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Encyclopedia > Median

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 it, or if the median is not unique, one often takes the mean of the two middle values. Look up median in Wiktionary, the free dictionary. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... This article is about the field of statistics. ... In probability theory, every random variable may be attributed to a function defined on a state space equipped with a probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied. ... In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ...


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.

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Popular explanation

The big difference between the median and mean is illustrated in a simple example.


Suppose 19 paupers and 1 billionaire are in a room. Everyone removes all money from their pockets and puts it on a table. Each pauper puts $5 on the table; the billionaire puts $1 billion (i.e. $109) there. The total is then $1,000,000,095. If that money is divided equally among the 20 people, each gets $50,000,004.75. That amount is the mean amount of money that the 20 people brought into the room. But the median amount is $5, since one may divide the group into two groups of 10 people each, and say that everyone in the first group brought in no more than $5, and each person in the second group brought in no less than $5. In a sense, the median is the amount that the typical person brought in. By contrast, the mean is not at all typical, since nobody in the room brought in an amount approximating $956,999,659,999 In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ...


Non-uniqueness

There may be more than one median: for example if there are an even number of cases, and the two middle values are different, then there is no unique middle value. Notice, however, that at least half the numbers in the list are less than or equal to either of the two middle values, and at least half are greater than or equal to either of the two values, and the same is true of any number between the two middle values. Thus either of the two middle values and all numbers between them are medians in that case.


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. In descriptive statistics, the range is the length of the smallest interval which contains all the data. ... In descriptive statistics, the interquartile range (IQR), also called the midspread and middle fifty is the range between the third and first quartiles and is a measure of statistical dispersion. ... The absolute deviation of an element of a data set is the absolute difference between that element and a given point. ... In statistics, the median absolute deviation (or MAD) is a resistant measure of the variability of a univariate sample. ... In descriptive statistics, a quartile is any of the three values which divide the sorted data set into four equal parts, so that each part represents 1/4th of the sample or population. ...


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.So if there is not a middle number and there is two numbers left that is an example


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 In probability theory, every random variable may be attributed to a function defined on a state space equipped with a probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... // Absolute continuity of real functions In mathematics, a real-valued function f of a real variable is absolutely continuous on a specified finite or infinite interval if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

operatorname{P}(Xleq m) geq frac{1}{2} quadandquad operatorname{P}(Xgeq m) geq frac{1}{2},!

or

int_{-infty}^m mathrm{d}F(x) geq frac{1}{2} quadandquad int_m^{infty} mathrm{d}F(x) geq frac{1}{2},!

in which a Riemann-Stieltjes integral is used. For an absolutely continuous probability distribution with probability density function f, we have In mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

operatorname{P}(Xleq m) = operatorname{P}(Xgeq m)=int_{-infty}^m f(x), mathrm{d}x=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 parameter λ is the natural log of 2 divided by the scale parameter: frac{ln 2}{lambda}The median of a Weibull distribution with shape parameter k and scale parameter λ is frac{(ln 2)^{1/k}}{lambda} The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In mathematics, the uniform distributions are simple probability distributions. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... In probability theory and statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous probability distribution with the probability density function where and is the shape parameter and is the scale parameter of the distribution. ...


Medians in descriptive statistics

The median is primarily used for skewed distributions, which it represents differently than the arithmetic mean. Consider the multiset { 1, 2, 2, 2, 3, 9 }. The median is 2 in this casecentral tendency lesser than the arithmetic mean of 3.166…. Example of experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... In mathematics, a multiset (or bag) is a generalization of a set. ... In statistics, central tendency is an average of a set of measurements, the word average being variously construed as mean, median, or other measure of location, depending on the context. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...


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. In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate as much as possible as simply as possible. ... Descriptive statistics is a branch of statistics that denotes any of the many techniques used to summarize a set of data. ... Figure 1. ... In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic 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-cright|),

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. In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ...


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. In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... For probability distributions having an expected value and a median, the mean (i. ...


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). In computer science and mathematics, a sorting algorithm is an algorithm that puts elements of a list in a certain order. ... For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ... In computer science, divide and conquer (D&C) is an important algorithm design paradigm. ... For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ... In computer science, a selection algorithm is an algorithm for finding the kth smallest number in a list, called order statistics. ...


See also

The geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. ... Probability distributions for the n = 5 order statistics of an exponential distribution with θ = 3. ... For probability distributions having an expected value and a median, the mean (i. ... In descriptive statistics, a quartile is any of the three values which divide the sorted data set into four equal parts, so that each part represents 1/4th of the sample or population. ... In descriptive statistics, a decile is any of the 9 values that divide the sorted data into 10 equal parts, so that each part represents 1/10th of the sample or population. ... A percentile is the value of a variable below which a certain percent of observations fall. ... The parties A and B want to catch the median voters and they will walk to the centre. ... This article is about bias of statistical estimators. ...

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


This article incorporates material from Median of a distribution on PlanetMath, which is licensed under the GFDL.
PlanetMath is a free, collaborative, online mathematics encyclopedia. ...


  Results from FactBites:
 
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whats good Median, just wanting to show some love on the page...the mixtape was crazy, play it every chance i get.
Median's Relief is one of the best CD's I have heard in such a long time.
I just got into you awhile ago, Median's Relief is like the constant joint playin the whip, and I just got up on the mixtape too.
CancerGuide: The Median Isn't the Message (2000 words)
If I line up five kids by height, the median child is shorter than two and taller than the other two (who might have trouble getting their mean share of the candy).
In short, we view means and medians as the hard "realities,"; and the variation that permits their calculation as a set of transient and imperfect measurements of this hidden essence.
If the median is the reality and variation around the median just a device for its calculation, the "I will probably be dead in eight months" may pass as a reasonable interpretation.
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