Pooled SD in ANOVA and calculation of the SE of the difference

 ANOVA (one- and two-way) assumes that all the groups are sampled from populations that follow a Gaussian distribution, and that all these populations have the same standard deviation, even if the means differ. Based on this assumption, ANOVA computes a pooled standard deviation. This value is used in multiple comparison tests.

The ANOVA results in Prism (and most programs) don't report this pooled standard deviation. But it is easy to calculate. As part of the ANOVA table, Prism reports several Mean Square values. One of these is the residual Mean Square (some programs use the term error rather than residual). The mean square values are essentially variances. The square root of the residual Mean Square is the pooled SD. 

How is this a pooled SD?

First, review how a SD of one group is computed: Calculate the difference between each value and the group mean, square those differences, add them up, and divide by the number of degrees of freedom (df), which equals n-1. That value is the variance. Its square root is the SD.

To compute the pooled SD from several groups, calculate the difference between each value and its group mean, square those differences, add them all up (for all groups), and divide by the number of df, which equals the total sample size minus the number of groups. That value is the residual mean square of ANOVA. Its square root is the pooled SD.

This case study uses the concept of pooled SD. 
 

The pooled SD is used to compute the standard error of the difference used to compute multiple comparison tests. To compute this SE of the difference, multiply the pooled SD by the square root of the sum of the reciprocals of the two sample sizes.