Studentized range

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The studentized range, q, is the difference between the largest and smallest values in a sample, divided by the sample standard deviation. It was introduced in 1927 by William Sealy Gosset (the mathematician known as "Student") and later discussed by Newman, Keuls, and Tukey. The distribution of q is called the studentized range distribution and it is used in several multiple comparison methods (such as Tukey’s range test, the Newman–Keuls method, and Duncan’s step-down procedure) and for building confidence intervals that stay valid after looking at the data.

What you need to know:
- q is built from n groups and a spread measure s that is independent of the data points. In practice, the setup often involves means x1, ..., xn from samples of size m, with s^2 as the pooled variance. The degrees of freedom are ν = n(m − 1).
- The critical value of q comes from the studentized range distribution with parameters n and ν.
- The distribution of q does not depend on the actual mean or standard deviation of the population; it is the same for any normal distribution after standardizing.
- “Studentized” means the data are scaled by an estimate of the population standard deviation. Since s varies from sample to sample, this adds extra uncertainty and makes the distribution of q more complex.

In short: the studentized range measures how far the sample’s extreme values are relative to the data’s spread, and its distribution helps compare multiple group means while accounting for sampling variability.