Median

For other uses, see the statistical concept.

In probability theory and statistics, a median is described as the number separating 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, then there is no single middle value, so one often takes the mean of the two middle values.

In a sample of data, or a finite population, there may be no member of the sample whose value is identical to the median (in the case of an even sample size) and, if there is such a member, there may be more than one so that the median may not uniquely identify a sample member. Nonetheless the value of the median is uniquely determined with the usual definition,

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. For example, if a < b < c, then the median of the list {a, b, c} is b, and if a < b < c < d, then the median of the list {a, b, c, d} is the mean of b and c, i.e. it is (b + c)/2.

The median can be used as a measure of location when a distribution is skewed, when end values are not known, or when one requires reduced importance to be attached to outliers, e.g. because they may be measurement errors. A disadvantage of the median is the difficulty of handling it theoretically.

Notation

The median of some variable x is denoted either as ${\tilde {x}}$ or as $\mu _{1/2}(x)\,\!.$

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. Median is the middle value after arranging data by any order.

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 inequalities

$\operatorname {P} (X\leq m)\geq {\frac {1}{2}}{\text{ and }}\operatorname {P} (X\geq m)\geq {\frac {1}{2}}\,\!$

or

$\int _{-\infty }^{m}\mathrm {d} F(x)\geq {\frac {1}{2}}{\text{ and }}\int _{m}^{\infty }\mathrm {d} F(x)\geq {\frac {1}{2}}\,\!$

in which a Lebesgue–Stieltjes integral is used. For an absolutely continuous probability distribution with probability density function ƒ, we have

$\operatorname {P} (X\leq m)=\operatorname {P} (X\geq m)=\int _{-\infty }^{m}f(x)\,\mathrm {d} x={\frac {1}{2}}.\,\!$

Medians of particular distributions: The medians of certain types of distributions can be easily calculated 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 rate parameter λ is the natural logarithm of 2 divided by the rate parameter: λ⁻¹ln 2. The median of a Weibull distribution with shape parameter k and scale parameter λ is λ(ln 2)^1/k.

Medians in descriptive statistics

The median is primarily used for skewed distributions, which it summarizes 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.

The sample median

Efficient computation of the sample median

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).

Easy explanation of the sample median

For an odd number of values

As an example, we will calculate the sample median for the following set of observations: 1, 5, 2, 8, 7.

Start by sorting the values: 1, 2, 5, 7, 8.

In this case, the median is 5 since it is the middle observation in the ordered list.

For an even number of values

As an example, we will calculate the sample median for the following set of observations: 1, 5, 2, 8, 7, 2.

Start by sorting the values: 1, 2, 2, 5, 7, 8.

In this case, the average is (2 + 5)/2 = 3.5. Therefore, the median is 3.5 since it is the average of the middle observations in the ordered list.

Other estimates of the median

If data are represented by a statistical model specifying a particular family of probability distributions, then estimates of the median can be obtained by fitting that family of probability distributions to the data and calculating the theoretical median of the fitted distribution. See, for example Pareto interpolation.

Median-unbiased estimators, and bias with respect to loss functions

Any mean-unbiased estimator minimizes the risk (expected loss) with respect to the squared-error loss function, as observed by Gauss. A median-unbiased estimator minimizes the risk with respect to the absolute-deviation loss function, as observed by Laplace. Other loss functions are used in statistical theory, particularly in robust statistics.

The theory of median-unbiased estimators was revived by George W. Brown in 1947:

An estimate of a one-dimensional parameter θ will be said to be median-unbiased, if for fixed θ, the median of the distribution of the estimate is at the value θ, i.e., the estimate underestimates just as often as it overestimates. This requirement seems for most purposes to accomplish as much as the mean-unbiased requirement and has the additional property that it is invariant under one-to-one transformation. [page 584]

Further properties of median-unbiased estimators have been noted by Lehmann, Birnbaum, van der Vaart and Pfanzagl. In particular, median-unbiased estimators exist in cases where mean-unbiased and maximum-likelihood estimators do not exist. Besides being invariant under one-to-one transformations, median-unbiased estimators have surprising robustness.

In image processing

Main article: Median filter

In monochrome raster images there is a type of noise, known as the salt and pepper noise, when each pixel independently become black (with some small probability) or white (with some small probability), and is unchanged otherwise (with the probability close to 1). An image constructed of median values of neighborhoods (like 3×3 square) can effectively reduce a noise in this case.

In discrete high-dimensional statistical inference

In discrete high-dimensional statistical inference, the value $c$ that minimizes

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

is also called a centroid. A medoid is sometimes used when a centroid cannot be defined, it relates to the centroid like the median to the mean.

History

Gustav Fechner popularized the median into the formal analysis of data, although it had been used previously by Laplace.

See also

• Order statistic
• Quantile
• A median is the 2nd quartile, 5th decile, and 50th percentile.
• A sample-median is median-unbiased but can be a mean-biased estimator.
• Absolute deviation
• Concentration of measure for Lipschitz functions

• An inequality on location and scale parameters
• Median voter theory
• Median graph
• The centerpoint is a generalization of the median for data in higher dimensions.
• Median search

References

• Brown, George W. ”On Small-Sample Estimation.” The Annals of Mathematical Statistics, Vol. 18, No. 4 (Dec., 1947), pp. 582–585.
• Lehmann, E. L. “A General Concept of Unbiasedness” The Annals of Mathematical Statistics, Vol. 22, No. 4 (Dec., 1951), pp. 587–592.
• Allan Birnbaum. 1961. “A Unified Theory of Estimation, I”, The Annals of Mathematical Statistics, Vol. 32, No. 1 (Mar., 1961), pp. 112–135
• van der Vaart, H. R. 1961. “Some Extensions of the Idea of Bias” The Annals of Mathematical Statistics, Vol. 32, No. 2 (Jun., 1961), pp. 436–447.
• Pfanzagl, Johann. 1994. Parametric Statistical Theory. Walter de Gruyter.

External links

• A Guide to Understanding & Calculating the Median
• 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
• Python script for Median computations and income inequality metrics
• Difference between mean and median

Median of a distribution at PlanetMath.