Introduction to

Parametric and
Non-Parametric
Statistics

BLOCK 2
PARAMETRIC STATISTICS

43
Inferential
Statistics: An
Introduction

44
 Test of
UNIT 3 TEST OF SIGNIFICANCE OF Significance of
Difference
DIFFERENCE BETWEEN TWO Between Two

MEANS* Means

Structure
3.0 Objectives
3.1 Introduction
3.2 Need and Importance of the Significance of the Difference between
Means
3.3 Fundamental Concepts in Determining the Significance of the
Difference between Means
3.3.1 Null Hypothesis
3.3.2 Standard Error
3.3.3 Degrees of Freedom (df)
3.3.4 Level of Significance
3.3.5 Two Tailed and One Tailed Tests of Significance
3.3.6 Errors in Testing
3.4 Methods to Test the Significance of Difference between the Means of
Two Independent Groups t-test
3.4.1 Testing Significance of Difference between Uncorrelated or Independent Means
3.5 Significance of the Difference Between two Correlated Means
3.5.1 The Single Group Method
3.5.2 Difference Method
3.5.3 The Method of Equivalent Groups
3.5.4 Matching by Pairs
3.5.5 Groups Matched for Mean and Standard Deviation
3.6 Let Us Sum Up
3.7 References
3.8 Key Words
3.9 Answers to Check Your Progress
3.10 Unit End Questions

3.0 OBJECTIVES
After reading this unit, you will be able to:
 discuss the need and importance of the significance of the difference
between means;

*
Dr. Vivek Belhekar, Faculty, Department of Psychology, Bombay University, Mumbai
45
Parametric
Statistics
 explain what is null hypothesis, standard error, degrees of freedom, level
of significance, two tailed and one tailed test of significance, type I error
and type II error; and
 calculate the significance of difference between mean ( t-test) when
groups are independent, when there are correlated groups, groups
matched by pair and groups matched by mean and standarddeviation.

3.1 INTRODUCTION
In psychology some times we are interested in research questions like Do the
AIDS patients who are given the drug AZT show higher T-4 blood cell
counts than patients who are not given that drug? Is the error rate of typist the
same when work is done in a noisy environment as in a quiet one? Whether
lecture method of teaching is more effective than discussion method?
Consider the question whether lecture method of teaching is more effective
than discussion method. For this investigator divides the class in to two
groups. One group is taught by lecture method and other by discussion
method. After a few months researchers administer an achievement test for
both the groups and find out the mean achievement scores of the two groups
say, M1 and M2. The difference between these two mean is then calculated.
Now the questions is whether the difference is a valid difference or it is
because of sampling fluctuation or error of sampling. Whether this difference
is significant or not significant. Whether on the basis of this difference, could
we conclude that one method of teaching is more effective than the other
method.
These questions can be answered by the statistical measures which we are
going to discuss in this unit. To test the significance of difference between
mean we can use either the t-test or z test. When the sample size is large, we
employ z test and when sample is small, then we use the t test. In this unit we
are concerned with t test. We will get acquainted with the various concepts
related, to computation and description of t test.

3.2 NEED AND IMPORTANCE OF THE

SIGNIFICANCE OF THE DIFFERENCE
BETWEEN MEANS
In psychology sometimes we are interested in knowing about the significance
of the differences between populations. For example we are interested to
discover whether ten year old boys and girls differ in their linguistic ability.
Or we want to find out if children from high SES (Socio Economic Status)
perform and score better academically than children from low SES. We may
also try to find out at times, if two groups of persons coming from different
background differ in their agility factor. Thus, many questions are asked and
to be answered in psychology for which one of the measures we use is the
Mean.

46
Let us take the first question on linguistic ability of boys and girls. First we Test of
Significance of
randomly select a large sample of boys and girls (large sample means the Difference
group comprises of 30 or more than 30 persons.). Then we administer a Between Two
Means
battery of verbal test to measure the linguistic ability of the two groups and
compute the mean scores on linguistic ability test of the two groups. Let us
say the obtained mean scores for boys and girls are M1and M2 respectively.
Now we try to find the difference between the two means. If we get a large
difference (M1 – M2) in favour of the girls then we can confidently say that
girls of 10 years of age are significantly more able linguistically than 10
years old boys. On the contrary if we find small difference between two
means then we would conclude that ten years old girls and boys do not differ
in linguistic ability.

An obtained mean is influenced by sampling fluctuation or error of sampling

and whatever differences are obtained in the means, it may be due to this
sampling error. Even mean of population 1 and mean of the population 2 may
be the same but because of sampling error we may find the difference in the
range of the two samples drawn from the two populations. In order to test the
significance of an obtained difference we must first have a standard error
(SE) of the difference. Then from the difference between the sample mean
and standard error of difference we can say whether the difference is
significant or not. Now the question arises what do we mean by significant
difference? According to Garrett (1981) a difference is called significant
when the probability is high and that it cannot be attributed to chance that is
(temporary and accidental factors) and hence represent a true difference
between population mean.

A difference is non significant when it appears reasonably certain that it

could easily have arisen from sampling fluctuation and hence imply no real
or true differences between the population means.

Check Your Progress I

1) When is a difference called significant?

…………………………………………………………………………….
…………………………………………………………………………….
…………………………………………………………………………….
…………………………………………………………………………….
…………………………………………………………………………….

3.3 FUNDAMENTAL CONCEPTS IN

DETERMINING THE SIGNIFICANCE OF THE
DIFFERENCE BETWEENMEANS
Let us discuss some of the fundamental concepts in determining the
significance of the difference between means. 47
Parametric
Statistics
3.3.1 Null Hypothesis
This is a useful tool in testing the significance of differences. Null hypothesis
asserts that there is no true difference between the two population means, and
the difference found between the sample mean is therefore, accidental or
unimportant (Garrett 1981). In the course of a study or an experiment, the
null hypothesis is stated so that it can be tested for possible rejection. For
example , to study the significant difference in linguistic ability of 8 years old
girls and boys we select random sample of girls and boys and administer a
battery of verbal test, compute the means of the two groups. In this study the
null hypothesis may be stated thus: There exists no significant difference
between the linguistic ability of boys and girls. If this null hypothesis is
rejected then we can say one group is superior to the other.

3.3.2 Standard Error

The primary objective of statistical inference is to make generalisation from a
sample to some population of which the sample is part. Standard error
measures (1) error of sampling and (2) error of measurement. Standard error
can be described as the standard deviation (SD) of all the standard deviations
that have been computed for given number of samples randomly drawn from
same population. Let us try to understand standard error with the help of an
example. Suppose a researcher wants to carry out a research on the
organisational citizenship behaviour of the employees in public sector banks.
Since it in not possible to take the whole population, the researcher randomly
takes around 1% of the whole population as the sample. Yet another
researcher is also carrying out similar study and also draws 1% from the
population. Thus, if many more researchers were carrying out such a study
then they would also draw 1% sample from the whole population. The
problem, though occurs when the 1% sample that is drawn by each researcher
is different from each other. And though each of the 1% sample is
representation of the population (that is heterogeneous in nature), they are
different from each other. The researchers will also compute means and
standard deviations for their respective sample. And in such a case, it is
expected that the means and standard deviations would be same, because the
sample has been randomly drawn from the same population. But in reality
that may not happen and the mean of one sample may be lower or higher than
the mean computed for another sample. The higher the standard error the less
the likelihood that the sample is representative of the population. Thus, the
difference between the sample needs to be close to zero so as to be sure that
they represent the population.

The standard error of the mean can be calculated by the following formula:
SEm or σm = σ/√N
Where
σ= The standard deviation of the sample mean
48 N= The number of cases in the sample.
If the standard error of measurement is large it shows considerable sampling Test of
error. Significance of
Difference
Between Two
3.3.3 Degrees of Freedom (df) Means

When a statistics is used to estimate a parameter the number of degrees of

freedom (df) available depends upon the restriction placed upon the
observations. One df is lost for each restriction imposed. For example we
have five scores as 5,6,7,8 and 9 the mean is 7 and deviation of our scores
from 7 are -2, -1, 0, 1 and 2. The sum of these deviations is zero. In
consequence if any four deviations are known the remaining deviation may
be automatically determined. In this way, out of the five deviations, only four
(N-1) are free to vary as, the condition that the sum equals to zero impose
restriction upon the independence of the 5thdeviation. Originally there were
5(N=5) degrees of freedom in computing the mean because all the
observation or scores were independent. But as we made use of the mean for
computing standard deviation we lost one degree of freedom.

Degrees of freedom varies with the nature of the population and the
restriction imposed. For example in the case of value calculated between
means of two independent variables, where we need to compute deviation
from two means, the number of restrictions imposed goes up to two
consequently df is (N- 1+N-2).

3.3.4 Level of Significance

Whether a difference between the means is to be taken as statistically
significant or not depends upon the probability that the given difference
could have arisen “by chance”. The researcher has to take a decision about
the level of significance at which he will test his hypothesis.

In social sciences .05 and .01 level of significance are most often used. When
we decide to use .05 or 5% level of significance for rejecting a null
hypothesis it means that the chances are 95 out of 100 that is not true and
only 5 chances out of 100 the difference is a true difference.

In certain types of data, the researcher may prefer to make it more exact and
use .01 or 1% level of significance. If hypothesis is rejected at this level it
shows the chances are 99 out of 100 that the hypothesis is not true and only 1
chance out of 100 it is true. The level of significance which the researcher
will accept should be set by researcher before collecting the data.

3.3.5 Two Tailed and One Tailed Tests of Significance

In many situations we are interested in finding the difference between
obtained mean and the population mean. Our null hypothesis states that the
M1 and M2 do not vary and the difference between them is zero. (that
is,Ho:M1- M2=0). Whether this difference is positive or negative we are not
interested in the direction of such a difference. All that we are interested is
whether there is a difference. For example we hypothesised that two groups 49
Parametric
Statistics
will differ from each other we don‟t know which group will have higher
mean scores and which group lower. This a non directional hypothesis and it
gives rise to a two-tailed hypotheses test. In other words the difference may
be in either direction and thus is said to be non directional.

In many experiments our primary concern is with the direction of the

difference rather than with its existence in absolute term. For example if we
are interested to determine the gain in vocabulary resulting from additions by
weekly reading assignment. Here we are interested in finding out the gain in
vocabulary. To take another example, if we say that training in yoga will
reduce the degree of tension in persons, then we are clearly stating that there
will be a reduction in the tension. In cases like this we make use of the one
tailed or non directional test to test the significance of difference between the
means.

3.3.6 Errors in Testing

If the null hypothesis is true and we retain it or if it is false and we reject it,
we had made a correct decision. But sometimes we make errors. There are
two types of errors Type I error and Type II error.

A type I error, also known as alpha error, is committed when null hypothesis
(Ho) is rejected when in fact it is true. A type II error, also known as beta
error, is committed when null hypothesis is retained and in fact it is not true.
For example suppose that the difference between two population means (μ1-
μ2) is actually zero, and if our test of significance when applied to the sample
mean shows that the difference in population mean is significant we make a
type I error. On the other hand if there is true difference between two
population mean, and our test of significance based on the sample mean
shows that the difference in population mean is “ not significant‟ we commit
a type II error.

Check Your Progress II

1) Given below are some statements. Indicate whether the statement is true
or false:

Sr. Statements True or False

No.

1 We commit a type I error when we reject a

null hypothesis when it is really true.

2 In testing a hypothesis, one can make three

types of error.

3 An exercise in hypothesis testing enables us

to draw conclusions about the estimated
parameters.
50
 Test of
4 For a given level of significance, we find Significance of
that as the sample size increases, the critical Difference
Between Two
values of t get closer to zero. Means

5 If the standardised sample mean exceeds

the critical clue, we should accept Ho.

3.4 METHODS TO TEST THE SIGNIFICANCE of

DIFFERENCE BETWEEN THE MEANS OF
TWO INDEPENDENT GROUPS T-TEST
3.4.1 Testing Significance of Difference between
Uncorrelated or Independent Means
The step used to find out significance of differences between independent
mean are as below:

Step 1:Computation of the mean.

Step 2: Computation of the combined standard deviation by using the

following formula:
x1=X1-M1 (deviation of scores of first sample from its mean)
x2=X2-M2 (deviation of scores of second sample from its mean)
SD = √ (Σx12+ Σx22)/(N1-1)+(N2-1)
Step 3: Computation of the standard error of the difference between two
means by using following formula:
SED= SD / √(1/N1+1/N2)
Step 4:To Compute the t value for the difference in two independent sample
mean. The following formula is used to determine t value is t = (M1-M2)-
0/SED
Step 5: Find out the degree to freedom. The (df) degree of freedom is
calculated using the following formula
df=(N1-1)+(N2-1)
Step 6: We then refer to table of t distributions (can be found in any Statistics
book) with the calculated degree of freedom df and read the t value given
under column 0.05 and 0.01of two tailed test. If our computed t value is equal
or greater than the critical t value given in table then we can say that t is
significant. If the computed value is lesser than given value then we will say
that it is non-significant.
Let us illustrate the whole process with the help of an example
Example 1:An interest test was administered to 6 boys and 10 girls. They
obtained following scores, is the mean difference between two groups
significant ? 51
Parametric
Statistics Table 3.1: Scores of boys and girls and the t value calculation
Scores x x² Scores y y2
obtained by obtained by
boys girls
X (Y)
28 -2 4 20 -4 16
35 5 25 16 -8 64
32 2 4 25 1 1
24 -6 36 34 10 100
36 -6 36 20 -4 16
35 5 25 28 4 16
31 7 49
24 0 0
27 3 9
15 -9 81
∑X=190 ∑x²= 130 ∑Y=240 ∑y² =352

Calculations
M = ΣX/ N = 190/ 6 = 31.66
X M = ΣY/ N = 240/ 10 = 24
Y

SD= √ [( ∑x²+∑y²)/(N1-1)+(N2-1)]
SD = √ [( 110+352)/(6-1)+(10-1)]
= √ 462/14 = √ 33 = 5.74
SE = SD [√( (N1+N2)/(N1N2)]
d

= 5.74 √ [ (16)/(60)] = 5.74 × .52 = 2.98

t = (M1-M2)-0/SED
= (31.66-24)-0/2.98 = 2.58
df=(N1-1)+(N2-1) =(6-1)+(10-1) = 14

Referring to the critical value in table for t test (that are given in the appendix
of a textbook on statistics) for df= 14, we get 2.58 at the .05 and 2.58 at
the.01 level, since our t is less than 2.14, therefore we will say that the mean
difference between boys and girls is non significant.

Let us take another example,

Example 2:On an academic achievement test 31 girls and 42 boys obtained

the following scores.

Table 3.2: Mean scores and Standard deviation

Mean SD N df
Boys 40.39 8.69 31 30
Girls 35.81 8.33 42 41
52
Is the mean difference in favour of boys and significant at the .05 level. Test of
Significance of
First we will compute the pooled standard deviation by the following formula Difference
Between Two
SD = √ [(SD1)2× (N1-1) + (SD2)2× (N2-1)]/(N1-1)+(N2-1) Means

SD1= Standard Deviation of group 1 i.e. 8.69

N1= Number of subject is group 1 i.e. 31
SD2= Standard devotion of groups 2 i.e. 8.33
N2= Number of subject in group 2 i.e. 42
SD = √[(8.69) × (31-1)+(8.33) × (42-1)/(31-1)+(42-1)] = 8.48
2 2

SE = SD [√(N1+N2)/(N1xN2)]
d

= 8.48 [√(31+42)/(31X42)] = 2.03

t= (M1-M2)-0/SED
= (40.39-35.81)-0/2.03
= 2.26
df=(N1-1)+(N2-1)
=(31-1)+(42-1)
=71

Referring to the critical value in table for t test (that are given in the appendix
of a textbook on statistics) for df= 71, we find t entries of 2.00 at the 0.05
level and 2.65 at the 0.01 level. The obtained t of 2.26 is significant at .05
level but not at the 0.01 level. We may say boys academic achievement is
better in comparison to that girls.

3.5 SIGNIFICANCE OF THE DIFFERENCE

BETWEEN TWO CORRELATED MEANS
3.5.1 The Single Group Method
In the previous section we discussed the problem of determining the
significance of difference between mean obtained by two independent groups
of boys and girls.

Some time we have single group and we administer the same test twice. For
example if we are intending to find out the effect of training on the students‟
educational achievement, first we take the measures of participants’
educational achievement before training , then we introduce the training
programme, and again we take the measure of educational achievement. In
this we have single group and administer educational achievement test twice.
Such type of design is known as single group design. In order to get the
significance of the difference between the means obtained in the before
training and after training we use the following method.

SED or σD = [σ2M1+ σ2M2- 2r σM1 σM2]

Where
53
Parametric
Statistics
σM1= Standard error of the initials test σM2=Standard error of the finals test
r= coefficient of correlation between scores on initial test and finaltest
t=(M1-M2)-0/σD
Let us illustrate the above formula will the help of following example.

Example: At the beginning of the session an educational achievement test in

maths was given to 100 IX grade students. Their mean was 55.4 and SD was
7.2. After six months an equivalent from of the same test was given and the
mean was 56.9 and SD was 8.0. The correlation between scores made on the
first testing and second testing was .64. Has the class made significant
progress in maths during six month? We may tabulate our data in the
following manner:

Table 3.3: Scores in the initial and final test of students

Initial Test Final Test

No. of students 100 100

Mean 55.4 56.9
SD 7.2. 8.0
Standard error of the 0.72 0.80
mean
r12 .64
Calculation
SED or σD= [σ2M1+ σ2M2- 2r σM1 σM2]
= [(.72)2 + (.80)2 – 2 x.64 x.72 x.80]
=.5184+.6400-.7373
= .42
t= (M1-M2)-0/SED
= 1.5-0/0.42
= 3.57
df=(N1-1)
=(100-1)
=99

Referring to the critical value in table for t test (that are given in the appendix
of a textbook on statistics) for df= 99, and find that the value at 0.05 level is
1.98 and at 0.01 level is 2.63. Our obtained value is 2.35 therefore this value
is significant at 0.05 level and not on 0.01 level. Here we can say that class
made substantial improvement in mathematical achievement in six months.

3.5.2 Difference Method

When groups are small, then we must prefer the difference method to that
given above. Let us illustrate the use of this method with the help of
following example.
54
Example: Twelve participants were tested on an attitude scale. Then they Test of
Significance of
were made to read some literature in order to change their attitude. Their Difference
attitude were again measured by the same scale. The results of the initials and Between Two
Means
final testing are as under.

Table 3.4: Results of initial and final testing of attitude

Initial Final Difference X = D-M x²

condition condition Cond 2- Cond 1
(Mean = 8)
50 62 12 4 (12-8) 16
42 40 -2 -10(-2--8) 100
35 30 -5 -13(-5-8) 169
51 61 10 2 (10-8) 4
42 52 10 2 (10-8) 4
26 35 9 -1 (9-8) 1
41 51 10 2 (10-8) 4
42 52 10 2 (10-8) 4
60 68 8 0 (8-8) 0
70 84 14 6 (14-8) 36
55 63 8 0 (8-8) 0
38 50 12 4 (12-8)) 16
Total ∑D = 96 ∑x² = 354

Note: The sum of scores of final condition is more than sum of initial
condition therefore we subtract scores of initial condition from scores of final
condition (Final condition –Initial condition) and add the score to find ΣD.

Mean = (∑D)/N
= 96/12=8
2
SDD= √[(∑x )/N-1]
=√[354/11]
=5.67
SEMD= √
= 5.67 √
= 1.64
t= MD-0/SEMD
=8-0/1.64
=4.88

df=12-1
= 11 55
Parametric
Statistics
Referring to the critical value in table for t test (that are given in the appendix
of a textbook on statistics) for df = 11, we find t values of 2.20 and 3.11 at
the 0.05 and at the 0.01 levels. Our t of 4.88 is far above the 0.01 level. We
can conclude that participants attitude changed significantly from initial to
final condition.

3.5.3 The Method of Equivalent Groups

In experiments when we want to compare the relative effect of one method of
treatment over another we generally take two groups, one is known as
experimental group and the other is known as control group. Here we have
two groups not a single group. For the desired results these two groups need
to be made equivalent. This can be done by (i) Matched pair technique or (ii)
Matched groups technique. These are explained below

i) Matched pair technique: In this techniques matching is done by pair.

Matching is done on variables which are going to affect the results of the
study like age, intelligence, interest, socio-economic status.

ii) Matched groups technique: In this technique instead of person to person

matching, matching of groups is carried out in terms of Mean and S.D.

3.5.4 Matching by Pairs

Formula for calculation of standard error of difference between mean is:
SED or σD = [(σM1)2+(σM2)2- 2r x σM1 x σM2]

Here,
σM1= Standard error of mean 1
σM2= Standard error of mean 2

r=correlation between the two groups scores

t = ( M1-M2 )-0 /SED

Example: There are two groups X and Y of Children .72 in each group are
paired child to child for age and intelligence. Both groups were given group
intelligence scale and scores were obtained. After three weeks experimental
group participants were praised for their performance and urged to try better.
The control groups did not get the incentive. Group intelligence scale again
was administered on the groups. The data obtained were as follows. Did the
praise affect the performance of the group or is there a significant difference
between the two groups.

56
 Table 3.5: Results of Experimental and Control Groups. Test of
Significance of
Difference
Experimental Control group Between Two
Means
group
No. of children in each group 72 72
Mean score of final test 88.63 83.24
SD of final test 24.36 21.62
Standard error of the mean of final test 2.89 2.57

Correlation between experimental and .65

control group scores

SED or σD = [(σM1)2+(σM2)2- 2r x σM1 x σM2]

= [(2.89)2+(2.57)2-2x.65x 2.89x2.57]

t=(M1-M2)-O/SED = 6.11
t= (88.63-83.24)-0/6.11

=0.88

df = 72-1

= 71

Referring to the critical value in table for t test (that are given in the appendix
of a textbook on statistics) for df = 71, the value at 0.05 is 2.00 and at 0.01 is
2.65. The obtained t is 0-88 therefore this value is not significant at 0.05 level
and not at 0.01 level. On the basis of the results is can be said that praise did
not have significant effect in stimulating the performance of children.

3.5.5 Groups Matched for Mean and Standard Deviation

When groups matched in terms of mean and SD, the following formula is
used to calculate ’t’.
SED = [(σM1)2 / N1] + [(σM2)2 / N2 ]-(1-r2 )
t = (M1-M2)-0/ SED

SED= Standard error of difference

σM1 = Standard error of mean 1
σM2 = Standard error of mean 2
r= Correlation between final scores of two tests

The above formula can be illustrated by the following example

Example: The 58 students of academic college and 72 students of technical

college were matched for mean and SD upon general intelligence test. Then
the achievement on a mechanical aptitude test was compared. The questions
is do the two groups enrolled in different courses differ in mechanical ability?
57
Parametric
Statistics Table 3.6: Results of the Academic and Technical Groups

Academic Technical
No. of children in each group 58 72
Mean on Intelligence GTest (Y) 102.50 102.80
SD on Intelligence Test Y 33.65 31.62
Mean on Mechanical Aptitude (X) 48.52 53.51
SD on Mechanical Aptitude X 10.60 15.36
r2 .50

SED = [(σM1)2 / N1] + [(σM2)2 / N2 ]-(1-r2 )

= [(10.60)2/58] + [( 15.35)2/72]- (1-.25)

= 4.46
t = (M1-M2)-0/ SED

t = (53.61-48.52)-0/4.46

= 0.63
df=(N1-1)+(N2-1)

=(58-1)+(72-1)

=128

Referring to the critical value in table for t test (that are given in the appendix
of a textbook on statistics) for df = 125(which is near to 128), the value are
1.98 at .05 level and 2.62 at 0.01 our obtained value is 0.63, which is not
significant at 0.05 level. We may say that two group do not differ in
mechanical aptitude.

Check Your Progress III

1) Ten persons are tested before and after the experimental procedure, their
scores are given below. Test the hypothesis that there is nochange.

Before After
60 72
52 50
61 71
36 45
45 40
52 62
70 78
58
 Test of
51 61 Significance of
Difference
80 94 Between Two
Means
65 73
72 82

3.6 LET US SUM UP

In the field of psychology some time we are interested in testing the
significance of difference between two sample means.
The sample may comprise of two independent groups and single groups
tested twice. Some time we have two groups matched by pair or matched for
means and standard deviations. The process of determining the significance
of difference between the means is different in different conditions. We may
broadly summarise the procedure of calculating significance of differences
between means as under.
 Establish a null hypothesis.
 Decide a suitable level of significance 0.05 or 0.01
 Determine the standard error of the difference between means of two
samples.
 Compute the value of t.
 Find out the degrees of freedom.
 Determine the critical value of t from the ’t’ table (that is given in
appendix of textbook on statistics).
 If the computed value is same or more than the value given in the table
then it is taken to be significant if the computed valued is less than the
given value it is considered as non significant.
 When the t-value is significant we reject the null hypothesis and when t-
value is not significant we retain the null hypothesis.

3.7 REFERENCES
Kerlinger, Fred N. (1963). Foundation of Behavioural Research, (2nd Indian
reprint) Surjeet Publication, New Delhi.
Garrett, H.E. (1971), Statistics in Psychology & Education, Bombay, Vakils,
Seffer & Simoss Ltd.
Guilford, J.P. (1973), Fundamental Statistics in Psychology & Education,
Newyork, McGraw Hill.

3.8 KEY WORDS

Null Hypothesis: A zero difference hypothesis. A statement about the status
quo about a population parameter that is being tested.
59
Parametric
Statistics
Alternative Hypotheses: A hypothesis that takes a value of population
parameter different from that used in the null hypothesis. It states that there is
a difference in the groups on a certain characteristic that is being tested.

Type I error:An error caused by rejecting a null hypothesis when it is true.

Type II error: An error caused by failing to reject a null hypotheses when it

is not true.

One tailed t test: A statistical hypothesis test in which the alternative

hypothesis is specified such that only one direction of the possible
distribution of values is considered. It would state there will be an increase in
the performance of students after training.

Two tailed t test: A statistical hypothesis test in which the alternative

hypothesis is stated in such way that it included both the higher and the lower
values of a parameter than the value specified in the null hypothesis, It would
state that there will be a difference (can be an increase or decrease) in the
group that undergoes training.

Significance level: The probability that a given difference arises because of a

chance factor or it is a true difference.

Standard error:The standard deviation of a sampling distributions.

3.9 ANSWERS TO CHECK YOUR PROGRESS

Check Your Progress I

1) When is a difference called significant?

According to Garrett (1981) a difference is called significant when the

probability is high and that it cannot be attributed to chance that is
(Temporary and accidental factors) and hence represent a true difference
between population mean.

Check Your Progress II

Sr. Statements True or False

No.

1 We commit a Type I error when we reject a True

null hypothesis when it is really true.

2 In testing a hypothesis, one can make three False

types of error.

3 An exercise in hypothesis testing enables us True

to draw conclusions about the estimated
parameters.
60
 Test of
4 For a given level of significance, we find True Significance of
Difference
that as the sample size increases, the critical Between Two
values of t get closer to zero. Means

5 If the standardised sample mean exceeds False

the critical clue, we should accept Ho.

Check Your Progress III

1) Ten persons are tested before and after the experimental procedure, their
scores are given below. Test the hypothesis that there is nochange.

Before After
60 72
52 50
61 71
36 45
45 40
52 62
70 78
51 61
80 94
65 73
72 82

t value = 4.37, significant at 0.01 level

3.10 UNIT END QUESTIONS

1) Differentiate between:

a) Null hypothesis and alternative hypothesis.

b) One tailed test and two tailed test.

c) Type I error and Type II error.

2) Explain the concept of Standard error and level of significance.

3) In the first trial of a practice period, 25 twelve-year-olds have a mean

score of 80.00 and a SD of 8.00 upon a digit-symbol learning test. On the
tenth trial, the mean is 84.00 and the SD is 10.00. The r between scores
on the first and tenth trials is .40. Our hypothesis is that practice leads to
gain.

61
Parametric
Statistics UNIT 4 TEST OF SIGNIFICANCE OF
DIFFERENCE BETWEEN MORE
THAN TWO MEANS*

Structure
4.0 Objectives
4.1 Introduction
4.2 Concept of Variance
4.3 Concept of Analysis of Variance (ANOVA)
4.4 Computation of One Way Analysis of Variance (ANOVA)
4.5 Factorial Design
4.6 Computation of Two Way Analysis of Variance (ANOVA)
4.7 Let Us Sum Up
4.8 References
4.9 Key Words
4.10 Answers to Check Your Progress
4.11 Unit End Questions

4.0 OBJECTIVES
After going through this unit, you will be able to:

 discuss the concept of variance;

 describe the computation of one way Analysis of Variance (ANOVA);
 discuss factorial designs; and
 describe the computation of two way Analysis of Variance (ANOVA).

4.1 INTRODUCTION
A researcher wanted to study the effect of stages of adolescence, early,
middle and late, on the emotional intelligence of adolescents in Chennai. For
the purpose, she used the standardised psychological tests and collected data
from a sample of 600 adolescents in Chennai, 200 each of early adolescents,
middle adolescents and late adolescents. As the data was organised, the
researcher then had to decide about which statistical technique to use for the
purpose of data analysis. Initially the researcher felt that t- test can be used.
But in the present case, there were three groups and as such t test is used
when there are more than two groups. The research then decided to use one
way ANOVA. Though it would be possible to use t test in this situation as

*
Prof. Suhas Shetgovekar, Faculty, Discipline of Psychology, IGNOU, Delhi
62
well, but then the researcher will have to first compare early and middle Test of
Significance of
adolescents, then early and late adolescents and then middle and late Difference
adolescents with regard to emotional intelligence. This will become further Between More
Than Two Means
cumbersome if there are more than three groups, for instance ten groups and
so on. Also, t test will not provide any information about the variance that
may exist from the mean values of the given groups. In such a situation, one
way ANOVA can be conveniently and effectively used.

In the present unit thus, we will mainly focus on the concept of variance and
then we will discuss the computation of one way ANOVA and two way
ANOVA. In this context we need to remember that both these techniques fall
under parametric statistics and thus the assumptions of parametric statistics
need to be met before these techniques are used for calculations.

4.2 CONCEPT OF VARIANCE

In BPCC104, we discussed about the concept of variance. Let us recapitulate
the concept.

The term variance was used to describe the square of the standard deviation
by R.A. Fisher in 1913. The concept of variance is of great importance in
advanced work where it is possible to split the total into several parts, each
attributable to one of the factors causing variations in their original series.
Variance is a measure of the dispersion of a set of data points around their
mean value. It is a mathematical expectation of the average squared
deviations. It mainly helps in separating the variability in to factors that are
random and factors that are systematic (Veraraghavan and Shetgovekar,
2016).

The variance (s2) or mean square (MS) is the arithmetic mean of the squared
deviations of individual scores from their means. In other words, it is the
mean of the squared deviation of scores.Variance is expressed as V = SD2.

The variance and the closely related standard deviation are measures that
indicate how the scores are spread out in a distribution. In other words, they
are measures of variability. The variance is computed as the average squared
deviation of each number from its mean.

Calculating the variance is an important part of many statistical applications

and analysis. It is a good absolute measure of variability and is useful in
computation of Analysis of Variance (ANOVA) to find out the significance
of differences between sample means.

In this context we also discuss about between group variance and within
group variance. Between groups variance can be explained as variance that
exists between the group means. For example, the variance in group means of
early, middle and late adolescents. Whereas, within group variance can be
explained as variance that exists amongst the members within a certain
group. For example, variance that may exist amongst the early adolescents. 63
Parametric
Statistics
Let us now focus on some of the characteristics of variance:
1) Variance can be termed as a measure of variability and it provides
information about variance in the scores in similar way as other measures
of variability. You may recall that we discussed about variance in the
context of measures of variability in BPCC104.
2) Variance is denoted in terms of an area. In normal probability curve, the
variance is denoted in terms of area either on the right or left side of the
curve. On the other hand, standard deviation is denoted by direction on
the normal probability curve.
3) The value of variance is always positive.
4) The variance will remain the same, even if a certain constant in a data is
subtracted or added.
The variance (s2) or mean square (MS) is the arithmetic mean of the squared
deviations of individual scores from their means. In other words, it is the
mean of the squared deviation of scores.Variance is expressed as V = SD2.
The formula thus is as follows:

Variance = SD2 or 2= XM N

Where,
X= Raw scores of a group
M= Mean of the raw scores
N= Total of the raw scores.
Variance is rigidly defined and based on all observations.It is also amenable
to further algebraic treatment and is not affected by sampling fluctuations.
Further, it is less erratic. The main disadvantage of variance is that it is
difficult to calculate and gives greater weight to extreme values.

Check Your Progress I

1) State any one characteristic of variance.

…………………………………………………………
…………………………………………………………
…………………………………………………………

4.4 CONCEPT OF ANALYSIS OF VARIANCE

(ANOVA)
In 1923, Ronald. A. Fisher reported about ANOVA, though the name F test was
given to it by George W. Snedecor in honour of Fisher (Mohanty and Misra,
2016). As the name suggests, ANOVA is mainly related to variance rather than
standard error or standard deviation. It is mainly used in order to measure the
differences between two or more sample means at the same time. And it is also
64
how it can be termed as more advantageous when compared to t test. ANOVA Test of
Significance of
can be computed for independent measures as well as repeated measures. Difference
Though the focus of the present unit will be on ANOVA for independent Between More
Than Two Means
measures.

To just focus on the terms independent measures and repeated measures. In

independent measures, for each treatment or condition there is a separate sample.
The design here can be referred to as between group design. Whereas, with
regard to repeated measures, the same sample is used for varied treatments or
conditions.

As stated by Mohanty and Misra (2016), the meaning of analysis is division in to

smaller parts and the process of carrying out the same is termed as Analysis of
Variance . In this context we discuss about two relevant components, namely,
between treatments variance and within treatments variance. The main focus of
between treatments variance is measuring how the sample means differ from
each other and this difference can either be attributed to the treatment effect or
chance factor. For instance, if we say that a significant difference exits in job
satisfaction of employees based on the intervention received by them namely,
technique 1, technique 2 and technique 3. In this case the occupational stress
may vary amongst the employees based on the intervention received by them or
it could also vary due to chance factors that could be due to individual
differences that may exist amongst the participants or even due to experimental
error.

In ANOVA we also focus on within treatment variance. This has mainly to do

with variance within a particular sample. For instance, in the example we
discussed earlier where the employees were give three different techniques to
measure their impact on job satisfaction of the employees. There was a sample
under each treatment that received the same technique. The scores of the
employees of the employees who received same technique may also vary and
this can be attributed to chance factor and thus within treatment variance will
help us understand to what extent a difference due to chance factor is reasonable
to expect (Mohanty and Misra, 2016).
As we know, ANOVA is parametric statistical technique and thus, the
assumptions of parametric statistics need to be met.
Some of the assumptions of ANOVA are as follows:
1) The F distribution of the dependent variable needs to be normal.
2) There needs to be homogeneity of variance indicating that the population
from which the sample for the research is taken displays equal variance.
3) The observations need to be independent.
4) The effect of various factors on the total variations needs to be additive (not
multiplicative). Thus, it can be said that in ANOVA, the observation can be
divided in to independent as well as additive parts and each part can be as a
result of a source that can be identified.
65
Parametric
Statistics Check Your Progress II

1) What is between treatment variance?

…………………………………………………………
…………………………………………………………
…………………………………………………………
…………………………………………………………

4.4 COMPUTATION OF ONE WAY ANALYSIS

OF VARIANCE (ANOVA)
As the concept of variance is clear, let us discuss one way ANOVA with help of
an example.

A researcher was interested in studying the achievement motivation of students

belonging to low, middle and high Socio Economic Status (SES). The researcher
wanted to find out the effect of Socio Economic Status (SES) on self-concept of
the students. In this case, the independent variable is SES with three categories
or levels and the dependent variable is self-concept that is continuous in nature.
Here it is not possible for the researcher to carry out t test and there are more
than two categories and thus in this case one way ANOVA can be used to find
out the effect of independent variable on dependent variable.

The data collected by the researcher is as follows:

Table 4.1: Example of One Way ANOVA

Low SES Middle SES High SES Total

(X1) (X2) (X3)
1 2 2 2 6
2 3 4 3 10
3 4 5 4 13
4 2 5 2 9
5 3 5 5 13
6 2 2 2 6
7 2 3 2 7
8 2 5 3 10
9 3 5 2 10
10 3 2 3 8
Total 26 38 28 92
Mean 2.6 3.8 2.8

Let us us now focus on the steps involved in computation of one way ANOVA.

66 Step 1: Mean is to be computed for each group

As can be seen in table 4.1, the mean has been computed for each group. Test of
Significance of
For low SES, the mean is 26/ 10= 2.6 Difference
Between More
For middle SES, the mean is 38/10 = 3.8 Than Two Means

For high SES, the mean is 28/10= 2.8

Step 2: Correction term (C) is to be computed
The formula for correction term is
C = (ΣX)2/ N
Thus, in our example,
(ΣX)2is (26+ 38+ 28)2 = (92)2
N = 30
C = (ΣX)2/ N
= (26+ 38+ 28)2/ 30
= (92)2/ 30
= 8464/ 30
= 282.13
Thus, C is obtained as 282.13.
Step 3: Sum of Squares (SSt) is to be computed.
Sum of Squares (SSt) is computed by squaring each score and subtracting
Correction sum.
For the purpose, we will first square each scores given in the three groups.
Table 4.2: Squaring of each score in the group
Low (X1)2 Middle (X2)2 High SES (X3)2
SES SES (X3)
(X1) (X2)
1 2 4 2 4 2 4
2 3 9 4 16 3 9
3 4 16 5 25 4 16
4 2 4 5 25 2 4
5 3 9 5 25 5 25
6 2 4 2 4 2 4
7 2 4 3 9 2 4
8 2 4 5 25 3 9
9 3 9 5 25 2 4
10 3 9 2 4 3 9
Total 26 72 38 162 28 88

Let us compute Sum of Squares (SSt) with the help of the formula
67
Parametric
Statistics
SSt =[(ΣX12 + ΣX22 + ΣX32) – C]
= [(72+ 162 + 88) – 282.13]
= 322- 282.13
= 39.87
Thus, SStis obtained as 39.87.
Step 4: The between groups Sum of Squares ( SSb) is to be computed
The between groups Sum of Squares ( SSb) is computed by squaring the total
of each group divided by respective number of cases, N1, N2, N3 and
subtracting Correction sum (C) from the obtained value.
SSb = (X12 /N1) + (X22 /N2)+( (X32 /N3) - C
= 26 x 26 +38 x 38 + 28 x 28 - 282.13
10 10 10
= (67.6 + 14.44 + 78.4) – 282.13
=290.4 - 282.13
= 8.27
Thus, SSbis obtained as 8.27.
Step 5: Sum of Squares (SSw) is to be computed
Sum of Squares (SSw) is to computed by subtracting the Between sum of
Square from the Total Sum of Square.
SSw = SSt - SSb
= 39.87 – 8.27
= 31.6
Thus, SSw is obtained as 31.6.
Step 6: The degree of freedom (df) are worked out.
For total Sum of Squares N- 1= 30-1= 29
For between group Sum of Squares K- 1 = 3-1 = 2
For within group Sum of Squares N- K = 29- 2 = 27
Step 7: Computation of F ratio
Table 4.3: Summary of ANOVA

Source of Sum of Squares Df Mean Square

Variance Variance (MSS)
Between Group 8.27 2 8.27/2 = 4.14
Within group 31.6 27 31.6/27 = 1.17
Total 543.47 29

68
Thus, F = MSS between groups / MSS within groups Test of
Significance of
F = 4.14/ 1.17 Difference
Between More
F= 3.54 Than Two Means
The F ratio is obtained for the degree of freedom (df) = 2,27

Step 8: Interpretation

Now we have computed the F ratio, but then the value as such is
meaningless, until and unless it is interpreted. For the purpose of
interpretation, you need to refer to certain tables. These tables are given in the
appendix of any Statistics books. The title of the table could be “F ratio for
0.01 and 0.05 levels of significance” (approximately) and may vary to some
extent in different books. In such a table, the degree of freedom for greater
mean square is given as headings for the column (on top) and degree of
freedom for smaller mean square is given as headings for the rows (left hand
side).

Based on the df smaller mean square and df for greater mean square, you can
identify the critical value given in the table for 0.01 and 0.05 levels of
significance.

In the case of our example, the df for smaller mean square as 2 and df for
greater mean square as 27, the critical value is 19.50 at 0.05 level of
significance and 99.50 at 0.01 level of significance. The F ratio obtained in
the example 3.54 is thus not significant as the value is less than the critical
values at 0.05 and 0.01 levels of significance. Thus, it can be said that there is
no significance difference between low, middle and high SES with regard to
achievement motivation.

Check Your Progress III

1. What is the formula for correction term?

…………………………………………………………………………….

…………………………………………………………………………….

…………………………………………………………………………….

4.5 FACTORIAL DESIGNS

In one way ANOVA, the independent variable is categorical in nature and has
more than two categories or levels and the dependent variable is continuous in
nature. With regard to two way ANOVA, there are two independent variables
that are both categorical and can have two or more categories or levels and there
is one dependent variable that is continuous in nature.

The design that is used here can be termed as factorial design. Let us discuss the
same before we go on to computation of two way ANOVA.

69
Parametric
Statistics
Factorial designs are mainly used to study the effectof more than two
independent variables on the dependent variable. The main effect (of each
variable separately) as well as interaction effect (of all the IVs) studied with
the help of this design.
Various types of factorial design are as follows:
 2 x 2 factorial design: Here there are two independent variables, each
with two categories or levels. For example, gender (male and females)
and managers (junior and senior).
 2 x 3 factorial design:Here there are two independent variables. One
with two categories or levels and the other with three categories or
levels. For example, gender (male and females) and Socio Economic
Status (high, middle and low).
 3 x 3 factorial design:Here there are two independent variables. each
with three categories or levels. For example, Socio Economic Status
(high, middle and low) and stages of adolescence (early, middle and
late)
 n x k factorial design: Here there are two independent variables with n
and k categories or levels. n and k can take any number.
 2 x 2 x 2 factorial design: Here there are three independent variables,
each with two categories or levels. For example, gender (males and
females), managers (junior and senior) and Socio Economic Status (high
and low).
 3 x 3 x 3 factorial design: Here there are three independent variables,
each with three categories or levels. For example, Socio Economic Status
(high, middle and low), phases of adolescents (early, middle and late)
and religion (Hindus, Muslims and Christians).
 n x k x l factorial design: Here there are three independent variables
with n and k categories or levels. n, k and l can take any number.
Two way ANOVA can help in studying the main effect and the interaction
effect of the independent variables on the dependent variables. Main effect
can be described as “ the mean difference amongst the levels of one factor”
(Mohanty and Misra, 2016, page 572). Factor means independent variable.
Thus, when we display the independent variables, they would be form of
columns and rows (as can be seen in table 4.4). Thus, the mean different
between the rows would denote the main effect of one variable and mean
difference in columns will denote the mean difference of the other variable.

70
 Table 4.4: Hypothetical example for two way ANOVA Test of
Significance of
Difference
Stages of adolescence (Independent Variable B) Between More
Than Two Means
Gender Early Middle Late
(Independent adolescence (b1) adolescence adolescence
variable A) (b2) (b3)
Male (a1) Mean for a1 and Mean a1and b2= Meana1and b3= Meana1=
b1= 60 55 70 61.7
Female (a2) Mean a2 and b1= Mean a2 and b3= Mean a2and b3= Meana2=
40 65 80 61.7
Mean b1= 50 Mean b2= 60 Meanb3= 75 61.7

As can be seen in table 4.4, the two independent variables are Stages of
adolescence and gender. And in the cells we have mentioned means that
would be based on the dependent variables. The mean difference between
a1and a2 will constitute the main effect for the independent variable A and the
mean difference between b1 and b2 and b3 will constitute the main effect for
independent variable A. Besides main effect, we also need to discuss about
the interaction effect. This is the effect that interaction between variable A
and B has on the dependent variable.

Two way ANOVA can also be effectively displayed in form of line graph.
The interaction effect in this regard can be of three types, that are discussed
as follows:

• Parallel: Parallel lines indicate that there is no interaction between the

independent variables in their effect on dependent variable. For example,
if we have two independent variables, gender (males and females) and
SES (High and low) for dependent variable self-concept (that would be
continuous in nature), then the line graph obtained could be parallel
indicating that there is no interaction between the two independent
variables with regard to their effect on the dependent variable that is self-
concept (refer to figure 4.1). Though, when the two way ANOVA is
computed, the interpretation will be based on whether the F ratio
obtained is significant at 0.01 or 0.05 levels of significance or not.

High SES

Low SES

Males Females
Fig 4.1: Parallel
71
Parametric
Statistics • Additive: Here there is some interaction between the independent
variables with regard their effect on dependent variables. In this, as you
can see in figure 4.2, the lines are not parallel but as such, there is no
complete interaction.

High SES

Low SES

Males Females

Fig 4.2: Additives

This denotes that there is some interaction between the independent

variables with regard to their effect on the dependent variable. Thus, as
can be seen that the self-concept of males with high SES is low when
compared to females with high SES, though it is quite close to the scores
obtained by males from low SES. However there may not be difference
between the scores obtained by male and female high SES, but there
seems to be a difference (mostly significant, that will be apparent from
the F ratio) between the females of high and low SES.

• Crossover interaction: Here there is complete interaction between the

independent variables with regard to their effect on dependent variable.
As can be seen in the figure 4.3, there is a crossover interaction and the
males belonging to high SES have lower self-concept as opposed to
females having high SES and the males of low SES have higher self-
concept compared to females of low SES.

High SES

Low SES

Males Females

Fig 4.3: Cross over interaction

72
Note: The example discussed here is hypothetical and may not hold true in Test of
Significance of
reality, but has been used in the context for clarity of the concept. Difference
Between More
Check Your Progress IV Than Two Means

1) What is crossover interaction?

…………………………………………………………………………….
…………………………………………………………………………….
…………………………………………………………………………….

4.6 COMPUTATION OF TWO WAY ANALYSIS

OF VARIANCE (ANOVA)
Let us us now discuss about two way ANOVA with the help of an example.

A researcher was interested in studying the perceived parental behaviour of

adolescents with regard to gender, and stages of adolescence . In this case, the
first independent variable is is gender with two categories or levels and the other
independent variable is stages of adolescence, early, middle and late. The
dependent variable is perceived parental behaviour that is continuous in nature.

Here it is not possible for the researcher to carry out a one way ANOVA as there
are two independent variables that are categorical and the researcher may also be
interested in finding out the interaction between the two independent variables
with regard to their effect on the dependent variables. Thus in this case, the two
way ANOVA can be used.

Table 4.5: Example of Two Way ANOVA-1

Stages of Adolescence (Variable B) Details of rows

Gender Early Middle Late
(Variable adolescence adolescence adolescence
A) (b1) (b2) (b3)
Males 2 2 2
(a1) 3 4 3
4 5 4
2 5 2
3 5 5
n 5 5 5 na1= 15
ΣX 14 21 16 ΣXa1= 51
Mean 2.8 4.2 3.2 Mean a1= 8.07
ΣX2 42 95 58 ΣXa12= 195
Females 2 2 2
(a2) 2 3 2
2 5 3
73
Parametric
Statistics 3 5 2
3 2 3
n 5 5 5 na2= 15
ΣX 12 17 12 ΣXa2= 41
Mean 2.4 3.4 2.4 Mean a2= 7.6
2
ΣX 30 67 30 ΣXa22= 127
Details of columns
n 10 10 10 nb2=30
ΣX 26 38 28 ΣΣX= 92
Mean 2.6 3.8 2.8 Mean= 3.07
2
ΣX 72 162 88 ΣΣX2= 322

As you can see, there are a total of six groups in the above example, which are as
follows:
- early adolescence who are males
- middle adolescence who are males
- late adolescence who are males
- early adolescence who are females
- middle adolescence who are females
- late adolescence who are females

The sums of rows and columns will be as follows:

Gender Pases of Adolescents Rpw Total

Early Middle Late
Males 14 21 16 51
Females 12 17 12 41
Column 26 38 28 92
total

In table 4.5, we have calculated n, ΣX, mean and ΣX2 for both rows and
columns.
Let us now try to understand the computation of ANOVA stepwise
Step 1: Compute the correction term (C) with the help of the formula
C = (ΣX)2/ N
In this formula N = n1 + n2 + n3 =10+ 10+ 10= 30.
Thus, N = 30
C = (ΣX)2/ N

74
 = (92)2/ 30 Test of
Significance of
Difference
= 8464/ 30 Between More
Than Two Means
= 282.13
Thus, C is obtained as 282.13.
Step 2: Compute total Sum of Squares (SSt)
Let us compute total Sum of Squares (SSt) with the help of the formula

SSt =ΣΣX2 – C
= 322-282.13
= 39.7
Thus, SSt is obtained as 39.7
Step 3: Between group Sum of Squares ( SSbetween) is to be computed.
Between group Sum of Squares (SSbetween) is to be computedwith the help of
the following formula:
SSbetween = Σ (ΣX)2/n - C
= (ΣX1)2+(ΣX2)2+ (ΣX3)2 +(ΣX4)2+(ΣX5)2+(ΣX6)2- C
n1 n2 n3 n4 n5 n6
= (14)2+(21)2+ (16)2+(12)2 +(17)2 +(12)2- 282.13
5 5 5 5 5 5
= 196 + 441+ 256+ 144+ 289+ 144 - 282.13
5 5 5 55 5
= 1470/5 - 282.13
= 294 - 282.13
= 11.87
Thus, SSbetweenis obtained as 11.87
Step 4:Within group Sum of Squares ( SSw) for columns is to be
computed.
Within group Sum of Squares ( SSw) for columns is to be computed with the
help of the following formula:
SSw=SSt- SSb
= 39.7- 12.87
= 27.43
Thus, SSwis obtained as 27.43
Step 5: ‘A’ Sum of Squares (SSa) is to be computed.
This is mainly the main effect of variable A or it can also be mentioned as the
effect of the row. 75
Parametric
Statistics
„A‟ Sum of Squares (SSa) is to computed
SSa = (ΣXa1)2 +(ΣXa2)2 - C
n1 n2
SSa = (51) +(41)2 - 282.13
2

15 15
= 2601 + 1681/15- 282.13
= 4282/15- 282.13
285.47- 282.13
= 3.34
Thus, SSa is obtained as 3.34.
Step 6: ‘B’ Sum of Squares (SSb) is to be computed.
This is mainly the main effect of variable B or it can also be mentioned as the
effect of the column.
„B‟ Sum of Squares (SSb) is to computed

SSb =
 Xb1 2 +  Xb2 2 +  Xb3 2 - C
n1 n2 n3

SSb =
 26 2 +  38 2 +  28 2 - 282.13
10 10 10
= 67.6 + 144.4 +78.4 - 282.13
= 290.4 - 282.13
= 8.27
Thus, SSb is obtained as 8.27.
Step 7: AB interactionSum of Squares (SSab) is to be computed.
(SSab) = SSbetween - SSa- SSb
= 11.87 - 3.34- 8.27
= 11.87 - 11.61
= 0.26
Thus, SSab is obtained as 0.26
Step 8: Degree of freedom
In our example, variable A, that is, gender has two levels (represented by
number of rows) and variable B, that is, stages of adolescence has three
levels (represented by number of columns). Thus, r (rows) = 2 and c
(columns) = 3. Further, there are six conditions (k) and the number of
observations in each is 5 (n). The degree of free would be as follows:
df for SSt = dft = N-1 = kn- 1= 30-1= 29
76
 df for SSbetween = dfbetween = k-1= 6-1= 5 Test of
Significance of
Difference
df for SSw = dfw = N-k= 30-6= 24 Between More
Than Two Means
The df for SSbetween can be in three parts as given below:
df for SSa = dfa = r-1 = 2-1= 1
df for SSb = dfb = c-1= 3-1= 2
df for SSab = dfab= (r-1) (c-1)= (2-1)(3-1)= 1x2= 2
Step 9: Variance estimate or mean squares (MS) is to be computed.
Variance estimate or mean squares is to be computed with the help of
following formula:
MS = SS/df
We need to compute MS for the following:
MS for variable A
MSa = SSa/dfa
= 3.34/1
= 3.34
MS for variable B
MSb = SSb/dfb
= 8.27/ 2
= 4.14
MS for AB interaction
MSab = SSab/dfab
= 0.26/ 2
= 0.13
MS for within groups
MSw = SSw/dfw
= 27.43/ 24
= 1.14
Step 10: Computation of F ratios
F ratios are computes as follows for variable A, variable B and AB:
Fa= MSa/MSw
= 3.34/ 1.14
=2.93
Fb= MSb/MSw
4.14/ 1.14
77
Parametric
Statistics
=3.63
Fab= MSab/MSw
0.13/ 1.14
=0.11
The summary of two way ANOVA is given in table 4.6

Table 4.6: Summary of Two Way ANOVA

Source of Sums of Degrees of Mean F p-value

Variation Squares freedom Squares
SS DF MS

A SSa=3.34 a-1=1 MSR= 0.1039

3.34/1 3.34/ 1.14

=3.34 =2.93

B SSb=8.27 b-1=2 MSC= 4.14/ 1.14 0.0449

8.2667/ 2 =3.63

=4.14

AB SSab=0.27 (a-1)(b- MSAB= 0.13/ 1.14 0.8925

1)=2
0.26/ 2 =0.11

=0.13

Error SSE=28 rab-ab=24 MSE=

(residual)
27.43/ 24
=1.14

Total SST=39.8667 rab-1=29

Step 11: Interpretation

The F ratio obtained can be interpreted based on the table of two way
ANOVA that is normally given in any statistics or research methodology
books.

Consulting such a table the table value for degree of freedom (df) 1, 24 , that
is in the context of variable A, is 4.26 at 0.05 level of significance and 7.82 at
0.01 level of significance. The F ratio for variable A is obtained as 2.93
78 which is less than the table value and thus the F ratio for variable A is not
significant and null hypothesis can be accepted and it can be said that no Test of
Significance of
gender difference exists with regard to perceived parental behaviour. Difference
Between More
With regard to variable B, the F ratio is obtained as 3.63. The df here is 2, 24 Than Two Means

and the table value is 3.40 for 0.05 level of significance and 5.61 for 0.01
level of significance. The F ratio obtained is less than the table value at 0.01
level, but more than the table value at 0.05 level. Thus, it can be said that F
ratio is significant at 0.05 level of significance and it can be said that
significant difference exists in perceived parental behaviour with regard to
stages of adolescence .

With regard to AB interaction, the F ratio is obtained as 0.11. The df here is

2, 24 and the table value is 3.40 for 0.05 level of significance and 5.61 for
0.01 level of significance. The F ratio obtained is less than the table value at
both 0.01 and 0.05 levels of significance. Thus, it can be said that F ratio is
not significant.

If the AB interaction is significant, it can be shown using a line graph with

mean scores on the y axis and the levels of one of the variables on the axis,
with the lines of the graph showing the levels of the other variables, as shown
in figures 4.1, 4.2,and 4.3.

Check Your Progress V

1) What is the formula for total Sum of Squares?

4. 7 LET US SUM UP
To sum up, in the present unit we mainly discussed about the concept of
variance. The variance (s2) or mean square (MS) is the arithmetic mean of the
squared deviations of individual scores from their means. In other words, it is
the mean of the squared deviation of scores. Variance is expressed as V =
SD2. The characteristics of variance were also discussed. Further, in the unit
we explained the concept of ANOVA. In 1923, Ronald. A. Fisher reported
about ANOVA, though the name F test was given to it by George W. Snedecor
in honour of Fisher. The assumptions of ANOVA were also described. The unit
then focused on the computation of one way ANOVA with the help of steps and
example. The unit later discussed about factorial designs. Factorial designs are
mainly used to study the effect of more than two independent variable(s) on the
dependent variable. The main effect (of each variable separately) as well as
interaction effect (of all the IVs) studied with the help of this design. Various
types of factorial design were also discussed. Lastly, the unit described the
computation of two way ANOVA with the help of steps and example.
79
Parametric
Statistics 4.8 REFERENCES
King, Bruce. M; Minium, Edward. W. (2008). Statistical Reasoning in the
Behavioural Sciences. Delhi: John Wiley and Sons, Ltd.
Mangal, S. K. (2002). Statistics in Psychology and Education. new Delhi: Phi
Learning Private Limited.
Minium, E. W., King, B. M., & Bear, G. (2001). Statistical Reasoning in
Psychology and Education. Singapore: John-Wiley.
Mohanty, B and Misra, S. (2016). Statistics for Behavioural and Social
Sciences. Delhi: Sage.
Veeraraghavan, V and Shetgovekar, S. (2016). Textbook of Parametric and
Non-parametric Statistics. Delhi: Sage.

4.9 KEY WORDS

Factorial designs: Factorial designs are mainly used to study the effectof
more than two independent variable (s)on the dependent variable. The main
effect (of each variable separately) as well as interaction effect (of all the
IVs) studied with the help of this design.

Variance:The variance (s2) or mean square (MS) is the arithmetic mean of

the squared deviations of individual scores from their means. In other words,
it is the mean of the squared deviation of scores.Variance is expressed as V =
SD2.

4.10 ANSWERS TO CHECK YOUR PROGRESS

Check Your Progress I

1) State any one characteristic of variance.

Variance can be terms as a measure of variability and it provides
information about variance in the scores in similar way as other measures
of variability.

Check Your Progress II

1) What is between treatment variance?

The main focus of between treatments variance is measuring how the
sample means differ from each other.

Check Your Progress III

1) What is the formula for correction term?

The formula for correction term is
C = (ΣX)2/ N

80