BETWEEN-GROUPS VARIABILITY

We noted earlier that a relationship exists between the variability of the observations in a population and the variability of sample means from that population. If you divide the standard deviation of the observations by the square root of the number of observations, you have an estimate of the standard deviation of the sample means, also known as the standard error. So if you know what the standard error of the mean is, you can estimate what the standard deviation of the original observations must be. You just multiply the standard error by the square root of the number of cases to get an estimate of the standard deviation of the observations. You square this to get an estimate of the variance.

Using this insight, you can obtain an estimate of the variance based on between-groups variability . You have a sample mean for each of the groups, and you can compute how much these means vary. If the population mean is the same in all three groups, you can use the variability between the sample means (and the sizes of the sample groups) to estimate the variability of the original observations. Of course, this estimate depends on whether the population means really are the same in all three groups ” which is the null hypothesis. If the null hypothesis is true, the between-groups estimate is correct. However, if the groups have different means in the population, then the between groups estimate (the estimate of variability based on the group means) will be too large.