PlanetMath: median of a distribution

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median of a distribution

(Definition)

Given a probability distribution (density) function on $\Omega$ over a random variable , with the associated probability measure , a median of is a real number such that

The median is also known as the $50^{\text{th}}$ -percentile or the second quartile.

Examples:

An example from a discrete distribution. Let $\Omega=\mathbb{R}$ . Suppose the random variable has the following distribution: and . Then we can easily see the median is 0.
Another example from a discrete distribution. Again, let $\Omega=\mathbb{R}$ . Suppose the random variable has distribution and . Then we see that the median is not unique. In fact, all real values in the interval are medians.
In practice, however, the median may be calculated as follows: if there are numeric data points, then by ordering the data values (either non-decreasingly or non-increasingly),
1. the $(\frac{N+1}{2})$ -th data point is the median if is odd, and
2. the midpoint of the th and the th data points is the median if is even.
The median of a normal distribution (with mean $\mu$ and variance $\sigma^2$ ) is $\mu$ . In fact, for a normal distribution, mean = median = mode.
The median of a uniform distribution in the interval is .
The median of a Cauchy distribution with location parameter t and scale parameter s is the location parameter.
The median of an exponential distribution with location parameter $\mu$ and scale parameter $\beta$ is the scale parameter times the natural log of 2, $\beta\operatorname{ln}2$ .
The median of a Weibull distribution with shape parameter $\gamma$ , location parameter $\mu$ , and scale parameter $\alpha$ is $\alpha(\operatorname{ln}2)^{1/\gamma}+\mu$ .