- the values in each of the groups (as a whole) follow the normal curve,
- with possibly different population averages (though the null hypothesis is that all of the group averages are equal) and
- equal population standard deviations.

The assumption that the groups follow the normal curve is the usual one made in most significance tests, though here it is somewhat stronger in that it is applied to several groups at once. Of course many distributions do not follow the normal curve, so here is one reason that ANOVA may give incorrect results. It would be wise to consider whether it is reasonable to believe that the groups' distributions follow the normal curve.

Of course the different population averages imposes no restriction on the
use of ANOVA; the null hypothesis, as usual, allows us to do the computations
that yield *F*.

The third assumption, that the populations' standard deviations are equal, is
important in principle, and it can only be approximately checked by using
as bootstrap estimates the sample standard deviations. In practice,
statisticians feel safe in using ANOVA if the largest sample *SD* is
not larger than twice the smallest.