Median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic feature of the median in describing data compared to the mean (often simply described as the "AVERAGE") is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of the center. Median income, for example, may be a better way to describe the center of the income distribution because increases in the largest incomes alone have no effect on the median. For this reason, the median is of central importance in robust statistics.
This article is about the statistical concept. For other uses, see Median (disambiguation).Formal definition and notation[edit]
Formally, a median of a population is any value such that at least half of the population is less than or equal to the proposed median and at least half is greater than or equal to the proposed median. As seen above, medians may not be unique. If each set contains more than half the population, then some of the population is exactly equal to the unique median.
The median is well-defined for any ordered (one-dimensional) data and is independent of any distance metric. The median can thus be applied to school classes which are ranked but not numerical (e.g. working out a median grade when student test scores are graded from F to A), although the result might be halfway between classes if there is an even number of classes. (For odd number classes, one specific class is determined as the median.)
A geometric median, on the other hand, is defined in any number of dimensions. A related concept, in which the outcome is forced to correspond to a member of the sample, is the medoid.
There is no widely accepted standard notation for the median, but some authors represent the median of a variable x as med(x), x͂,[3] as μ1/2,[1] or as M.[3][4] In any of these cases, the use of these or other symbols for the median needs to be explicitly defined when they are introduced.
The median is a special case of other ways of summarizing the typical values associated with a statistical distribution: it is the 2nd quartile, 5th decile, and 50th percentile.
Properties[edit]
Optimality property[edit]
The mean absolute error of a real variable c with respect to the random variable X is
[edit]
Interpolated median[edit]
When dealing with a discrete variable, it is sometimes useful to regard the observed values as being midpoints of underlying continuous intervals. An example of this is a Likert scale, on which opinions or preferences are expressed on a scale with a set number of possible responses. If the scale consists of the positive integers, an observation of 3 might be regarded as representing the interval from 2.50 to 3.50. It is possible to estimate the median of the underlying variable. If, say, 22% of the observations are of value 2 or below and 55.0% are of 3 or below (so 33% have the value 3), then the median is 3 since the median is the smallest value of for which is greater than a half. But the interpolated median is somewhere between 2.50 and 3.50. First we add half of the interval width to the median to get the upper bound of the median interval. Then we subtract that proportion of the interval width which equals the proportion of the 33% which lies above the 50% mark. In other words, we split up the interval width pro rata to the numbers of observations. In this case, the 33% is split into 28% below the median and 5% above it so we subtract 5/33 of the interval width from the upper bound of 3.50 to give an interpolated median of 3.35. More formally, if the values are known, the interpolated median can be calculated from
History[edit]
Scientific researchers in the ancient near east appear not to have used summary statistics altogether, instead choosing values that offered maximal consistency with a broader theory that integrated a wide variety of phenomena.[56] Within the Mediterranean (and, later, European) scholarly community, statistics like the mean are fundamentally a medieval and early modern development. (The history of the median outside Europe and its predecessors remains relatively unstudied.)
The idea of the median appeared in the 6th century in the Talmud, in order to fairly analyze divergent appraisals.[57][58] However, the concept did not spread to the broader scientific community.
Instead, the closest ancestor of the modern median is the mid-range, invented by Al-Biruni[59]: 31 [60] Transmission of his work to later scholars is unclear. He applied his technique to assaying currency metals, but, after he published his work, most assayers still adopted the most unfavorable value from their results, lest they appear to cheat.[59]: 35–8 [61] However, increased navigation at sea during the Age of Discovery meant that ship's navigators increasingly had to attempt to determine latitude in unfavorable weather against hostile shores, leading to renewed interest in summary statistics. Whether rediscovered or independently invented, the mid-range is recommended to nautical navigators in Harriot's "Instructions for Raleigh's Voyage to Guiana, 1595".[59]: 45–8
The idea of the median may have first appeared in Edward Wright's 1599 book Certaine Errors in Navigation on a section about compass navigation.[62] Wright was reluctant to discard measured values, and may have felt that the median — incorporating a greater proportion of the dataset than the mid-range — was more likely to be correct. However, Wright did not give examples of his technique's use, making it hard to verify that he described the modern notion of median.[56][60][b] The median (in the context of probability) certainly appeared in the correspondence of Christiaan Huygens, but as an example of a statistic that was inappropriate for actuarial practice.[56]
The earliest recommendation of the median dates to 1757, when Roger Joseph Boscovich developed a regression method based on the L1 norm and therefore implicitly on the median.[56][63] In 1774, Laplace made this desire explicit: he suggested the median be used as the standard estimator of the value of a posterior PDF. The specific criterion was to minimize the expected magnitude of the error; where is the estimate and is the true value. To this end, Laplace determined the distributions of both the sample mean and the sample median in the early 1800s.[26][64] However, a decade later, Gauss and Legendre developed the least squares method, which minimizes to obtain the mean. Within the context of regression, Gauss and Legendre's innovation offers vastly easier computation. Consequently, Laplaces' proposal was generally rejected until the rise of computing devices 150 years later (and is still a relatively uncommon algorithm).[65]
Antoine Augustin Cournot in 1843 was the first[66] to use the term median (valeur médiane) for the value that divides a probability distribution into two equal halves. Gustav Theodor Fechner used the median (Centralwerth) in sociological and psychological phenomena.[67] It had earlier been used only in astronomy and related fields. Gustav Fechner popularized the median into the formal analysis of data, although it had been used previously by Laplace,[67] and the median appeared in a textbook by F. Y. Edgeworth.[68] Francis Galton used the English term median in 1881,[69][70] having earlier used the terms middle-most value in 1869, and the medium in 1880.[71][72]
Statisticians encouraged the use of medians intensely throughout the 19th century for its intuitive clarity. However, the notion of median does not lend itself to the theory of higher moments as well as the arithmetic mean does, and is much harder to compute. As a result, the median was steadily supplanted as a notion of generic average by the arithmetic mean during the 20th century.[56][60]
This article incorporates material from Median of a distribution on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.