The F ratio is the ratio of two mean square values. The square root of the mean square residual can be thought of as the pooled standard deviation. Adjusted for the size of each group, this becomes the treatment sum-of-squares.Įach sum-of-squares is associated with a certain number of degrees of freedom (df, computed from number of subjects and number of groups), and the mean square (MS) is computed by dividing the sum-of-squares by the appropriate number of degrees of freedom. Variation among groups (due to treatment) is quantified as the sum of the squares of the differences between the group means and the grand mean (the mean of all values in all groups). Variability within groups (within the columns) is quantified as the sum of squares of the differences between each value and its group mean. The P value is computed from the F ratio which is computed from the ANOVA table.ĪNOVA partitions the variability among all the values into one component that is due to variability among group means (due to the treatment) and another component that is due to variability within the groups (also called residual variation). Look at the results of post tests to identify where the differences are. This doesn't mean that every mean differs from every other mean, only that at least one differs from the rest. You can reject the idea that all the populations have identical means. If the overall P value is small, then it is unlikely that the differences you observed are due to random sampling. You just don't have compelling evidence that they differ. This is not the same as saying that the true means are the same. Even if the population means were equal, you would not be surprised to find sample means this far apart just by chance. If the overall P value is large, the data do not give you any reason to conclude that the means differ.
If all the populations really have the same mean (the treatments are ineffective), what is the chance that random sampling would result in means as far apart (or more so) as observed in this experiment? Therefore, the P value answers this question: The P value tests the null hypothesis that data from all groups are drawn from populations with identical means. One-way ANOVA compares three or more unmatched groups, based on the assumption that the populations are Gaussian.