# Difference between revisions of "Mode"

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− | One of the numerical characteristics of the [[ | + | 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. [[ | + | 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> | + | <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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## Latest revision as of 22:49, 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