NECESSARY ASSUMPTIONS

The data must meet two conditions for you to use analysis of variance:

1. Each of the groups must be a random sample from a normal population.

2. In the population, the variances in all groups must be equal.

You can visually check these conditions by making a histogram of the data for each group and seeing whether the data are approximately normal. To check whether the groups have the same variance in the population, you can examine the histograms as well as compute the variances for each of the groups and compare them. In practice, analysis of variance gives good results even if the normality assumption does not quite hold. If the number of observations in each of the groups is fairly similar, the equal variance assumption is also not too important. The assumption of random samples, however, is always important and cannot be relaxed .

In analysis of variance, the observed variability in the sample is divided, or partitioned, into two parts : variability of the observations within a group (around the group mean) and variability between the group means. Why are we talking about variability? Are we not testing hypotheses about means? Previously we mentioned that a relationship exists between variability of observations (in the population) and variability of sample means. If you know the standard deviation of the observations, you can estimate how much the sample means should vary. In your study, you have several different groups (for example, in our earlier example of the ride, we had individuals who found it comfortable, normal and rough). If the null hypothesis is true (that is, if all three groups have the same mean in the population), you can estimate how much observed means should vary due to sampling variation alone. If the means you actually observe vary more than you would expect from sampling variation, you have reason to believe that this extra variability is due to the fact that some of the groups do not have the same mean in the population.