The data sets in the following two individual value plots have exactly the same factor level means. Therefore, the variability in the data because of the factor is the same for both data sets. When you examine the plots, you might be tempted to conclude that the means are different in both cases. Notice, however, that the variability within factor levels is much greater in the second data set than in the first.

To assess the differences between means, you must compare these differences with the spread of the observations about the means. This is exactly what an analysis of variance does. Using analysis of variance, the p-value corresponding to the first plot is 0.000, whereas the p-value corresponding to the second plot is 0.109.

Therefore, using a significance level of 0.05, the test indicates that the means in the first data set are significantly different. The differences in the sample means for the second data set, however, could very well be a random result of the large overall variability in the data.