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Median

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Median

In statistics and probability theory, the median is the numerical value separating the higher half of a data 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 (e.g., the median of {3, 3, 5, 9, 11} is 5). If there is an even number of observations, then there is no single middle value; the median is then usually defined to be the mean of the two middle values [1] [2] (the median of {3, 5, 7, 9} is (5 + 7) / 2 = 6), which corresponds to interpreting the median as the fully trimmed mid-range. The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data is contaminated, the median will not give an arbitrarily large result. A median is only defined on ordered one-dimensional data, and is independent of any distance metric. A geometric median, on the other hand, is defined in any number of dimensions.

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); 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. A related concept, in which the outcome is forced to correspond to a member of the sample, is the medoid. At most, half the population have values strictly less than the median, and, at most, half have values strictly greater than the median. If each group contains 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 {abc} is b, and, if a < b < c < d, then the median of the list {abcd} 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.

In terms of notation, some authors represent the median of a variable x either as \tilde{x} or as \mu_{1/2},[1] sometimes also M.[3] There is no widely accepted standard notation for the median,[4] so the use of these or other symbols for the median needs to be explicitly defined when they are introduced.

The median is the 2nd quartile, 5th decile, and 50th percentile.

Contents

• Measures of location and dispersion 1
• Probability distributions 2
• Medians of particular distributions 2.1
• Descriptive statistics 3
• Populations 4
• Optimality property 4.1
• Unimodal distributions 4.2
• Inequality relating means and medians 4.3
• Jensen's inequality for medians 5
• Medians for samples 6
• The sample median 6.1
• Efficient computation of the sample median 6.1.1
• Easy explanation of the sample median 6.1.2
• Variance 6.1.3
• Other estimators 6.2
• Coefficient of dispersion 6.3
• Multivariate median 7
• Marginal median 7.1
• Spatial median (L1 median) 7.2
• Other multivariate medians 7.3
• Other median-related concepts 8
• Pseudo-median 8.1
• Variants of regression 8.2
• Median filter 8.3
• Cluster analysis 8.4
• Median-Median Line 8.5
• Median-unbiased estimators 9
• History 10
• References 12

Measures of location and dispersion

The median is one of a number of ways of summarising the typical values associated with members of a statistical population; thus, it is a possible location parameter. Since the median is the same as the second quartile, its calculation is illustrated in the article on quartiles.

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.

For practical purposes, different measures of location and dispersion are often compared on the basis of how well the corresponding population values can be estimated from a sample of data. The median, estimated using the sample median, has good properties in this regard. While it is not usually optimal if a given population distribution is assumed, its properties are always reasonably good. For example, a comparison of the efficiency of candidate estimators shows that the sample mean is more statistically efficient than the sample median when data are uncontaminated by data from heavy-tailed distributions or from mixtures of distributions, but less efficient otherwise, and that the efficiency of the sample median is higher than that for a wide range of distributions. More specifically, the median has a 64% efficiency compared to the minimum-variance mean (for large normal samples), which is to say the variance of the median will be ~50% greater than the variance of the mean—see Efficiency (statistics)#Asymptotic efficiency and references therein.

Probability distributions

For any probability distribution on the real line R with cumulative distribution function F, regardless of whether it is any kind of continuous probability distribution, in particular an absolutely continuous distribution (which has a probability density function), or a discrete probability distribution, a median is by definition any real number m that satisfies the inequalities

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

or, equivalently, the inequalities

\int_{(-\infty,m]} dF(x) \geq \frac{1}{2}\text{ and }\int_ (\nu - m) where \nu is the population median—has been studied by several authors. The standard 'delete one' jackknife method produces inconsistent results.[15] An alternative—the 'delete k' method—where k grows with the sample size has been shown to be asymptotically consistent.[16] This method may be computationally expensive for large data sets. A bootstrap estimate is known to be consistent,[17] but converges very slowly (order of n^{-\frac{1}{4}}).[18] Other methods have been proposed but their behavior may differ between large and small samples.[19]

Efficiency

The efficiency of the sample median, measured as the ratio of the variance of the mean to the variance of the median, depends on the sample size and on the underlying population distribution. For a sample of size N = 2n + 1 from the normal distribution, the ratio is[20]

\frac{ 4n }{ \pi ( 2n + 1 ) }

For large samples (as n tends to infinity) this ratio tends to \frac{2}{\pi} .

Other estimators

For univariate distributions that are symmetric about one median, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population median.[21]

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. Pareto interpolation is an application of this when the population is assumed to have a Pareto distribution.

Coefficient of dispersion

The coefficient of dispersion (CD) is defined as the ratio of the average absolute deviation from the median to the median of the data.[22] It is a statistical measure used by the states of Iowa, New York and South Dakota in estimating dues taxes.[23][24][25] In symbols

CD = \frac{ 1 }{ n } \frac{ \sum| m - x | }{ m }

where n is the sample size, m is the sample median and x is a variate. The sum is taken over the whole sample.

Confidence intervals for a two sample test where the sample sizes are large have been derived by Bonett and Seier[22] This test assumes that both samples have the same median but differ in the dispersion around it. The confidence interval (CI) is bounded inferiorly by

\exp \left[ \log \left( \frac{ t_a } { t_b } \right) - z_\alpha \left( var \left[ \log \left( \frac{ t_a } { t_b } \right) \right] \right)^{ 0.5 } \right]

where tj is the mean absolute deviation of the jth sample, var() is the variance and zα is the value from the normal distribution for the chosen value of α: for α = 0.05, zα = 1.96. The following formulae are used in the derivation of these confidence intervals

var [ \log ( t_a ) ] = \frac{ \left( \frac{ s_a^2 } { t_a^2 } + \left( \frac{ x_a - \bar{ x } } { t_a } \right) ^2 - 1 \right) } { n }
var[ \log( t_a / t_b ) ] = var[ \log( t_a ) ] + var[ \log( t_b )] - 2r ( var[ \log( t_a ) ] var[ \log( t_b ) ] )^{0.5}

where r is the Pearson correlation coefficient between the squared deviation scores

d_{ia} = | x_{ia} - \bar{x}_{a} | and d_{ib} = |x_{ib} - \bar{x}_{b} |

a and b here are constants equal to 1 and 2, x is a variate and s is the standard deviation of the sample.

Multivariate median

Previously, this article discussed the concept of a univariate median for a one-dimensional object (population, sample). When the dimension is two or higher, there are multiple concepts that extend the definition of the univariate median; each such multivariate median agrees with the univariate median when the dimension is exactly one. In higher dimensions, however, there are several multivariate medians.[21]

Marginal median

The marginal median is defined for vectors defined with respect to a fixed set of coordinates. A marginal median is defined to be the vector whose components are univariate medians. The marginal median is easy to compute, and its properties were studied by Puri and Sen.[21][26]

Spatial median (L1 median)

In a normed vector space of dimension two or greater, the "spatial median" minimizes the expected distance

a \mapsto \mathrm{E}(\left\|X-a\right\|), \,

where X and a are vectors, if this expectation has a finite minimum; another definition is better suited for general probability-distributions.[9][21] The spatial median is unique when the data-set's dimension is two or more.[9][10][21] It is a robust and highly efficient estimator of a central tendency of a population.[27][21]

The Geometric median is the corresponding estimator based on the sample statistics of a finite set of points, rather than the population statistics. It is the point minimizing the arithmetic average of Euclidean distances to the given sample points, instead of the expectation. Note that the arithmetic average and sum are interchangeable since they differ by a fixed constant which does not alter the location of the minimum.

Other multivariate medians

An alternative generalization of the spatial median in higher dimensions that does not relate to a particular metric is the centerpoint.

Other median-related concepts

Pseudo-median

For univariate distributions that are symmetric about one median, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population median; for non-symmetric distributions, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population pseudo-median, which is the median of a symmetrized distribution and which is close to the population median. The Hodges–Lehmann estimator has been generalized to multivariate distributions.[28]

Variants of regression

The Theil–Sen estimator is a method for robust linear regression based on finding medians of slopes.

Median filter

In the context of image processing of monochrome raster images there is a type of noise, known as the salt and pepper noise, when each pixel independently becomes 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 noise in this case.

Cluster analysis

In cluster analysis, the k-medians clustering algorithm provides a way of defining clusters, in which the criterion of maximising the distance between cluster-means that is used in k-means clustering, is replaced by maximising the distance between cluster-medians.

Median-Median Line

This is a method of robust regression. The idea dates back to Wald in 1940 who suggested dividing a set of bivariate data into two halves depending on the value of the independent parameter x: a left half with values less than the median and a right half with values greater than the median.[29] He suggested taking the means of the dependent y and independent x variables of the left and the right halves and estimating the slope of the line joining these two points. The line could then be adjusted to fit the majority of the points in the data set.

Nair and Shrivastava in 1942 suggested a similar idea but instead advocated dividing the sample into three equal parts before calculating the means of the subsamples.[30] Brown and Mood in 1951 proposed the idea of using the medians of two subsamples rather the means.[31] Tukey combined these ideas and recommended dividing the sample into three equal size subsamples and estimating the line based on the medians of the subsamples.[32]

Median-unbiased estimators

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:[33]

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 reported.[34][35][36][37] In particular, median-unbiased estimators exist in cases where mean-unbiased and maximum-likelihood estimators do not exist. Median-unbiased estimators are invariant under one-to-one transformations.

History

The idea of the median originated in Edward Wright's book on navigation (Certaine Errors in Navigation) in 1599 in a section concerning the determination of location with a compass. Wright felt that this value was the most likely to be the correct value in a series of observations.

In 1757, Roger Joseph Boscovich developed a regression method based on the L1 norm and therefore implicitly on the median.[38]

In 1774, Laplace suggested the median be used as the standard estimator of the value of a posterior pdf. The specific criteria was to minimize the expected magnitude of the error; |α - α*| where α* is the estimate and α is the true value. Laplaces's criterion was generally rejected for 150 years in favor of the least squares method of Gauss and Legendgre which minimizes < (α - α*)2 > to obtain the mean. [39] The distribution of both the sample mean and the sample median were determined by Laplace in the early 1800s.[12][40]

Antoine Augustin Cournot in 1843 was the first 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.[41] 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.[41]

Francis Galton used the English term median in 1881,[42] having earlier used the terms middle-most value in 1869 and the medium in 1880.

References

1. ^ a b Weisstein, Eric W., "Statistical Median", MathWorld.
2. ^ http://www.stat.psu.edu/old_resources/ClassNotes/ljs_07/sld008.htm Simon, Laura J.; "Descriptive statistics", Statistical Education Resource Kit, Pennsylvania State Department of Statistics
3. ^ David J. Sheskin (27 August 2003). Handbook of Parametric and Nonparametric Statistical Procedures: Third Edition. CRC Press. pp. 7–.
4. ^ Derek Bissell (1994). Statistical Methods for Spc and Tqm. CRC Press. pp. 26–.
5. ^ Stroock, Daniel (2011). Probability Theory. Cambridge University Press. p. 43.
6. ^ http://www.se16.info/hgb/cheb2.htm#3unimodalinequalities
7. ^ Mallows, Colin (August 1991). "Another comment on O'Cinneide". The American Statistician 45 (3): 257.
8. ^ Piché, Robert (2012). Random Vectors and Random Sequences. Lambert Academic Publishing.
9. ^ a b c Kemperman, Johannes H. B. (1987). Dodge, Yadolah, ed. "Statistical data analysis based on the L1-norm and related methods: Papers from the First International Conference held at Neuchâtel, August 31–September 4, 1987". Amsterdam: North-Holland Publishing Co. pp. 217–230.
10. ^ a b Milasevic, Philip; Ducharme, Gilles R. (1987). "Uniqueness of the spatial median".
11. ^ Merkle, M. (2005). "Jensen’s inequality for medians". Statistics & Probability Letters 71 (3): 277–281.
12. ^ a b
13. ^ Rider, Paul R. (1960). "Variance of the median of small samples from several special populations".
14. ^ Stuart, Alan; Ord, Keith (1994). Kendall's Advanced Theory of Statistics. London: Arnold.
15. ^ Efron, B. (1982). The Jackknife, the Bootstrap and other Resampling Plans. Philadelphia: SIAM.
16. ^ Shao, J.; Wu, C. F. (1989). "A General Theory for Jackknife Variance Estimation".
17. ^ Efron, B. (1979). "Bootstrap Methods: Another Look at the Jackknife".
18. ^ Hall, P.; Martin, M. A. (1988). "Exact Convergence Rate of Bootstrap Quantile Variance Estimator". Probab Theory Related Fields 80 (2): 261–268.
19. ^ Jiménez-Gamero, M. D.; Munoz-García, J.; Pino-Mejías, R. (2004). "Reduced bootstrap for the median". Statistica Sinica 14 (4): 1179–1198.
20. ^ Kenney, J. F.; Keeping, E. S. (1962). "The Median". Mathematics of Statistics, Pt. 1 (3rd ed.). Princeton, NJ: Van Nostrand. pp. 211–212.
21. ^ a b c d e f Hettmansperger, Thomas P.; McKean, Joseph W. (1998). Robust nonparametric statistical methods. Kendall's Library of Statistics 5. London: John Wiley and Sons.
22. ^ a b Bonett DG, Seier E (2006) Conﬁdence interval for a coeﬃcient of dispersion in non-normal distributions. Biometrical Journal 48 (1) 144-148
23. ^ http://www.iowa.gov/tax/locgov/Statistical_Calculation_Definitions.pdf
24. ^ http://www.tax.ny.gov/research/property/reports/cod/2010mvs/reporttext.htm
25. ^ http://www.state.sd.us/drr2/publications/assess1199.pdf
26. ^ Puri, Madan L.; Sen, Pranab K.; Nonparametric Methods in Multivariate Analysis, John Wiley & Sons, New York, NY, 197l. (Reprinted by Krieger Publishing)
27. ^ Vardi, Yehuda; Zhang, Cun-Hui (2000). "The multivariate l1-median and associated data depth".
28. ^ Oja, Hannu (2010). Multivariate nonparametric methods with R: An approach based on spatial signs and ranks. Lecture Notes in Statistics 199. New York, NY: Springer. pp. xiv+232.
29. ^ Wald, A. (1940). "The Fitting of Straight Lines if Both Variables are Subject to Error".
30. ^ Nair, K. R.; Shrivastava, M. P. (1942). "On a Simple Method of Curve Fitting". Sankhyā: The Indian Journal of Statistics 6 (2): 121–132.
31. ^ Brown, G. W.; Mood, A. M. (1951). "On Median Tests for Linear Hypotheses". Proc Second Berkeley Symposium on Mathematical Statistics and Probability. Berkeley, CA: University of California Press. pp. 159–166.
32. ^ Tukey, J. W. (1977). Exploratory Data Analysis. Reading, MA: Addison-Wesley.
33. ^ Brown, George W. (1947). "On Small-Sample Estimation".
34. ^
35. ^
36. ^ van der Vaart, H. Robert (1961). "Some Extensions of the Idea of Bias".
37. ^ Pfanzagl, Johann; with the assistance of R. Hamböker (1994). Parametric Statistical Theory. Walter de Gruyter.
38. ^ Stigler, S. M. (1986). The History of Statistics: The Measurement of Uncertainty Before 1900. Harvard University Press.
39. ^ Jaynes, E.T. (2007). Probability theory : the logic of science (5. print. ed.). Cambridge [u.a.]: Cambridge Univ. Press. p. 172.
40. ^ Laplace PS de (1818) Deuxième supplément à la Théorie Analytique des Probabilités, Paris, Courcier
41. ^ a b Keynes, J.M. (1921) A Treatise on Probability. Pt II Ch XVII §5 (p 201) (2006 reprint, Cosimo Classics, ISBN 9781596055308 : multiple other reprints)
42. ^ Galton F (1881) "Report of the Anthropometric Committee" pp 245-260. Report of the 51st Meeting of the British Association for the Advancement of Science
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