## Saturday, August 25, 2007

### Median

Median

In probability theory and statistics, a median is a type of average that is described as the number dividing the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one. If there is an even number of observations, the median is not unique, so one often takes the mean of the two middle values.

At most half the population have values less than the median and at most half have values greater than the median. If both groups contain less than half the population, then some of the population is exactly equal to the median.
Contents
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* 1 Popular explanation
* 2 Non-uniqueness
* 3 Measures of statistical dispersion
* 4 Medians of probability distributions
o 4.1 Medians of particular distributions
* 5 Medians in descriptive statistics
* 6 Theoretical properties
o 6.1 An optimality property
o 6.2 An inequality relating means and medians
* 7 Efficient computation

 Popular explanation

The big difference between the median and mean is illustrated in a simple example.

Suppose 19 paupers and 1 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 (i.e.£109) there. The total is then £1,000,000,095. If that money is divided equally among the 20 people, each gets £50,000,004.75. That amount is the mean amount of money that the 20 people brought into the room. But the median amount is £5, since one may divide the group into two groups of 10 people 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 is not at all typical, since nobody in the room brought in an amount approximating £50,000,004.75.

 Non-uniqueness

There may be more than one median: for example if there are an even number of cases, and the two middle values are different, then there is no unique middle value. Notice, however, that at least half the numbers in the list are less than or equal to either of the two middle values, and at least 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, the mean absolute deviation, and the median absolute deviation. Since the median is the same as the second quartile, its calculation is illustrated in the article on quartiles.

Working with computers, a population of integers should have an integer median. Thus, for an integer population with an even number of elements, there are two medians known as lower median and upper median. For floating point population, the median lies somewhere between the two middle elements, depending on the distribution.

 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 inequality

P(X\leq m) \geq 1/2 \leq P(X\geq m) \,\!

or

\int_{-\infty}^m \mathrm{d}F(x) \geq 1/2 \leq \int_m^{\infty} \mathrm{d}F(x)\,\!

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

P(X\leq m)=P(X\geq m)=\int_{-\infty}^m f(x)\, \mathrm{d}x=0.5.\,\!

 Medians of particular distributions

The medians of certain types of distributions can be easily estimated from their parameters:

* 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, which is also the mean.
* The median of a Cauchy distribution with location parameter x0 and scale parameter y is x0, the location parameter.
* The median of an exponential distribution with parameter λ is the natural log of 2 divided by the scale parameter, (1/λ)ln 2.
* The median of a Weibull distribution with shape parameter k and scale parameter λ is (1/λ)[ln(2)]1/k.

 Medians in descriptive statistics

The median is primarily used for skewed distributions, which it represents differently than the arithmetic mean. Consider the multiset { 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 minimizes 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

E(\left|X-c\right|)\,

is the median of the probability distribution of the random variable X. Note, however, that c is not always unique, and therefore not well defined in general.

 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 "divide and conquer" algorithm the median of n items can be computed with only O(n) operations (in fact, you can always find the k-th element of a list of values with this method; this is called the selection problem).

* Geometric median
* Order statistic
* An inequality on location and scale parameters
* The median is the 2nd quartile, 5th decile, and 50th percentile.
* Median voter theory
* The median in general is a biased estimator.

* Median as a weighted arithmetic mean of all Sample Observations
* On-line calculator
* Calculating the median
* A problem involving the mean, the median, and the mode.
* mathworld: Statistical Median

Category: Means
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