Tukey's range test

Tukey's range test, also known as the Tukey's test, Tukey method, Tukey's honest significance test, Tukey's HSD (honest significant difference) test, or the Tukey–Kramer method, is a single-step multiple comparison procedure and statistical test. It can be used on raw data or in conjunction with an ANOVA (post-hoc analysis) to find means that are significantly different from each other. Named after John Tukey, it compares all possible pairs of means, and is based on a studentized range distribution (q) (this distribution is similar to the distribution of t from the t-test. See below). The Tukey HSD tests should not be confused with the Tukey Mean Difference tests (also known as the Bland–Altman diagram).

Tukey's test compares the means of every treatment to the means of every other treatment; that is, it applies simultaneously to the set of all pairwise comparisons

and identifies any difference between two means that is greater than the expected standard error. The confidence coefficient for the set, when all sample sizes are equal, is exactly 1 − α. For unequal sample sizes, the confidence coefficient is greater than 1 − α. In other words, the Tukey method is conservative when there are unequal sample sizes.

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 family-wise error rate (when there are multiple comparisons being made, the probability of making a Type I error within at least one of the comparisons, increases — Tukey's test corrects for that, and is thus more suitable for multiple comparisons than a number of t-tests would be).

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