Dictionary Definition

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 most 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
or
in which a Riemann-Stieltjes integral is used. For an absolutely continuous probability distribution with probability density function f, we have
\operatorname(X\leq m) = \operatorname(X\geq m)=\int_^m f(x)\, \mathrmx=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 rate parameter \lambda is the natural log of 2 divided by the rate parameter: \ln 2 /\lambda. The median of a Weibull distribution with shape parameter k and scale parameter \lambda is \lambda (\ln 2)^.

Medians in descriptive statistics

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

Easy explanation (Statistics)

As an example, we will calculate the median of the following population of numbers: 1, 5, 2, 8, 7.
Start by sorting the numbers: 1, 2, 5, 7, 8.
In this case, 5 is the median, because when the numbers are sorted, it is the middle number. If there is an even amount of numbers, the median is the arithmetic mean of the two middle numbers.

median in Arabic: وسيط حسابي
median in Bulgarian: Медиана (статистика)
median in Catalan: Mediana
median in Czech: Medián
median in Danish: Median
median in German: Median
median in Estonian: Mediaan
median in Esperanto: Mediano (statistiko)
median in Basque: Mediana
median in Persian: میانه (آمار)
median in French: Médiane (centre)
median in Galician: Mediana
median in Croatian: Mediana
median in Icelandic: Miðgildi
median in Italian: Mediana (statistica)
median in Hebrew: חציון
median in Lithuanian: Mediana
median in Hungarian: Medián
median in Dutch: Mediaan (statistiek)
median in Japanese: 中央値
median in Norwegian: Median
median in Polish: Mediana
median in Portuguese: Mediana (estatística)
median in Russian: Медиана (статистика)
median in Sicilian: Mediana
median in Simple English: Median
median in Slovak: Medián
median in Slovenian: Mediana
median in Serbian: Медијана (статистика)
median in Sundanese: Median
median in Finnish: Mediaani
median in Swedish: Median
median in Tamil: இடைநிலையளவு
median in Thai: มัธยฐาน
median in Vietnamese: Số trung vị
median in Tajik: Медиана
median in Turkish: Medyan
median in Chinese: 中位數