educational statistics new lecture1_1stsem2010-2011

8/7/2019 educational statistics new lecture1_1stsem2010-2011

1/85

EDUCATIONAL STATISTICS

Dr. Joseph Mercado

Special Lecturer

PUP GS


2/85

Research Process

Identification of the Research ProblemFormulation of HypothesesIdentification of Necessary DataData CollectionAnalysisSummarizing Results

Drawing Conclusions and ImplicationsExpanded, Revised, and New TheoryNew Knowledge


3/85

Research Problems

Classroom Leadership, Attitude Towards TeachingScience and other Correlates of Teaching Competenceof Science High School Teachers in the NCR

Principals Attributes : Their Effects on TeachersEmpowerment and Learners Achievement in PublicSchools in the NCR

Evaluation of the Quality of Research and ExtensionPrograms of Selected Higher Education Institutions inthe CALABARZON Region


4/85

Ethical Responsibility in the Work Behavior of SchoolManagers in Four selected Universities in the LuzonArea

An Assessment of the Expanded ROTC Program s andCivic Welfare Service and Its Implications to theNational Peace and Development Plan (NPDP)

Total Quality Management Practices in SelectedHigher Education Institutions in Metro Manila

Principal Empowerment in public Secondary Schools:Basis For The Development of a Primer

Prediction Models of School Based Management,Teaching Behavior and Professionalism


5/85

Factors Affecting the Results of the National AchievementTest (NAT) in Selected City Schools in the NCR

Teachers Beliefs Regarding The Implementation of Constructivism in the Classroom

Analysis of Leadership Behavior and Self-Efficacy of Principals of Catholic Secondary Schools

Study Habits and Academic Achievements of IntermediatePupils in Guadalupe Nuevo Elementary School in Makati:An Assessment

Teachers Behavior, Teachers Collective Efficacy, StudentsBehavior and Their Relationship to Students AcademicAchievement in Selected Public Secondary Schools inBulacan


6/85

Personal Needs, Job Satisfaction and Work-Related Characteristics as Correlates of TeacherCommitment Among Mathematics FacultyMembers of State Universities and Colleges inthe NCR

Teacher s Behavioral Pattern and TeachingStyles: Their Influence on Pupils AcademicAchievement in the District of Tanza

Leadership Styles of Administrators in ThreeSelected State Universities in Manila and TheirEffect on Faculty Performance


7/85

An Analysis of the Effectiveness of Field Study Coursesof the Revised Teacher Education Curriculum AmongSelected Higher Education Institutions (HEIs) in the

National Capital Region

An Evaluation of the International Training Program inthe Hospitality Education of Selected Higher EducationInstitutions (HEIs) in Metro Manila

Personological Attributes Affecting the ManagementStyles of Educational Managers in Selected Schoolsin the Province of Laguna

Relationship Between Some Selected Variables and

Conflict Management Styles of the Administrators of the Polytechnic University of the Philippines System

Quality Assessment of Student Development andServices Program in Local Colleges and Universities inMetro Manila


8/85

Nature of Statistics

Statistics may be defined as the science of artof collecting, presenting, analyzing andinterpreting data in a certain field, such aseducation, science, psychology, business,economics, engineering, medicine, or any otherarea.


9/85

Statistics can also be used in makingcorrect decisions during the time of uncertainty. One may apply the differentstatistical methods so as to arrive at thecorrect result with appropriate criticaljudgment.

Meaning of Statistics


10/85

Branches of Statistics

Descriptive Statistics the branch thatsummarizes and organizes raw data into ameaningful information.

To calculate the average score of your studentsor can summarize the percentages of the score.


11/85

Branches of Statistics

Inferential Statistics statistical inference isthe process of obtaining information about a

larger group from the study of a smaller group.The total group of people, things, or characteristics you are interested instudying, understanding, or predicting is called population .

A sample is a group of representative items chosen from the populationand used to predict the behavior or characteristics of the total populationwith help of inferential statistics.


12/85

The Statistical Treatment

The design of the studydeterminesdetermines what statisticaltechniques should be employed,not vice versa.


13/85


The kind of statistical treatmentthat can be done in a study is

mainly constrainedconstrained by the level(or scale) of data measurementemployed in the study.


14/85


Scales of Data Measurement

Nominal scale


15/85



1. Nominal scale2. Ordinal Scale


16/85



1. Nominal scale2. Ordinal Scale3. Interval and Ratio Scale


17/85


Scale of Data MeasurementNominal data (the least sophisticated):

data assumes no natural ordering;

largely allied to measuring qualitativecharacteristics such as eye colo r, hair colo r, g en d e r, nati on a l it y o r even l if es t ylegr oups , i.e ., si ng les , youn g marri e d,r e tir e d.


18/85

The Statistical TreatmentScale of Data Measurement

Nominal data: Example

Eye Color Number of Men %

Blue 60 30Brown 80 40Green 30 15Grey 20 10Hazel 10 5

TOTAL 200 100


19/85


20/85


Ordinal data: Example

Rail travelers might be asked to give theirviews on the quality of the MRT service

according to a scale of 1 5 where:1 = very poor2 = poor3 = adequate

4 = good5 = very good


21/85


22/85


Interval and Ratio Data

(the most sophisticated)data where values progress bothin order and according to a seriesof equal steps.


23/85


Interval and Ratio DataExamples:

number of babies born to different

familiesnumber of people pre-purchasingtheater tickets at a particular venueover different periodsage, weight and height of a sample of human beings


24/85


Statistical procedure is selected on thebasis of its appropriateness for

answering the question involved in thestudy.Nothing is gained by using a complicatedprocedure when a simple one will do just aswell. Statistics are to serve research, not todominate it.


25/85


Interval and Ratio Data

Examples:number of babies born to differentfamiliesnumber of people pre-purchasingtheater tickets at a particular venue

over different periodsage, weight and height of a sample of human beings


26/85

Sources of Data

1. S econ dar y Data : Data which arealready available. An example:statistical abstract of USA.

Advantage : less expensive.Disadvantage : may not satisfyyour needs.2. P rimar y Data : Data which must becollected.


27/85

P reliminary Steps in Statistical Study

Define the problemDetermine the population/subject of

the studyDevise the set of questionsDetermine the sampling design


28/85

G uidelines in the Selection of a Research P roblem

or Topic

The research problem must be chosen by theresearcher himself so that he will not make

excuses for all the obstacles he will encounter.The problem must be within the interest of the

researcher so that he will give all the time and

effort in the research work.


29/85

G uidelines in the Selection of a Research P roblemor Topic

The problem must be within the specialization of the researcher. It will make the work easier for theresearcher because he is familiar in the area and it

will help him improve his specialization, skill andcompetence in his own area.The research problem must be within thecompetence of the researcher. The researcher must

know the procedures in making research and howto apply them. He must have a workableunderstanding of his study.


30/85


The researcher must have the ability and capacityto finance the research problem to make sure thatthe study will be completed on the target time.

The research problem must be manageable. Thedata must be available or within the capacity of theresearcher to gather data. The data must beaccurate, objective and not biased. The data should

help the researcher answer the question beinginvestigated.


31/85


The research problem must be completedwithin the period set by the researcher.

The research problem must be significant,important and relevant to the present time aswell as to the future. This means that theresearch problem must have an impact to the

situation and people it is intended for.


32/85


The results of the study must be practical andimplementable.

The study must contribute to the humanknowledge. The facts and knowledge must be aproduct of research.


33/85

Different Ways to Draw a Sample

(2 Types of Sampling)

1 . Random Sampling or Probability Sampling

In probability sampling, the sample is aproportion of the population and such sample isselected from the population by means of systematic way in which every element of thepopulation has a chance of being included in thesample.

2 . Non-Random Sampling or Non-Probability Sampling

In a non-probability sampling, the sample is nota proportion of the population and there isno system in selecting the sample. The selectiondepends on the situation.


34/85


35/85

b. Systematic Random Sampling - Select some startingpoint and then select every K th element in thepopulation

Random Sampling or Probability Sampling


36/85

c. Stratified Sampling - subdivide the population intosubgroups that share the same characteristic, thendraw a sample from each stratum



37/85

d. Cluster Sampling - divide the population into sections(or clusters); randomly select some of those clusters;choose all members from selected clusters



38/85


39/85

N on-Random Sampling or N on-ProbabilitySampling

1 . Accidental SamplingIn this type of sampling, there is no system of selection butonly those whom the researcher or interviewer meets bychance are included in the sample. This type of samplinglacks representativeness where the sample may be biased. If the interviewer goes to a business section, most people whowill be interviewed are likely from the business and probablyrich people hence the respondents will be from well-to-dopeople. But if the interviewer stays in a slum area, then it ispossible that the respondents are poor people. In a research,

every section of the population must be equally representedin the sample. This method is being used when there is noalternative.


40/85

2 . Quota SamplingIn this type of sampling, specified number of personsof certain types is included in the sample. Supposethe reactions of the people for a particular issue, suchas the effects of drug addiction in a certain locality,can be decided from a sample that constitutes 10doctors, 9 lawmakers, 15 parents and 20 drugaddicts.In quota sampling, many sectors of the populationare represented. However, the representation is

doubtful because there is no proportionalrepresentation since there are no guidelines in theselection of the respondents. Anyone who is selectedto participate will do. Quota sampling may be usedonly when any of the more desirable types of

sampling will not do.

N on-Random Sampling or N on-Probability Sampling


41/85

N on-Random Sampling or N on-Probability Sampling

3 . Convenience SamplingConvenience sampling is a process of picking out peoplein the most convenient and fastest way to get reactionsimmediately. This method can be done by telephoneinterview to get the immediate reactions of a certaingroup of sample for a certain issue.

This kind of method is biased and not representative.This is quite different from gathering data by interviewwhereby the interview can be done through the

telephone. In the interview method, people who areinterviewed through the telephone are properlyselected to be included in the sample.


42/85


43/85

Sample Size Determination

The extent of the population will depend on the nature of theproblem. The census survey will require all individual in the populationthat is considered while the sample survey will consider a fewrepresentative of the population.

In determining the sample size, the formula which can be applied is asfollows: (Slovins Formula)

n = sample size

N = population sizee = desired margin for error

(per cent allowance for non-precision because a sample is used)

21 NeN

n u


44/85

Example 1A researcher wants to make use of a student

population of 3 , 000 for his study in the mathematicalachievement test. If he allows a 5% margin of error,how many students must he take for his sample?

Solution:The formula given can be used:

21 N eN n u

2)05.0(300013000

un

)0025.0(300013000

un

5.713000

un

5.83000

un

35394.352 or n u


45/85

DESCRIPTIVE STATISTICS

Summarizes or describes the important characteristicsof a known set of data.

Measures of Central Tendency

A measure of tendency for a collection of datavalues is number that is meant to convey the idea ofcentralness for the data set.

Numerical values that are indicative of the centralpoint or the greatest frequency concerning a set ofdata. The most common measures of centraltendency are the mean, median, and mode.


46/85

Measures of Central Tendency and Scales ofmeasurement

T he mode requires only nominal data - and you cancompute it for ordinal, interval, and ratio.

T he Median requires ordinal data - and you can compute itfor interval and ratio. You cannot compute the Median for nominal data.

T he Mean requires interval or ratio data - you cannotcompute it for either nominal or ordinal data.


47/85

Example

1. What is the mean for the following samplevalues? 3, 8, 6, 14, 0, -4, 0, 12, -7, 0, -10.

Solution:

211

)10(0)7(120)4(014683!!

X


48/85


49/85

2. Find the median for the ages of the following

eight college students:23 19 32 25 26 22 24 20

Solution: First order the values, The orderedarray is

19 20 22 23 24 25 26 32

median = (23 + 24)/2 = 23.5


50/85

What is the mode for the following sample

values?3 5 1 4 2 9 6 10The data set has no mode

What is the mode for the following samplevalues?

3 5 1 4 2 9 6 10 5

3 4 3 9 3 6 1The value of the is mode is 3,Unimodal


51/85

What is the mode for the following sample values?6 10 5 3 4 3 9 3 6 1 6

The values of the mode are 3 and 6, bimodal


52/85

Mean: Grouped DataT he calculation of mean from a frequencydistribution is almost the same as that from anungrouped data (raw data), only in a distribution, theindividual values are not known. When the number of items is too large, it is best to compute for themeasures of central tendency using a frequencydistribution.

n

fX n

ii

X

!! 1


53/85

Ex ample:

Calculation of Mean from Frequency Distribution of Sample Test Scores of 40 Students in E ducational Statistics

ifX

Scores f Xi fXi

70 74 2 72 144

65 69 2 67 134

60 64 3 62 186

55 59 2 57 114

50 54 8 52 41645 49 9 47 423

40 44 2 42 84

35 39 4 37 148

30 34 5 32 160

25 29 3 27 81

n = 40 = 1,890

25.4740

18901 !!!

!

n

f X n

ii

X


54/85

Median: Grouped Data

if

F n

LMdnm

m

m

!12

Formula:

Where:Mdn = median

lower class boundary of the median class= less than cumulative frequency of the class immediately preceding the

median classi = the size of the intervaln = the total number of scores

!mL

1mF


55/85

Scores f


56/85

Mode: Grouped Data

id d

d LM

m oo

!

21

1

Formula:

Where:lower class boundary of the modal class

= numerical difference between the frequency of the modal classand the frequency of the adjacent lower class

= numerical difference between the frequency of the modal classand the frequency of the adjacent higher class

i = size of the class interval

!m oL

1d

2d

Ex ample:


57/85

Scores f

70 74 2

65 69 2

60 64 3

55 59 2

50 54 8

45 49 9

40 44 235 39 4

30 34 5

25 29 3

n = 40

Ex ample:Calculation of Mode from Frequency Distribution of Sample Test Scores of 40Students in E ducational Statistics

88.48517

75.44

21

1 !

!

! i

d d

d LM

m oo

P roperties of the different central tendency measures:


58/85

P roperties of the different central tendency measures:

1. T he mean is the standard measure of central tendency in statistics. Itis most frequently used.

2. T he mean is not necessarily equal to any score in the data set

3. T he mean is the most s table measure from s ample to s ample .

4.T

he mean is very influenced by Outlier s

--T

hat is, the mean will bestrongly influenced by the presence of e x treme scores.

5. T he median is not s en s itive to outlier s .

6. T he mean is based on all scores from the sample but the mode and

the median are not.

7. T he M ode is the lea s t s table measure from sample to sample.

8. T he median is the best measure of central tendency if the distributionis skewed.


59/85

Measures of P ositionT he Quartiles

Quartiles are the points which divide the total number of scores into four equal parts. E ach set of scores has three quartiles. T wenty-five percentfalls below the first quartile (Q 1), fifty percent is below the second quartile(Q 2), and seventy-five percent is below the third quartile (Q 3).

T he steps in finding the quartiles of raw scores can be summarized asfollows:

1. Arrange the scores from highest to lowest or lowest to highest.

2. Determine Q k, where Q k is the k th quartile and k = 1, 2, 3

2.1 If is an integer, scores

2.2 If is not an integer, Q k = ith score where I is the closestinteger greater than

4nk

2

14

thth

k

nk k n

Q

!

4nk

4nk


60/85

Ex ample

Calculation of Quartiles from Sample Raw Scores of E ight Students inE ducational Statistics and Nine Students in Applied Statistics

Edu catio nal Statistics Applie d Statistics

17 15

17 19

26 20

28 24

30 28

30 30

31 32

37 32

40

5.212

26172

3224

1811 !!!p!! scor esQQ

th


61/85

5.30

23130

276

64

3833 !!!p!! scor esQQ

th

For Applied Statistics:

20325.2

4

1911

!!p!! scor eQQ r d

32775.6

439

33 !!p!! scor eQQth


62/85

For Grouped Data:

Formula: First Quartile

if

F n

LQQ

m

Q

!1

1

1

1 4

1Q

1QL 1Q

1mF 4n

1Qf 1Q

Where:

= the First Quartile

= lower class boundary where lies

= less than cumulative frequency approaching or equal to but not e x ceeding

= the frequency where

i = the size of the interval

n = the total number of scores

lies


63/85

For Grouped Data:Formula: T hird Quartile

if

F n

LQQ

mQ

!3

3

13

4

3

3Q

13QL

3Q

1mF

43n

3Qf 3Q

Where:

= the T hird Quartile

= lower class boundary where lies

= less than cumulative frequency approaching or equal to but not e x ceeding

= the frequency where

I = the size of the interval

n = the total number of scores

lies


64/85

Scores f


65/85

3754

8105.344

1

1

1

1!

!!

! i

f

F n

LQQ

m

Q

88.53582330

5.494

3

3

3

1

3 !

!

! if

F n

LQ Q

m

Q


66/85

Measures of Variability

A measure of variability for a collection of datavalues is a number that is meant to convey theidea of spread for the data set. The mostcommonly used measures of variability forsample data are the range, the interquartilerange, the mean absolute deviation, the

variance or standard deviation, and thecoefficient of variation.


67/85


68/85


69/85

Mean Absolute Deviation isthe average of the absolutedeviation values from themean.

Mean Absolute Deviationutilizes deviations of thedata values from the meanin its computation.

nxMAD x! //


70/85

What is the MAD for the following sample values?

3 8 6 12 0 -4 10

Data Values, x Absolute Deviations /x-mean/

3 2

8 36 1

12 7

0 5

-4 9

10 5


71/85

The average (absolute) distance of the samplevalues from the mean.

n

xMAD x! //

57.47

32 !!MAD


72/85

The Variance and Standard Deviation

The Variance and Standard Deviation are the most commonand useful measures of variability. These two measuresprovide information about how the data vary about the mean.

If the data are clustered around the mean, then the varianceand standard deviation will somewhat small.

There is small variability when the data values are clusteredabout the mean


73/85

Sample Variance

The formula says that you subtract the mean

from each data value and square thedifferences, then you add these values anddivide by the sample size minus 1.

1

2( )2 ! n

x xS

Do not let the formula frighten you. We will build a table to helpcompute the variance.


74/85

What is the variance for the following sample

values?3 8 6 14 0 11

Solution: First of all, we need to compute the

sample mean:

7642

611014683 !!!X


75/85

Table Used in Helping to Compute the SampleVariance

Data Deviations (x-mean) Squared Deviations(x-mean)

3 3 7 = -4 16

8 8 7 = -1 16 6 7 = -1 1

14 14 7 = 7 49

0 0 7 = -7 49

11 11 7 = -4 16

Total 0 132


76/85

The sample variance is

1

2( )2 ! n

x xS

4.26

5

132

16

1322 !!!S


77/85

Wh i h d d d i i f h f ll i l


78/85

What is the standard deviation for the following samplevalues?

3 8 6 14 0 11

Solution:

1

2( )!

n

xS x

14.54.265

132 !!!S


79/85

Coefficient of Variation

The Coefficient of Variation (CV) allows us to compare thevariation of two (or more) different variables.The sample coefficient of variation is defined as samplestandard deviation divided by the sample mean of the dataset. Usually, the result expressed as a percentage .

%100x

S

CV X !


80/85

The sample coefficient of variation standardizes the variation bydividing it by the sample mean. The CV has no units, since thestandard deviation and the mean have the same units, and thus theunits cancel each other. Because of this property, we can use thismeasure to compare variations for different variables with differentunits.

We said that standard deviation measures the variation in a set of

data. For distributions having the same mean, the distribution withthe largest standard deviation has the greatest variation. But whenconsidering distributions with different means, decision makerscan't compare the uncertainty in distribution only by comparingstandard deviations. In this case, the coefficient of variation is used,i.e., the coefficients of variation for different distributions arecompared, and the distribution with the largest coefficient of variation value has the greatest relative variation.


81/85


82/85

%56.5%100905

)( !! xticket sCV

%35.14

%100400,5

775)(

!!x

r evenue sCV

Since the CV is larger for the revenues, there is

more variability in the recorded revenues than in thenumber of tickets issued.


83/85

For e x ample, Mark teaches two sections of statistics. He gives eachsection a different test covering the same material. T he mean score onthe test for the day section is 27, with a standard deviation of 3.4. T hemean score for the night section is 94 with a standard deviation of 8.0.Which section has the greatest variation or dispersion of scores?

D ay Section.................... N ight Section

M ean .......27.......................94S. D ...........3.4.....................8.0

Direct comparison of the twostandard deviations

shows that the nightsection has the greatest variation. But comparing the coefficient of variations show quite different results:

C.V.(day) = (3.4/27) x 100 = 12.6% and C.V.(night) = (8/94) x 100 = 8.5%

Example 2

E mpirical Rule:


84/85

E mpirical Rule:

T his rule generally applies to mound-shaped data, but specifically to thedata that are normally distributed, i.e., bell shaped. T he rule is as follows:

Appro x imately 68% of the measurements (data) will fall within onestandard deviation of the mean (One-Sigma Rule), 95% fall within twostandard deviations ( T wo-Sigma Rule), and 99.7% (or almost 100% ) fallwithin three standard deviations ( T hree-Sigma Rule). See the following

figure:


85/85

For e x ample, in the height problem, the mean height was 70 inches with astandard deviation of 3.4 inches. T hus, 68% of the heights fall between 66.6and 73.4 inches, one standard deviation, i.e., (mean + 1 standard deviation) =

(70 + 3.4) = 73.4, and (mean - 1 standard deviation) = 66.6. Ninety fivepercent (95%) of the heights fall between 63.2 and 76.8 inches, two standarddeviations. Ninety nine and seven tenths percent (99.7%) fall between 59.8and 80.2 inches, three standard deviations. See the following figure:

Documents

educational statistics new lecture1_1stsem2010-2011