To look for differences in more than 2 sample means an ANOVA is used. SST (between groups) and SSE (within groups) is calculated. F = MST/MSE. I get all this, however why can we say that there are differences between sample means when MST > MSE? When the variance between groups is larger than variance within groups?
Exactly. When the variance of the means of the groups is greater than the variance within the groups, we have evidence that the population means are spread out and not equal. If the group means have about as much variability as the data do in general, then of course the observed group means are spread out some.
Further explanation: Under the null hypothesis, the group means are the same. Thus, you’re essentially sampling from the same distribution three times. I just sampled 10 observations from a standard normal three times and got means of 0.143, 0.451, and -0.248. There’s variability, but those three values have a variance of 0.116. Sticking those values into the ANOVA equations, we get an F-statistic of 1.65 and a p-value of 0.211. There is no evidence that the three groups have different means, not should there be.
However, as one of the groups gets a different mean, the spread of the observed means is going to get to be bigger. Imagine that the observed means had been 0.143, 0.451, and 600. Suddenly the spread of those three values is very large, not enough that the sampling variability accounts for the spread, and I would reject the null hypothesis of equal means.
Here is a very simple explanation.
Basically, if there's more variation between groups than within groups, that's evidence that the groups are different.