A probability density function characterizes continuous random variables by assigning a probability density, rather than an exact probability, to possible values. Since the exact probability of any single continuous value is zero, determining the probability that the variable falls within a specific interval requires calculating the definite integral of the density function over that interval.

Probability Density Function

The continuous probability distribution of the studentized range of an i.i.d. sample from a normally distributed population.
Differentiating the cumulative distribution function with respect to q gives the probability density function.

University of Michigan - Ann Arbor

Single-step multiple comparison procedure and statistical test. It can be used to find means that are significantly different from each other.
Compares all possible pairs of means, and is based on a studentized range distribution

Tukey-Kramer Method

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (Vol. 112, pp. 3-7). New York: springer.

An Introduction to Statistical Learning with Applications in R

Conducting multiple independent-samples $$t$$-tests to compare every pair of group means in an experiment creates a statistical problem. While a single $$t$$-test has a $$5\%$$ chance of mistakenly rejecting a true null hypothesis (a Type I error), conducting several $$t$$-tests simultaneously causes this risk to compound. As more tests are run, the overall probability of mistakenly rejecting at least one true null hypothesis becomes unacceptably high, necessitating specialized statistical procedures to control the error rate.

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