Comparing several groups
When data are sampled from more than two groups, we usually want to test a null hypothesis that all group means are equal. The alternative hypothesis is that at least two of the groups have different means.
Variation between and within groups
If the model means are all equal, it would be expected that the sample means would be similar. However they are unlikely to be identical. We therefore need to assess whether the variation between the group means is unusually great. To do this, we must also take account of the variation within the groups.
Signal and noise
Variation between and within groups correspond to the ideas of signal and noise in the data.