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In ANOVA the assumption is made that the population variances of the different groups are the same. Assuming $H_0$ is correct, all population means are also the same. In this case, would it not be easier to simply assume that all observations come from the same single population? I do not see how this makes a difference in calculation, but conceptually a single population is easier to imagine and work with. Did I miss something?

Johannes Titz
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  • I don't really understand what you ask about. Could uou clarify? But see https://stats.stackexchange.com/questions/76151/what-is-an-intuitive-explanation-of-why-we-want-homoskedasticity-in-a-regression/409074#409074 – kjetil b halvorsen Apr 08 '21 at 13:36
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    "Come from the same single population" will work but is (far) more restrictive unless you also assume all distributions are Normal. – whuber Apr 08 '21 at 14:28
  • @whuber: and the normal distribution assumption is not always made? (I implicitly assumed it is.) – Johannes Titz Apr 08 '21 at 18:02
  • It is rare for the Normal distribution assumption to hold exactly. What matters is that the *sampling distributions* of the ANOVA statistics be close enough to their reference distributions (usually F ratio distributions) to produce reliable p-values. This permits a (much) broader application of ANOVA than would otherwise be possible. – whuber Apr 08 '21 at 18:07

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