# Median

In probability theory and statistics, the median is a number that separates the highest half of a sample, a population, or a probability distribution from the lowest half. More precisely 1/2 of the population will have values less than or equal to the median and 1/2 of the population will have values equal to or greater than the median.

To find the median of a finite list of numbers, arrange all the observations from lowest value to highest value and pick the middle one. If there are an even number of observations, one often takes the mean of the two middle values.

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## Popular explanation

Suppose 19 paupers and one billionaire are in a room. Everyone removes all money from their pockets and puts it on a table. Each pauper puts $5 on the table; the billionaire puts$1 billion (that is, $109) there. The total is then$1,000,000,095. If that money is divided equally among the 20 persons, each gets $50,000,004.95. That amount is the mean (or "average") amount of money that the 20 persons brought into the room. But the median amount is$5, since one may divide the group into two groups of 10 persons each, and say that everyone in the first group brought in no more than $5, and each person in the second group brought in no less than$5. In a sense, the median is the amount that the typical person brought in. By contrast, the mean (or "average") is not at all typical, since no one present—pauper or billionaire—brought in an amount approximating \$50,000,004.95.

## Non-uniqueness: there may be more than one median

There may be more than one median: for example if there are an even number of cases then there is no unique middle value. Notice, however, that half the numbers in the list are less than or equal to either of the two middle values, and half are greater than or equal to either of the two values, and the same is true of any number between the two middle values. Thus either of the two middle values and all numbers between them are medians in that case.

## Measures of statistical dispersion

When the median is used as a location parameter in descriptive statistics, there are several choices for a measure of variability: the range, the interquartile range, and the absolute deviation. Since the median is the same as the second quartile, its calculation is illustrated in the article on quartiles.

## Medians of probability distributions

For any probability distribution on the real line with cumulative distribution function F, regardless of whether it is any kind of continuous probability distribution, in particular an absolutely continuous distribution (and therefore has a probability density function), or a discrete probability distribution, a median m satisfies the equality

[itex]P(X\leq m)=P(X\geq m)=\int_{-\infty}^m dF(x)[itex]

in which a Riemann-Stieltjes integral is used. For an absolutely continuous probability distribution with probability density function f, we have

[itex]P(X\leq m)=P(X\geq m)=\int_{-\infty}^m f(x)\, dx=0.5.[itex]

### Medians of particular distributions

• The median of a normal distribution with mean μ and variance σ2 is μ. In fact, for a normal distribution, mean = median = mode.
• The median of a uniform distribution in the interval [a, b] is (a + b) / 2.
• The median of a Cauchy distribution with location parameter x0 and scale parameter y is the location parameter.
• The median of an exponential distribution with parameter λ is the scale parameter times the natural log of 2, λln 2.
• The median of a Weibull distribution with shape parameter k and scale parameter λ is λ(log 2)1/k.

## Medians in descriptive statistics

The median is primarily used for skewed distributions, which it represents more accurately than the arithmetic mean. Consider the set { 1, 2, 2, 2, 3, 9 }. The median is 2 in this case, as is the mode, and it might be seen as a better indication of central tendency than the arithmetic mean of 3.166….

Calculation of medians is a popular technique in summary statistics and summarizing statistical data, since it is simple to understand and easy to calculate, while also giving a measure that is more robust in the presence of outlier values than is the mean.

## Theoretical properties

### An optimality property

The median is also the central point which minimises the average of the absolute deviations; in the example above this would be (1 + 0 + 0 + 0 + 1 + 7) / 6 = 1.5 using the median, while it would be 1.944 using the mean. In the language of probability theory, the value of c that minimizes

[itex]E(\left|X-c\right|)\,[itex]

is the median of the probability distribution of the random variable X.

### An inequality relating means and medians

For continuous probability distributions, the difference between the median and the mean is less than or equal to one standard deviation. See an inequality on location and scale parameters.

## Efficient computation

Even though sorting n items takes in general O(n log n) operations, by using a recursive "divide-and-conquer" algorithm the median of n items can be computed with only O(n) operations (in fact, you can always find the n-th element of a list of values with this method, this is called the selection problem).

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