10.4. An Introduction to Analysis of Variance

Analysis of Variance (ANOVA): a statistical technique can be used to test

the hypothesis that the means of 3 or more populations are equal.

General Case:

Suppose there are K populations. The data are the following

Populations	Samples
1
2
k

3 assumptions for the above problem:

1. Normal populations.

2. Equal variance for all populations.

3. The data must be independent of any other one.

Let

: the i’th sample value form population k.

: the sample mean for population j.

the overall mean.

: the sample variance for population j.

: F distribution with degrees of and .

F test:

Let

vs.

Then,

,

where

,

can be obtained by

,

MSB=Mean Square Between, and

MSW=Mean Square Within.

In addition,

Example 5:

Five observations were selected from each of three populations. The data obtained follow.

Observation	Sample 1	Sample 2	Sample 3
1	32	44	33
2	30	43	36
3	30	44	35
4	26	46	36
5	32	48	40
Sample mean	30	45	36
Sample variance	6	4	6.5

(i) Set up the ANOVA table for this problem.

(ii) At the level of significance, test the null hypothesis that

the three population means are equal?

[solution:]

Therefore,

Source	DF	SS	MS	F
Between	2	570	285	51.82
Within	12	66	5.5
Total	14	636

(ii)

.

JavaStatSoft:

F test:

Statistics -> Tests -> One-Way ANOVA