I am trying to understand ANOVA. In one-way ANOVA to test equality of mean of several population groups, each group has a normal distribution with the same variance. I wonder if the common variance of each group is known and used somewhere or unknown and estimated from samples in ANOVA? Thanks and regards!

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#### Best Answer

Standard ANOVA estimates the common variance from the data. It uses a "pooled" estimate by taking the differences between each data point and the group mean for that data point, squaring those differences, summing, then dividing by the degrees of freedom (which is the number of data points minus the number of groups, for simple one-way ANOVA).

It would be possible to derive an equivalent of ANOVA where the common variance is known (probably based on the $chi^2$ distribution ranther than the F), but the liklihood of finding a real world situation where the variance is known but the means are not is low enough that most people don't worry about that situation.

In truth it is probably never true that the populations are exactly normal or that the variances are exactly equal. The Central Limit Theorem covers the normality assumption for normal enough data and big enough sample sizes. The ANOVA tests have been shown to be fairly robust to the assumption that the variances are equal as long as they are at least similar (a common rule of thumb is that ANOVA is ok as long as the ratio of the largest variance and smallest variance is less than 4).