Analysis of Variance (ANOVA)

The analysis of variance (ANOVA) is the statistical method used for testing the null hypothesis that the means of several populations are equal. The comparison in means of 3 or more populations which follow normal distributions can be taken simultaneously in just one application of this test. This test, therefore, is a generalization of the z and t tests of two normal population means.

In order to use ANOVA, certain conditions must be met:

1. The sample(s) must be randomly selected from normal populations.
2. The populations should have equal variances.
3. The distance from one value to its group’s mean should be independent of the distances of other values to that mean (independence of error).

ANOVA is reasonably robust, so that minor variations from normality and equal variance are tolerable.
ANOVA uses squared deviations or variances so that computation of distances of individual data points from their own mean or from the grand mean can be summed.

The test statistic for ANOVA is the F ratio. It compares the variance from the two sources:

F = Between – groups variance / Within – groups variance
F = Mean Square between (MSB) / Mean Square within (MSW)
F = MSB / MSW

Where:
MSB = Sum of Squares between / Degrees of Freedom between
MSB = (SSB) / (dfB)
MSW = Sum of Square within / Degrees of Freedom within
MSW = (SSW) / (dfw)
The degree of freedom for SSB = k – 1 and for SSW = k (n – 1)

Where:
k = number of groups or samples
n = no. of items per column (size of each sample)