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One of the numerical characteristics of the probability distribution of a random variable. For a random variable with density (cf. Density of a probability distribution), a mode is any point where is maximal. A mode is also defined for discrete distributions: If the values of a random variable with distribution are arranged in increasing order, then a point is called a mode if and .

Distributions with one, two or more modes are called, respectively, unimodal (one-peaked or single-peaked), bimodal 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 , the mode is equal to and to the median and to the mathematical expectation, if the latter exists.


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References

[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. A.V. Prokhorov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Mode&oldid=16879
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098