Tukey's Range Test - The Test Statistic

The Test Statistic

Tukey's test is based on a formula very similar to that of the t-test. In fact, Tukey's test is essentially a t-test, except that it corrects for experiment-wise error rate (when there are multiple comparisons being made, the probability of making a type I error increases — Tukey's test corrects for that, and is thus more suitable for multiple comparisons than doing a number of t-tests would be).

The formula for Tukey's test is:

where Y_A is the larger of the two means being compared, Y_B is the smaller of the two means being compared, and SE is the standard error of the data in question.

This q_s value can then be compared to a q value from the studentized range distribution. If the q_s value is larger than the q_critical value obtained from the distribution, the two means are said to be significantly different.

Since the null hypothesis for Tukey's test states that all means being compared are from the same population (i.e. μ₁ = μ₂ = μ₃ = ... = μ_n), the means should be normally distributed (according to the central limit theorem). This gives rise to the normality assumption of Tukey's test.