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One of the numerical characteristics of the probability distribution of a random variable. For a random variable with density $p(x)$ (cf. Density of a probability distribution), a mode is any point $x_0$ where $p(x)$ is maximal. A mode is also defined for discrete distributions: If the values $x_k$ of a random variable $X$ with distribution $p_k = \mathsf{P}(X = x_k)$ are arranged in increasing order, then a point $x_m$ is called a mode if $p_m \ge p_{m-1}$ and $p_m \ge p_{m+1}$.

Distributions with one, two or more modes are called, respectively, unimodal (one-peaked or single-peaked), bimodal (doubly-peaked) or multimodal. The most important in probability theory and mathematical statistics are the unimodal distributions (cf. Unimodal distribution). Along with the mathematical expectation and the median (in statistics) the mode acts as a measure of location of the values of a random variable. For distributions which are unimodal and symmetric with respect to some point $a$, the mode is equal to $a$ and to the median and to the mathematical expectation, if the latter exists.



[a1] A.M. Mood, F.A. Graybill, "Introduction to the theory of statistics" , McGraw-Hill (1963)
[a2] L. Breiman, "Statistics with a view towards applications" , Houghton Mifflin (1973) pp. 34–40
How to Cite This Entry:
Mode. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article