Numerical Example of One-Way ANOVA

A

1

18

A

2

18

A

3

18

change a sample sizeor generate a new dataset

sample

A

1

sample

A

2

sample

A

3

1

2

3

1 ANOVA complex

2 ANOVA table

3 ANOVA summary

sample

A

1

{1,4,1,4,3,4,1,1,1,1,3,1,1,1,2,2,3,2}

sample

A

2

{2,2,4,2,4,6,2,4,5,4,3,3,4,3,6,2,5,5}

sample

A

3

{1,1,2,2,2,3,3,5,4,1,1,2,3,4,2,5,1,3}

samples	A 1	A 2	A 3	formulasfor row	total, a = 3
n j	18	18	18	a ∑ j=1 n j	N = 54
n j ∑ i=1 X i	36	66	45	a ∑ j=1 n j ∑ i=1 X i	147
2 n j ∑ i=1 X i	1296	4356	2025	—	—
1 n j 2 n j ∑ i=1 X i	72.00	242.0	112.5	a ∑ j=1 2 n j ∑ i=1 X i n j	430.0
n j ∑ i=1 2 X i	96	274	143	a ∑ j=1 nj ∑ i=1 2 x i	513
H= 2 a ∑ j=1 n j ∑ i=1 X i N		H = 2401 6
H	400.17
D y	-400.17+513
D x	-400.17+430.0

This Demonstration illustrates some basic principles of one-way ANOVA (factor analysis of variance) and shows how it works so you can analyze the statistical variability of a statistical complex.

You can vary the sample sizes of the three groups

A

1

,

A

2

,

A

3

separately from 3 to 18, and observe how the ANOVA table changes. The total variation depends not only on the sizes but also on the variability within each sample.

An ANOVA table helps in understanding the overall relationship of random variables and in learning the principle of calculating the variations.

This Demonstration considers the calculation of the ANOVA table manually and illustrates the repeated-measures design. Repeated measures analysis of variance (rANOVA) is one of the most commonly used statistical approaches to repeated measures designs.