median of a distribution

Given a probability distribution (density) function fX(x) on Ω over a random variableMathworldPlanetmath X, with the associated probability measureMathworldPlanetmath P, a median m of fX is a real number such that

  1. 1.


  2. 2.


The median is also known as the 50th-percentile or the second quartile.


  • An example from a discrete distribution. Let Ω=. Suppose the random variable X has the following distributionPlanetmathPlanetmath: P(X=0)=0.99 and P(X=1000)=0.01. Then we can easily see the median is 0.

  • Another example from a discrete distribution. Again, let Ω=. Suppose the random variable X has distribution P(X=0)=0.5 and P(X=1000)=0.5. Then we see that the median is not unique. In fact, all real values in the interval [0,1000] are medians.

  • In practice, however, the median may be calculated as follows: if there are N numeric data points, then by ordering the data values (either non-decreasingly or non-increasingly),

    1. (a)

      the (N+12)-th data point is the median if N is odd, and

    2. (b)

      the midpoint of the (N-1)th and the (N+1)th data points is the median if N is even.

  • The median of a normal distributionMathworldPlanetmath (with mean μ and varianceMathworldPlanetmath σ2) is μ. In fact, for a normal distribution, mean = median = mode.

  • The median of a uniform distributionMathworldPlanetmath in the interval [a,b] is (a+b)/2.

  • The median of a Cauchy distributionMathworldPlanetmath with location parameter t and scale parameter s is the location parameter.

  • The median of an exponential distributionMathworldPlanetmath with location parameter μ and scale parameter β is the scale parameter times the natural log of 2, βln2.

  • The median of a Weibull distributionMathworldPlanetmath with shape parameter γ, location parameter μ, and scale parameter α is α(ln2)1/γ+μ.

Title median of a distribution
Canonical name MedianOfADistribution
Date of creation 2013-03-22 14:24:10
Last modified on 2013-03-22 14:24:10
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 12
Author CWoo (3771)
Entry type Definition
Classification msc 60A99
Classification msc 62-07
Synonym second quartile
Defines median