ANOVA: Analysis of Variance

1. Analysis Of
Variance:
ANOVA
Submitted To:-
Dr. Madhu Gupta
Submitted By:-
Priyamvada

2. content
Content
 Introduction of ANOVA
 Concept and Meaning
 Definitions of ANOVA
 Classification Of
ANOVA
 One-Way ANOVA
 Two-Way ANOVA
 Uses of ANOVA
 Advantages of ANOVA
 Limitations of ANOVA

3. Introduction
 ANOVA is commonly used to test
hypothesis of equality between two
variances and among different means.
 It is one of the standard and extremely
useful technique for experiments in the
field of psychology, Sociology,
Education, Commerce and several
other disciplines.
 It is an advanced technique for the
experimental treatment of testing
differences among all of the means
which is not possible in case of t-test.
 This test is known as F test of
significance in which we can find out
more than two experimental variables
and their interaction in the experiment.

4. Concept
 This technique of ANOVA was first developed by Ronald A.
Fisher, a British Scientist in 1923.
 This method was widely used in the experiments of behavioral
and social sciences to test the significance of differences of
means in different group of a varied populations.
 For this technique fisher is called the father of Modern Statistics.
 Through his technique, it is possible to determine the significance
of difference of different means in a single test rather than many.
 It minimizes the Type 1 error unlike in case of t-test.
 The value of F ratio is computed through
Variance between groups/ Larger estimate of variance
Variance within groups/ Smaller estimate of variance
The value of F is always more than 1.
F =

5. Meaning
The object of F-test is to find out whether the two independent estimates of population variance
differ significantly or whether the two sample may be regarded as drawn from the normal
population. That’s why also known as Variance Ratio Test.
ANOVA is one of the most powerful techniques available in the field of statistical teaching.
It is a statistical technique in which, significance of more than two mean sample can be
analyzed.
This method is mostly used in research process. Because for conclusion or interpretation more
than two variables are required.
In earlier times, this method was used in agricultural researches. But in modern times it is also
used in natural, social and agricultural researches.

6. Definitions of ANOVA
 “The analysis of variance is
essentially a procedure for
testing the difference between
different groups of data for
homogeneity.”
Yule and Kendal
 “ Analysis of variance is the
separation of the variance
ascribe to one group of cause
from the variance ascribe to
other group.”
R.A.Fisher

7. Some Features
A substitute of t-test & z-test.
A Parametric test.
Determining several differences between several means.
Data are treated at once/at the same time.
A general hypothesis among means of different group is tested.
Minimizing type- 1 error.
Breaks the total variance of large sample and makes it two groups.
- Variance within Groups (Average Variance of each & their means)
- Variance Between Groups (Variance of the group means and group mean of all groups)

8. Assumptions of Analysis of
Variance
Normal distribution
Independent samples
Same population variance
Random Selection
Additivity of effects

9. Techniques of Analysis of
Variance
One way
classification
Two way
classification

11. One Way ANOVA
In one way ANOVA only one factor is computed. It is
also called single classification ANOVA. For example,
in an experiment, three groups are selected for an
experimental treatment on one factor i.e., evaluation of
performance of the three groups on the basis of three
attitude scales.
In one-way ANOVA, the total variance is equal to the
variance within groups and variance between groups.

12. Example:-
Set up an analysis of variance table for the following per acre
production data for three varieties of wheat, each grown on
four plots. Consider variety differences to be significant.
Plot of Land A B C
1 6 5 5
2 7 5 4
3 3 3 3
4 8 7 4
Per Area Production data Variety of Wheat

13. A
X1 (x1)
B
X2 (x2)
C
X3 (x3)
6 36 5 25 5 25
7 49 5 25 4 16
3 9 3 9 3 9
8 64 7 49 4 16
∑X =24 ∑x =158 ∑X =20 ∑x =108 ∑X =16 ∑x =66
1
2
1
2 3
2
2
2
3
• Solution
2 22

14.  Step 1:-
Null Hypothesis:- Let us take the hypothesis
that there is no significant difference in
production of three varieties.
Step 2:- Correlation Factor (c)
C.F= (∑X)
N
C.F= (60) /12=300
2
2

15.  Step 3:-
Sum of Square Total= ∑X – C
SSt=(158+108+66) – 300
SSt= 32
 Step 4:-
Sum of Square Between
SSb= (∑x ) (∑x ) (∑x )
n1 n2 n3
SSb= (24)/4 + (20)/4 + (16)/4- 300
SSb= 144+ 100+ 64- 300
2
1
2
2 3
2 2
+ + - C
2 2 2

16. SSb=8
 Step 5:-
Sum of Square With in Groups
SSw= SSt- SSb
SSw=32-8=24
SSw= 24
Step 6:-
Degree of Freedom
D.O.F for sum of square total= (N-1)
= 12-1=11
D.O.F for sum of square between= (k-1)
= 3-1=2
D.O.F for sum of square with in groups= (N-k)
= 12-3=9

17. ANOVA Table
Source of
variation
Sum of
Square
Degree of
Freedom
Mean Sum
of Square
F-Ratio Table
value at
5% level
Between
Sample
SSB= 8 K-1
3-1=2
8/2=4 4/2.67=1.5 F
(2,9)=19.45
H0=
Accepted
Within
Sample
SSW= 24 N-K
12-3=9
24/9=2.67
Total SST= 32 N-1
12-1=11

18.  Step 8:-
Interpretation:-
The computed value 1.5 is less than table
value 19.45 at 5% level of significance. So
the null hypothesis that there is no significant
difference in production of three varieties is
accepted.

20. Two-Way ANOVA
In two-way ANOVA there is more than one experimental factor
and one or more control factors. In case of the above one way
ANOVA the three attitude scales for the test of performance are
experimental factor where as the examiner is the control factor.
If the tests are conducted by two examiners so the over all
factors will be 3×2=6.
It is also called double classification ANOVA. This procedure can
be used for the application of three-way or larger class of
ANOVA.

21.  Illustration:-
A farmer applied three types of fertilizers
on 4 separate plots. The figure on yield per
acre are tabulated below:
Fertilizers
Plots
A B C D Total
Nitrogen 6 4 8 6 24
Potash 7 6 6 9 28
phosphates 8 5 10 9 32
Total 21 15 24 24 84
Yield

22. Plots
Fertilizer
A
X1
B
X2
C
X3
D
X4
Row
Total
(x1) (x2) (x3) (x4)
Nitrogen 6 4 8 6 24 36 15 64 36
Potash 7 6 6 9 28 49 36 36 81
Phosphat
es
8 5 10 9 32 64 25 100 81
Column
Total
21 15 24 24 T=84 149 77 200 198
2 2 2 2
Yield Square of Data

23.  Find out if the plots are materially different in fertility,
as also, if the three fertilizers make any material
difference in yields.
 Solution:-
 Step 1:-
Null Hypothesis:- Let us take the hypothesis that:
1. All plots are not significantly differ in fertility (Column
wise analysis)
2. All the fertilizers are not significantly differ in
yields.(Row wise analysis)
 Step 2:- Correlation Factor (c)
C= (∑X)
N
C= (84) /12= 588
2
2

24.  Step 3:-
Sum of Square Total= ∑X – C
∑X=149+77+200+198=624
SSt= 624- 588
SSt= 36
 Step 4:-
Sum of Square Between Columns:
SSc= (∑x ) (∑x ) (∑x ) (∑x )
n1 n2 n3 n4
2
2
1 2 3 4
2
2 2 2
+ + + - C

25. SSc= (21)/3+ (15)/3 + (24)/3 + (24)/3-588
SSc= 147+75+192+192-588=18
SSc=18
 Step 5:-
Sum of Square Between Rows:
SSr= (∑x ) (∑x ) (∑x )
n1 n2 n3
SSr=(24)/4+ (28)/4+ (32)/4- 588
SSr=144+196+256-588=8
 Step 6:-
Sum of Square With in Groups:
2 2 2 2
1 2 3
2 2 2
+ + - c

26. SSw= SSt- (SSc+SSr)
SSw= 36- (18+8)=10
SSw= 10
 Step 7:-
Degree of Freedom:
D.O.F for total sum of square= (N-1)
= 12-1=11
D.O.F for sum of square between columns= c-1
= 4-1=3
D.O.F for sum of square between rows= r-1
= 3-1=2
D.O.F for sum of square with in groups= (r-1) (c-1)
= 2×3=6
Here: r= no. of rows
c= no. of columns

27. Source of
Variance
Sum of
Square
Degree of
Freedom
Mean Sum
of Square
F-Ratio Table value
Between
Columns
SSc=18 c-1
4-1=3
18/3=6 SSc/SSw
6/1.667=
3.6
F (3,6)=8.94
H0=
Accepted
Between
Rows
SSr= 8 r-1
3-1=2
8/2=4 SSr/SSw
4/1.667=
2.4
F (2,6)
=19.33
H0=
Accepted
With in
Groups
SSw= 10 (r-1) (c-1)
3×2=6
10/6=1.667
Total SSt= 36 N-1= 11
ANOVA Table

28.  Step 9:-
Interpretation
1. Columns wise analysis:- The computed value
of F=3.6 is less than table value 8.94, hence
the null hypothesis is accepted, it means the
plots are not significantly differ in fertility.
2. Row wise analysis:- the calculated value of
F=2.4 is less than table value 19.33, hence
the null hypothesis is accepted. It means the
fertilizers are alike so far as productivity
concern.

29. Uses of ANOVA
To test the significance between variance of two
samples.
Used to study the homogeneity in case of Two-
way classification.
It is used in testing of correlation & regression.
ANOVA is used to test the significance of
multiple correlation coefficient.
The linearity of regression is also tested with
the help of Analysis of Variance.
Interpretation of significance of means & their
interactions.

30. Advantages of ANOVA
It is improved technique over t-test & z-test.
Suitable for multi-dimensional variables.
Analysis various factors at a time.
Can be used in three and more than three groups.
Economical and good method of Parametric testing.
It involve more than one independent variables in studying the main impact & interaction
effect.
The experimental design ( simple random design & level treatment design) are based on
one way ANOVA technique.

31. Limitations of ANOVA
It is difficult to analyze ANOVA under strict assumptions regarding the nature of
data.
It is not so helpful in comparison with t-test that there is no special interpretation
of the significance of two means.
It is not always easy to interpret the cases of multiple interactions and their
significance level.
It has a fixed and difficult set for designing experiments for the researcher.
Requirement of post- ANOVA t-test for further testing.
Sometimes, time consuming & also time requires knowledge & skills for solving
numerical problems.
It provides no additional information as compared to t-test.