Tukey Test

When the decision from the One-Way Analysis of Variance is to reject the null hypothesis, it means that at least one of the means isn't the same as the other means. What we need is a way to figure out where the differences lie, not just that there is a difference.
This is where the Scheffe' and Tukey tests come into play. They will help us analyze pairs of means to see if there is a difference -- much like the difference of two means covered earlier.

Hypotheses

Both tests are set up to test if pairs of means are different. The formulas refer to mean i and mean j. The values of i and j vary, and the total number of tests will be equal to a combination of k objects, 2 at a time C(k,2), where k is the number of samples.

Tukey Test

The Tukey test is only usable when the sample sizes are the same.
The Critical Value is looked up in a table. It is Table N in the Bluman text. There are actually several different tables, one for each level of significance. The number of samples, k, is used as a index along the top, and the degrees of freedom for the within group variance, v = N-k, are used as an index along the left side.
The test statistic is found by dividing the difference between the means by the square root of the ratio of the within group variation and the sample size.
Reject the null hypothesis if the absolute value of the test statistic is greater than the critical value (just like the linear correlation coefficient critical values).