          # Median

• finding the median in sets of data with an odd and even number of values

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. for example, in the data set [1, 3, 3, 6, 7, 8, 9], the median is 6, the fourth largest, and also the fourth smallest, number in the sample. for a continuous probability distribution, the median is the value such that a number is equally likely to fall above or below it.

the basic advantage of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed so much by a small proportion of extremely large or small values, and so it may give a better idea of a "typical" value. for example, in understanding statistics like household income or assets, which vary greatly, the mean may be skewed by a small number of extremely high or low values. median income, for example, may be a better way to suggest what a "typical" income is. because of this, 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 are contaminated, the median will not give an arbitrarily large or small result.

• finite data set of numbers
• probability distributions
• populations
• jensen's inequality for medians
• medians for samples
• multivariate median
• other median-related concepts
• median-unbiased estimators
• history
## This article is about the statistical concept. For other uses, see Median (disambiguation). Finding the median in sets of data with an odd and even number of values 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. For example, in the data set [1, 3, 3, 6, 7, 8, 9], the median is 6, the fourth largest, and also the fourth smallest, number in the sample. For a continuous probability distribution, the median is the value such that a number is equally likely to fall above or below it. The basic advantage of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed so much by a small proportion of extremely large or small values, and so it may give a better idea of a "typical" value. For example, in understanding statistics like household income or assets, which vary greatly, the mean may be skewed by a small number of extremely high or low values. Median income, for example, may be a better way to suggest what a "typical" income is. Because of this, 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 are contaminated, the median will not give an arbitrarily large or small result. Contents 1 Finite data set of numbers 2 Probability distributions 2.1 Medians of particular distributions 3 Populations 3.1 Optimality property 3.2 Unimodal distributions 3.3 Inequality relating means and medians 4 Jensen's inequality for medians 5 Medians for samples 5.1 The sample median 5.1.1 Efficient computation of the sample median 5.1.2 Easy explanation of the sample median 5.1.3 Sampling distribution 5.2 Other estimators 5.3 Coefficient of dispersion 6 Multivariate median 6.1 Marginal median 6.2 Geometric median 6.3 Centerpoint 7 Other median-related concepts 7.1 Interpolated median 7.2 Pseudo-median 7.3 Variants of regression 7.4 Median filter 7.5 Cluster analysis 7.6 Median–median line 8 Median-unbiased estimators 9 History 10 See also 11 References 12 External links  