whose weight is 65kg would be included in next class 65 70.

Construction of Frequency Distribution

Following steps are involved in the construction of a frequency distribution.

(1) Find the range of the data: The range is the difference between the largest and the


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smallest values.

(2) Decide the approximate number of classes: Which the data are to be grouped.There are no hard and first rules for number of classes. Most of the cases we have 5 to

20 classes. H.A. Sturges has given a formula for determining the approximation numberof classes.

Where = Number of Classes

Where = Logarithm of the total number of observationsFor Example: If the total number of observations is 50, the number of classes would be

Or classes approximately.

(3) Determine the approximate class interval size: The size of class interval is

obtained by dividing the range of data by number of classes and denoted by class

interval sizeIn case of fractional results, the next higher whole number is taken as the size of theclass interval.

(4) Decide the starting point: The lower class limits or class boundary should coverthe smallest value in the raw data. It is a multiple of class interval.For Example: 0, 5, 10, 15, 20 etcare commonly used.

(5) Determine the remaining class limits (boundary): When the lowest classboundary of the lowest class has been decided, then by adding the class interval size tothe lower class boundary, compute the upper class boundary. The remaining lower andupper class limits may be determined by adding the class interval size repeatedly till the

largest value of the data is observed in the class.

(6) Distribute the data into respective classes: All the observations are marked intorespective classes by using Tally Bars (Tally Marks) methods which is suitable fortabulating the observations into respective classes. The number of tally bars is countedto get the frequency against each class. The frequency of all the classes is noted to getgrouped data or frequency distribution of the data. The total of the frequency columnsmust be equal to the number of observations.

Example Construction of Frequency Distribution


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Construct a frequency distribution with suitable class interval size of marksobtained by 50 students of a class are given below:23, 50, 38, 42, 63, 75, 12, 33, 26, 39, 35, 47, 43, 52, 56, 59, 64, 77, 15, 21, 51, 54, 72,68, 36, 65, 52, 60, 27, 34, 47, 48, 55, 58, 59, 62, 51, 48, 50, 41, 57, 65, 54, 43, 56, 44,30, 46, 67, 53

Solution: Arrange the marks in ascending order as12, 15, 21, 23, 26, 27, 30, 33, 34, 35, 36, 38, 39, 41, 42, 43, 43, 44, 46, 47, 47, 48, 48,50, 50, 51, 51, 52, 52, 53, 54, 54, 55, 56, 56, 57, 58, 59, 59, 60, 62, 63, 64, 65, 65, 67,68, 72, 75, 77

Minimum Value = 12 Maximum = 77Range = Maximum Value Minimum Value =77 12 = 65

Number of Classes =

=

== = or approximate

Class Interval Size ( ) = = = or

Marks

Class Limits

C.L

Tally

Marks

Number of

StudentsClass

Boundary

C.B

Class

Marks


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Note: For finding the class boundaries, we take half of the difference between lower

class limit of the 2nd class and upper class limit of the 1st class .

This value is subtracted from lower class limit and added in upper class limit to get therequired class boundaries.

Frequency Distribution by Exclusive Method

Class

Boundary

C.B

Tally

Marks

Frequency

Frequency Distribution of Discrete Data

Discrete data is generated by counting; each and every observation is exact.When an observation is repeated. It is counted the number for which the observationis repeated is called frequency of that observation. The class limits in discrete data aretrue class limit; there are no class boundaries in discrete data.

Example:

The following are the number of female employees in different branches ofcommercial banks. Make a frequency distribution.2, 4, 6, 1, 3, 5, 3, 7, 8, 6, 4, 7, 4, 4, 2, 1, 3, 6, 4, 2, 5, 7, 9, 1, 2, 10, 1, 8, 9, 2, 3, 1, 2, 3,

4, 4, 4, 6, 6, 5, 5, 4, 5, 8, 5, 4, 3, 3, 2, 5, 0, 5, 9, 9, 8, 10, 0, 4, 10, 10, 1, 1, 2, 2, 1, 8, 6,9, 10

Solution:

The involved variable is the number of female employees which is a discretevariable. The largest and smallest values of the given data are 10 and 0 respectively.

Number of Employees

(Classes)Tally

Marks

Branches

(Frequency)


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Cumulative Frequency Distribution

The total frequency of all classes less than the upper class boundary of a given class is called thecumulative frequency of that class. A table showing the cumulative frequencies is called acumulative frequency distribution. There are two types of cumulative frequency distributions.

Less than cumulative frequency distribution: It is obtained by adding successively thefrequencies of all the previous classes including the class against which it is written. Thecumulate is started from the lowest to the highest size.

More than cumulative frequency distribution: It is obtained by finding the cumulate total offrequencies starting from the highest to the lowest class. The less than cumulative frequencydistribution and more than cumulative frequency distribution for the frequency distribution given

below are:

Less than C.F More than C.F

Class

LimitC.B Marks C.F Marks C.F

Less than ormore

Less than ormore

Less than ormore

Less than ormore

Less than or

more


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Less than ormore

Less than ormore

Diagrams and Graphs of Statistical Data

We have discussed the techniques of classification and tabulation that help us in organizing thecollected data in a meaningful fashion. However, this way of presentation of statistical datadoes not always prove to be interesting to a layman. Too many figures are often confusing andfail to convey the massage effectively.

One of the most effective and interesting alternative way in which a statistical data may be

presented is through diagrams and graphs. There are several ways in which statistical data maybe displayed pictorially such as different types of graphs and diagrams. The commonly useddiagrams and graphs to be discussed in subsequent paragraphs are given as under:

Types of Diagrams/Charts:

1. Simple Bar Chart2. Multiple Bar Chart or Cluster Chart3. Staked Bar Chart or Sub-Divided Bar Chart or Component Bar Chart

Simple Component Bar Chart Percentage Component Bar Chart Sub-Divided Rectangular Bar Chart Pie Chart

Types of Diagrams/Charts:

1. Histogram2. Frequency Curve and Polygon3. Lorenz Curve

4. Historigram


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Simple Bar Chart

A simple bar chart is used to represents data involving only one variable classified on

spatial, quantitative or temporal basis. In simple bar chart, we make bars of equal width butvariable length, i.e. the magnitude of a quantity is represented by the height or length of the

bars. Following steps are undertaken in drawing a simple bar diagram:

Draw two perpendicular lines one horizontally and the other vertically at anappropriate place of the paper.

Take the basis of classification along horizontal line (X-axis) and the observedvariable along vertical line (Y-axis) or vice versa.

Marks signs of equal breath for each class and leave equal or not less than half breathin between two classes.

Finally marks the values of the given variable to prepare required bars.

Example: Draw simple bar diagram to represent the profits of a bank for 5 years.

Years

Profit

(million $)

Simple bar chart showing the profit of a bank for 5 years.

Multiple Bar Chart


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By multiple bars diagram two or more sets of inter-related data are represented (multiple bardiagram facilities comparison between more than one phenomena). The technique of simple

bar chart is used to draw this diagram but the difference is that we use different shades,colours, or dots to distinguish between different phenomena. We use to draw multiple barcharts if the total of different phenomena is meaningless.

Example: Draw a multiple bar chart to represent the import and export of Canada (values in $)for the years 1991 to 1995.

Years Imports Exports


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The required diagram is given below:

Percentage Component Bar Chart

Sub-divided bar chart may be drawn on percentage basis. To draw sub-divided bar

chart on percentage basis, we express each component as the percentage of its respectivetotal. In drawing percentage bar chart, bars of length equal to 100 for each class are drawn atfirst step and sub-divided in the proportion of the percentage of their component in the secondstep. The diagram so obtained is called percentage component bar chart or percentage staked

bar chart. This type of chart is useful to make comparison in components holding thedifference of total constant.

Example: The table below shows the quantity in hundred kgs of Wheat, Barley and Oatsproduced on a certain form during the years 1991 to 1994.

Years Wheat Barley Oats

Construct a percentage component bar chart to illustrate this data.

Solution:

Necessary computations for the construction of percentage bar chart given below:


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Item

cum cum cum cum

Wheat

Barley

Oats

Total

indicates Percentage of each item

Cum indicates the cumulative percentage.

Pie Chart


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Pie chart can used to compare the relation between the whole and its components.Pie chart is a circular diagram and the area of the sector of a circle is used in pie chart.Circles are drawn with radii proportional to the square root of the quantities because the

area of a circle is .To construct a pie chart (sector diagram), we draw a circle with radius (square root

of the total). The total angle of the circle is . The angles of each component arecalculated by the formula.

Angle of SectorThese angles are made in the circle by mean of a protractor to show different components.The arrangement of the sectors is usually anti-clock wise.

Example:

The following table gives the details of monthly budget of a family. Represent these

figures by a suitable diagram.

Item of Expenditure Family Budget

Food

Clothing

House Rent

Fuel and Lighting

Miscellaneous

Total

Solution:

The necessary computations are given below:

Angle of Sector

ItemsFamily Budget

Expenditure $ Angle of Sectors Cumulative Angle

Food

Clothing

House Rent

Fuel and Lighting

Miscellaneous


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Total


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Average and Types of Averages

Average:A single value which can represent the whole set of data is called an

average. If the average tends to lie or indicating the center of the distribution iscalled measure of central tendency or sometimes they locate the general position ofthe data, so they are also called measure of location.

Desirable Qualities of a Good Average:An average possesses all or most of the following qualities (characteristics) is

considered a good average:(1) It should be easy to calculate and simple to understand.

(2) It should be clearly defined by a mathematical formula.

(3) It should not be affected by extreme values.

(4) It should be based on all the observations.

(5) It should be capable of further mathematical treatment.

(6) It should have sample stability.

Types of Averages:Mathematical averages are:

(1) Arithmetic Mean(2) Geometric Mean(3) Harmonic Mean(4) Median(5) Mode

Arithmetic Mean


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It is the most commonly used average or measure of the central tendency applicable only in caseof quantitative data. Arithmetic mean is also simply called mean. Arithmetic mean is defined as:

Arithmeticmean is quotient of sum of the given values and number of the given values .

The arithmetic mean can be computed for both ungroup data (raw data: a data without anystatistical treatment) and grouped data (a data arranged in tabular form containing different groups).

If is the involved variable, then arithmetic mean of is abbreviated as of and

denoted by . The arithmetic mean of can be computed by any of the following methods.

Methods NameNature of Data

Ungrouped Data Grouped Data

Direct Method

Indirect orShort-Cut Method

Method ofStep-Deviation

Where

Indicates values of the variable .

Indicates number of values of .

Indicates frequency of different groups.

Indicates assumed mean.

Indicates deviation from i.e,

Step-deviation and Indicates common divisor

Indicates size of class or class interval in case of grouped data.

Summation or addition.

Example (1):

The one-sided train fare of five selected BS students is recorded as follows , , ,

and . Calculate arithmetic mean of the following data.Solution:

Let train fare is indicated by , then

Arithmetic mean of , we decide to use above-mentioned formula. Form

the given data, we have and . Placing these two quantities in aboveformula, we get the arithmetic mean for given data.


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Example (2): Given the following frequency distribution of first year students of a particular college.

Age (Years)

Number of Students

Solution: The given distribution belongs to a grouped data and the variable involved is ages of first yearstudents. While the number of students Represent frequencies.

Ages (Years) Number of Students

Total

Now we will find the Arithmetic Mean as years.

Example (3):

The following data shows distance covered by persons to perform their routine jobs.

Distance (Km)

Number of Persons

Solution: The given distribution belongs to a grouped data and the variable involved is ages of distancecovered. While the number of persons Represent frequencies.

Distance (Km)Number of Persons Mid Points


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Total

Now we will find the Arithmetic Mean as Km.

Examples of Arithmetic Mean

Example (4):

The following data shows distance covered by persons to perform their routine jobs.

Distance (Km)

Number of Persons

Calculate Arithmetic Mean by Step-Deviation Method; also explain why it is better than direct method in thisparticular case.Solution: The given distribution belongs to a grouped data and the variable involved is ages of distance covered.While the number of persons Represent frequencies.

Distance Coveredin (Km)

Number of Persons Mid Points

Total

Now we will find the Arithmetic Mean asWhere

, , and

Km

Explanation:

Here from the mid points ( ) it is very much clear that each mid point is multiple of and there is

also a gap of from mid point to mid point i.e. class size or interval ( ). Keeping in view this, we should

prefer to take method of Step-Deviation instead of Direct Method.


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Example (5):

The following frequency distribution showing the marks obtained by students in statistics at acertain college. Find the arithmetic mean using (1) Direct Method (2) Short-Cut Method (3) Step-Deviation.

Marks

Frequency

Solution:

DirectMethod

Short-CutMethod

Step-DeviationMethod

Marks

Total

(1) Direct Method:

or Marks(2) Short-Cut Method:

Where

Marks(3) Step-Deviation Method:

Where

Marks

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Meanings of Statistics