Tukey's range test

Tukey's range test, also known as Tukey's test, Tukey method, Tukey's honest significance test, or Tukey's HSD (honestly significant difference) test, is a single-step multiple comparison procedure and statistical test. It can be used 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).

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

${\displaystyle \mu _{i}-\mu _{j}\,}$

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 ${\displaystyle 1-\alpha }$ for any ${\displaystyle 0\leq \alpha \leq 1}$. 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.

This test is often followed by the Compact Letter Display (CLD) statistical procedure to render the output of this test more transparent to non-statistician audiences.