2. INTRODUCTION
• It is a statistical method used to test whether the
effects of several factors are equal or not.
• It tests whether the given samples can be
considered as having been drawn from
populations with same mean.
• It is designed to test whether a significant
difference exists among three or more sample
means.
• In this method the total variation in a set of data
is divided into variation within groups and
variation between group.
3. • The statistical test used is F test.
• The null hypothesis is sample means are
equal.
• Means the effects of different factors which
cause variation are same.
4. Suppose we drew 3 samples from the same population.
Our results might look like this:
10
0
-10
-20
4
3
2
1
0
10
0
-10
-20 10
0
-10
-20
Ra w Sc o re s (X)
10
0
-10
-20
Three Samples fromthe Same Population
M ean 1
M ean 2
M ean 3
Standard Dev Group 3
Note that the means from
the 3 groups are not
exactly the same, but they
are close, so the variance
among means will be
small.
5. Suppose we sample people from 3 different populations.
Our results might look like this:
20
10
0
-10
-20
4
3
2
1
0
Three Samples from3 DiffferentPopulations
20
10
0
-10
-20
Three Samples from3 DiffferentPopulations
20
10
0
-10
-20
Three Samples from3 DiffferentPopulations
20
10
0
-10
-20
Raw Scores (X)
Three Samples from3 DiffferentPopulations
Mean 1
Mean 2
Mean 3
SD Group 1
Note that the sample
means are far away from
one another, so the
variance among means
will be large.
6. Suppose we complete a study and find the following results
(either graph). How would we know or decide whether
there is a real effect or not?
10
0
-10
-20
4
3
2
1
0
10
0
-10
-20 10
0
-10
-20
Ra w Sc o re s (X)
10
0
-10
-20
Three Samples fromthe Same Population
M ean 1
M ean 2
M ean 3
Standard Dev Group 3
20
10
0
-10
-20
4
3
2
1
0
Three Samples from3 DiffferentPopulations
20
10
0
-10
-20
Three Samples from3 DiffferentPopulations
20
10
0
-10
-20
Three Samples from3 DiffferentPopulations
20
10
0
-10
-20
Raw Scores (X)
Three Samples from3 DiffferentPopulations
Mean 1
Mean 2
Mean 3
SD Group 1
To decide, we can compare our observed
variance in means to what we would expect
to get on the basis of chance given no true
difference in means.
7. Why ANOVA not t-test
• Tedious when many groups are present
(type 1 error will accumulate).
• Using all data increases stability
• Large number of comparisons some
may appear significant by chance
8. ONE WAY ANOVA
• ANOVA used for studying the differences
among the influence of various categories of
independent variables on a dependent
variable is called One way ANOVA.
• Observations are classified into groups on the
basis of single criterion.
9. PROCEDURE
1. State the Null and Alternate hypothesis.
2. Compute Mean squares of variation between the samples say MSC and Mean
squares of variation within the samples say MSE.
For computing MSC and MSE, following calculations are made.
i. T=Sum of all observations
ii. SST=(x1
2+ x2
2+ x3
2+…+xN) - T2/N (total sum of squares of variation)
iii. SSC= (∑X1)2/n1+ (∑X2)2/n2+ (∑X3)2/n3+… - T2/N) where ∑X1, ∑X2, ∑X3….are the
column totals. (total sum of squares of variation between samples)
iv. SSE=SST-SSC (total sum of squares of error or sum of squares of variation with in the
samples)
10. v Then calculate MSC=SSC/k-1 where k is the
number of columns.
vi Calculate MSE=SSE/N-k
3 Calculate F ratio = MSC/MSE
4 Obtain the table value of F ie Fcritical(k-1, N-k)
degrees of freedom
5 Fcalculated< Fcritical ,Accept the hypothesis that
the sample means are equal.
11. ANOVA TABLE
SOURCE OF
VARIATION
SUM OF
SQUARES
DEGREE OF
FREEDOM
MEAN SQUARE
BETWEEN
SAMPLES
SSC k-1 MSC
WITHIN
SAMPLES
SSE N-k MSE F =MSC/MSE
TOTAL SST N-1
12. • Below are given the yield (in kg per acre for 5
trial plots of 4 varieties of treatment)
TREATMENT 1 2 3 4
PLOT NO.
1 42 48 68 80
2 50 66 52 94
3 62 68 76 78
4 34 78 64 82
5 52 70 70 66
13. SOURCE OF
VARIATION
SUM OF
SQUARES
D.O.F MEAN SQUARE
BETWEEN
SAMPLES
SSC=2580 k-1=3 MSC=860
WITHIN
SAMPLES
SSE=1656 N-K=16 MSE=103.5 FCAL=860/103.5
=8.3
TOTAL SST=4236 N-1=19 FCRI(3,16) =3.24
16. Two way ANOVA
• Used for studying the difference among the
influence of various categories of two
independent variables on a dependent
variable is called two way ANOVA.
• Observations are classified into groups on the
basis of two criteria.
17. PROCEDURE
1. Assume the means of the samples are equal. ie the
effects of all factors in one kind of treatment are
equal.
Assume the means of all rows are equal. ie, the effect
of all factors in the second kind of treatment are
equal.
2. Compute T=Sum of all values.
3. Find SST=sum of squares of all observations -T2/N
4. Find SSC = (∑X1)2/n1+ (∑X2)2/n1+ (∑X3)2/n1+… - T2/N
where ∑X1, ∑X2, ∑X3….are the column totals
5. Find SSR = (∑X1)2/n2+ (∑X2)2/n2+ (∑X3)2/n2+… - T2/N
where ∑X1, ∑X2, ∑X3….are the row totals.
18. 6. SSE=SST-SSC-SSR
7. MSC=SSC/c-1 ; MSR=SSR/r-1 ; MSE=SSE/(c-1)(r-1)
where c is the no. of columns and r is the no. of
rows.
8. FC=MSC/MSE and FR=MSR/MSE
9. Obtain the table value of Fc & FR
10. ie. FC,critical[c-1, (c-1) (r-1)] FR, critical[r-1, (c-1) (r-1)]
19. ANOVA TABLE FOR TWO WAY
ANALYSIS
SOURCE OF
VARIATION
SUM OF
SQUARES
D.O.F MEAN SQUARE F RATIO
BETWEEN
COLUMNS
SSC c-1 MSC FC =MSC/MSE
BETWEEN
ROWS
SSR r-1 MSR FR=MSR/MSE
RESIDUAL SSE (c-1)(r-1) MSE
TOTAL SST N-1
20. Three varieties of crops A,B,C are tested in a randomized block
design with four replications. The yield are given below.
Test whether there is difference between replications. Test also
whether varieties differ significantly.
Replication
variety
1 2 3 4
1 6 4 8 6
2 7 6 6 9
3 8 5 10 9
21. ANOVA TABLE FOR TWO WAY
ANALYSIS
SOURCE OF
VARIATION
SUM OF
SQUARES
D.O.F MEAN SQUARE F RATIO
BETWEEN
COLUMNS
18 3 6 FC =3.6
BETWEEN
ROWS
8 2 4 FR=2.4
RESIDUAL 10 6 1.667
TOTAL 36 11
