# All Pairwise Comparisons among Means: Newman-Keuls Procedure (1 of 4)

The Newman-Keuls method, like the Tukey HSD, is based on the studentized range distribution. Consider an experiment in which there are five treatment conditions. First the means are rank ordered from smallest to largest. Then, the smallest mean is compared to the largest mean using the studentized t. If the test is not significant, then no pairwise tests are significant and no more testing is done. So far, the Newman-Keuls method is exactly the same as the Tukey HSD. If the difference between the largest mean and the smallest mean is significant, then the difference between the smallest mean (Mean 1) and the second largest mean (Mean 4) as well as the difference between the largest mean (Mean 5) and the second smallest mean (Mean 2) are tested. Unlike the Tukey HSD, these comparisons are done using a critical value based on only four means rather than all five. The rationale is that the comparison of Mean 1 to Mean 4 only spans four means so the lower critical value associated with four rather than five means is used. The basic idea is that when a comparison that spans k means is significant, comparisons that span k-1 means within the original span of k means are performed.