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One of the numerical characteristics of the [[Probability distribution|probability distribution]] of a [[Random variable|random variable]]. For a random variable with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064340/m0643401.png" /> (cf. [[Density of a probability distribution|Density of a probability distribution]]), a mode is any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064340/m0643402.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064340/m0643403.png" /> is maximal. A mode is also defined for discrete distributions: If the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064340/m0643404.png" /> of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064340/m0643405.png" /> with distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064340/m0643406.png" /> are arranged in increasing order, then a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064340/m0643407.png" /> is called a mode if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064340/m0643408.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064340/m0643409.png" />.
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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 or multimodal. The most important in probability theory and mathematical statistics are the unimodal distributions (cf. [[Unimodal distribution|Unimodal distribution]]). Along with the [[Mathematical expectation|mathematical expectation]] and the [[Median (in statistics)|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064340/m06434010.png" />, the mode is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064340/m06434011.png" /> and to the median and to the mathematical expectation, if the latter exists.
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Distributions with one, two or more modes are called, respectively, [[Unimodal distribution|unimodal]] (one-peaked or single-peaked), [[Bimodal distribution|bimodal]] (doubly-peaked) or [[Multimodal distribution|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)|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.
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.M. Mood,  F.A. Graybill,  "Introduction to the theory of statistics" , McGraw-Hill  (1963)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Breiman,  "Statistics with a view towards applications" , Houghton Mifflin  (1973)  pp. 34–40</TD></TR></table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.M. Mood,  F.A. Graybill,  "Introduction to the theory of statistics" , McGraw-Hill  (1963)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Breiman,  "Statistics with a view towards applications" , Houghton Mifflin  (1973)  pp. 34–40</TD></TR>
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</table>
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Latest revision as of 21:49, 14 December 2015

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.


Comments

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. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Mode&oldid=16879
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article