# Two estimates of variance (1 of 5)

Analysis of variance tests the null hypothesis that all the population means are equal:
H0: µ1 = µ2 = ... = µa
by comparing two estimates of variance (σ²). (Recall that σ² is the variance within each of the "a" treatment populations.) One estimate (called the Mean Square Error or "MSE" for short) is based on the variances within the samples. The MSE is an estimate of σ² whether or not the null hypothesis is true. The second estimate (Mean Square Between or "MSB" for short) is based on the variance of the sample means. The MSB is only an estimate of σ² if the null hypothesis is true. If the null hypothesis is false then MSB estimates something larger than σ². (A later section discusses exactly what MSB estimates when the null hypothesis is false.) The logic by which analysis of variance tests the null hypothesis is as follows: If the null hypothesis is true, then MSE and MSB should be about the same since they are both estimates of the same quantity (σ²); however, if the null hypothesis is false then MSB can be expected to be larger than MSE since MSB is estimating a quantity larger then σ².