Median

The middle value of the collection once sorted.

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Median(collection)the middle value of the collection once sorted.

SageMathmedian(collection)

Median(matrix)the column-wise median, one value per column.

SageMathmedian(matrix)

Median(dist)the exact quantile at 1/2 of a distribution, where one exists; otherwise unevaluated.

SageMathmedian(dist)

Typed signatures

Type	Package	Applies
`((collection<nothing \| number> \| number)+) -> nan \| real \| signed_infinity`	`compute-engine`	always
`((collection<nothing \| number> \| list<list<any>> \| number)*) -> list \| nan \| real \| signed_infinity`	`combinatorics`replaces `compute-engine`	always
`((collection<nothing \| number> \| distribution \| list<list<any>> \| number)*) -> list \| nan \| number \| real \| signed_infinity`	`statistics`replaces `combinatorics`	always

Domain: Collections

Details

For an odd-length collection, the median is the middle element of Sort's result; for even length, it's the average of the two middle elements.
Much less sensitive to outliers than Mean — a single extreme value barely moves it.
Given a matrix (a list of equal-length rows), computes the median of each column.
See Mode for the most frequent value.

Examples

Basic #

In

\mathrm{median}(\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack)

Out

3

In

\mathrm{median}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack)

Out

\frac{5}{2}

In

\mathrm{median}(\bigl\lbrack1, 2, 3, 4, 5, 6, 7\bigr\rbrack)

Out

4

In

\mathrm{median}(\bigl\lbrack1, 2, 3, 4, 5, 6, 7, 8\bigr\rbrack)

Out

\frac{9}{2}

Scope #

Given a matrix, Median is column-wise

In

\mathrm{median}(\bigl\lbrack\bigl\lbrack1, 11, 3\bigr\rbrack, \bigl\lbrack4, 6, 7\bigr\rbrack\bigr\rbrack)

Out

\bigl\lbrack\frac{5}{2}, \frac{17}{2}, 5\bigr\rbrack

Unsorted data is sorted first; the middle pair 3 and 4 averages to $\frac{7}{2}$

In

\mathrm{median}(\bigl\lbrack3, 1, 4, 1, 5, 9, 2, 6\bigr\rbrack)

Out

\frac{7}{2}

Exact rationals

In

\mathrm{median}(\bigl\lbrack\frac{1}{2}, \frac{1}{3}, \frac{1}{6}\bigr\rbrack)

Out

\frac{1}{3}

Exact constants are ordered by value, putting $e$ in the middle

In

\mathrm{median}(\bigl\lbrack π, e, 2\bigr\rbrack)

Out

e

The median of a lazy Range of even length is the mean of the middle two

In

\mathrm{median}(1..100)

Out

\frac{101}{2}

The median of a 30-digit numeric list is a float

In

\mathrm{median}(N(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, 30))

Out

2.5

Columnwise over a three-dimensional array of data

In

\mathrm{median}(\bigl\lbrack\bigl\lbrack\bigl\lbrack3, 7\bigr\rbrack, \bigl\lbrack2, 1\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack5, 19\bigr\rbrack, \bigl\lbrack12, 4\bigr\rbrack\bigr\rbrack\bigr\rbrack)

Out

\bigl\lbrack\bigl\lbrack4, 13\bigr\rbrack, \bigl\lbrack7, \frac{5}{2}\bigr\rbrack\bigr\rbrack

Applications #

The same outlier-laden data as in Mean's example: the median of 1 barely moves, unlike the mean's $-\frac{38}{3}$

In

\mathrm{median}(\bigl\lbrack-100, 1, 1, 1, 1, 20\bigr\rbrack)

Out

1

Robust to the outlier $100000$ , unlike the mean

In

\mathrm{median}(\bigl\lbrack1, 5, 2, 6, 10, 100\,000, 5, 4\bigr\rbrack)

Out

5

Properties #

For an odd-length list, the median is the middle element of the sorted list. See Sort

In

\mathrm{median}(\bigl\lbrack5, 3, 1, 4, 2\bigr\rbrack)=\mathrm{Sort}(\bigl\lbrack5, 3, 1, 4, 2\bigr\rbrack)[3]

Out

\mathrm{True}

Even count: the average of the two middle values, $2$ and $3$

In

N(\mathrm{median}(\bigl\lbrack1, 2, 3, 4, 4, 3, 1, 1\bigr\rbrack))

Out

2.5

Stays an exact rational for exact data

In

\mathrm{median}(\bigl\lbrack1, 2, 3, 4, 4, 3, 1, 1\bigr\rbrack)

Out

\frac{5}{2}

About

Examples	16
Checked against	Wolfram Language 16 agreeSageMath 12 agree2 error
Crosswalk	Wolfram LanguageSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts

See also: Mean, Mode