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In probability theory and statistics, a median is described as the numeric value 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 the middle one. If there is an even number of observations, then there is no single middle value, so one often takes the mean of the two middle values.

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) and, 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.

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. 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. A disadvantage of the median is the difficulty of handling it theoretically.[citation needed]

Contents

Notation

The median of some variable x is denoted either as \tilde{x} or as \mu_{1/2}(x)\,\!.[1]

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[citation needed]. For floating point population, the median lies somewhere between the two middle elements, depending on the distribution[citation needed]. Median is the middle value after arranging data by any order[citation needed].

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

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

or

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

in which a Lebesgue–Stieltjes integral is used. For an absolutely continuous probability distribution with probability density function ƒ, we have

\operatorname{P}(X\leq m) = \operatorname{P}(X\geq m)=\int_{-\infty}^m f(x)\, \mathrm{d}x=\frac{1}{2}.\,\!

Medians of particular distributions: The medians of certain types of distributions can be easily calculated 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 [ab] 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 λ is the natural logarithm of 2 divided by the rate parameter: λ−1ln 2. The median of a Weibull distribution with shape parameter k and scale parameter λ is λ(ln 2)1/k.

Medians in descriptive statistics

The median is primarily used for skewed distributions, which it summarizes differently than the arithmetic mean. Consider the multiset { 1, 2, 2, 2, 3, 9 }. 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

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An optimality property

A median is also a central point which minimizes the average of the absolute deviations: In the above example, a median would be (1 + 0 + 0 + 0 + 1 + 7) / 6 = 1.5 using the minimum of the absolute deviations; in contrast, the minimizer of the sum of squares would be mean, which is 1.944. In the language of statistics, a value of c that minimizes

E(\left|X-c\right|)\,

is a median of the probability distribution of the random variable X.

However, a median c need not be uniquely defined. Where exactly one median exists, statisticians speak of "the median" correctly; even when no unique median exists, some statisticians speak of "the median" informally.

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.

The sample median

Efficient computation of the sample median

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 of the sample median

For an odd number of values

As an example, we will calculate the sample median for the following set of observations: 1, 5, 2, 8, 7.

Start by sorting the values: 1, 2, 5, 7, 8.

In this case, the median is 5 since it is the middle observation in the ordered list.

For an even number of values

As an example, we will calculate the sample median for the following set of observations: 1, 5, 2, 8, 7, 2.

Start by sorting the values: 1, 2, 2, 5, 7, 8.

In this case, the average is (2 + 5)/2 = 3.5. Therefore, the median is 3.5 since it is the average of the middle observations in the ordered list.

Other estimates of the median

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. See, for example Pareto interpolation.

Median-unbiased estimators, and bias with respect to loss functions

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:

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 noted by Lehmann, Birnbaum, van der Vaart and Pfanzagl. In particular, median-unbiased estimators exist in cases where mean-unbiased and maximum-likelihood estimators do not exist. Besides being invariant under one-to-one transformations, median-unbiased estimators have surprising robustness.

In image processing

In monochrome raster images there is a type of noise, known as the salt and pepper noise, when each pixel independently become 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.

In multidimensional statistical inference

In multidimensional statistical inference, the value c that minimizes E(\left|X-c\right|) is also called a centroid.[2] In this case \left|X-c\right| is indicating a norm for the vector difference, such as the Euclidean norm, rather than the one-dimensional case's use of an absolute value. (Note that in some other contexts a centroid is more like a multidimensional mean than the multidimensional median described here.)

Like a centroid, a medoid minimizes E(\left|X-c\right|), but c is restricted to be a member of specified set. For instance, the set could be a sample of points drawn from some distribution.

History

Gustav Fechner popularized the median into the formal analysis of data, although it had been used previously by Laplace. [3]

See also

References

  1. ^ http://mathworld.wolfram.com/StatisticalMedian.html
  2. ^ Carvalho, Luis; Lawrence, Charles (2008), "Centroid estimation in discrete high-dimensional spaces with applications in biology", Proc Natl Acad Sci U S A 105 (9): 3209-3214, doi:10.1073/pnas.0712329105 
  3. ^ Keynes, John Maynard; A Treatise on Probability (1921), Pt II Ch XVII §5 (p 201).
  • Brown, George W. ”On Small-Sample Estimation.” The Annals of Mathematical Statistics, Vol. 18, No. 4 (Dec., 1947), pp. 582–585.
  • Lehmann, E. L. “A General Concept of Unbiasedness” The Annals of Mathematical Statistics, Vol. 22, No. 4 (Dec., 1951), pp. 587–592.
  • Allan Birnbaum. 1961. “A Unified Theory of Estimation, I”, The Annals of Mathematical Statistics, Vol. 32, No. 1 (Mar., 1961), pp. 112–135
  • van der Vaart, H. R. 1961. “Some Extensions of the Idea of Bias” The Annals of Mathematical Statistics, Vol. 32, No. 2 (Jun., 1961), pp. 436–447.
  • Pfanzagl, Johann. 1994. Parametric Statistical Theory. Walter de Gruyter.

External links

This article incorporates material from Median of a distribution on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.


Study guide

Up to date as of January 14, 2010

From Wikiversity

In Statistics, median is the number in the middle of a set of numbers.The median of a a series of numbers is obtained by arranging the numbers in ascending order and then choosing the number in the 'middle'.If there are two'middle numbers the median is the average(mean) of these two numbers.

Examples and application

1. Find the median of the following set of numbers:

5, 4, 10, 3, 3, 4, 7, 4, 6, 5

So, first of all lets arrange the numbers in ascending order:

3, 3, 4, 4, 4, 5, 5, 6, 7, 10

Then you can use 'striking out' to find the median easily and without mixing as I did here:

3, 3, 4, 4, 4, 5,5, 6, 7, 10

Here, what I did was that I began canceling for the ends of the set: cancel the 3 with 10, the second three with 7 and so on.. So, now we found out that the median was two numbers: 4 and 5. In this case we will add them and then divide them by two, just as if we are finding their mean. So 4+5=9, and then 9\2 is 4.5, so the median here is 4.5.

See also

For further information about any school-level math, just contact me. Special thanks to my math teacher,--;Hiba;1 15:39, 23 February 2009 (UTC)


Wiktionary

Up to date as of January 15, 2010

Definition from Wiktionary, a free dictionary

See also median

Contents

English

Etymology

Mede +‎ -ian

Proper noun

Singular
Median

Plural
-

Median

  1. The northwestern Iranian language of the Medes, attested only by numerous loanwords in Old Persian; nothing is known of its grammar.

Synonyms

Anagrams


German

Noun

Median m.

  1. median (statistics: measure of central tendency)

This German entry was created from the translations listed at median. It may be less reliable than other entries, and may be missing parts of speech or additional senses. Please also see Median in the German Wiktionary. This notice will be removed when the entry is checked. (more information) December 2008


Bible wiki

Up to date as of January 23, 2010
(Redirected to Media article)

From BibleWiki


Heb. Madai, which is rendered in the Authorized Version (1) "Madai," Gen 10:2; (2) "Medes," 2Kg 17:6; 18:11; (3) "Media," Est 1:3; 10:2; Isa 21:2; Dan 8:20; (4) "Mede," only in Dan 11:1.

We first hear of this people in the Assyrian cuneiform records, under the name of Amada, about B.C. 840. They appear to have been a branch of the Aryans, who came from the east bank of the Indus, and were probably the predominant race for a while in the Mesopotamian valley. They consisted for three or four centuries of a number of tribes, each ruled by its own chief, who at length were brought under the Assyrian yoke (2Kg 17:6). From this subjection they achieved deliverance, and formed themselves into an empire under Cyaxares (B.C. 633). This monarch entered into an alliance with the king of Babylon, and invaded Assyria, capturing and destroying the city of Nineveh (B.C. 625), thus putting an end to the Assyrian monarchy (Nah 1:8; 2:5,6; 3:13, 14).

Media now rose to a place of great power, vastly extending its boundaries. But it did not long exist as an independent kingdom. It rose with Cyaxares, its first king, and it passed away with him; for during the reign of his son and successor Astyages, the Persians waged war against the Medes and conquered them, the two nations being united under one monarch, Cyrus the Persian (B.C. 558).

The "cities of the Medes" are first mentioned in connection with the deportation of the Israelites on the destruction of Samaria (2Kg 17:6; 18:11). Soon afterwards Isaiah (13:17; 21:2) speaks of the part taken by the Medes in the destruction of Babylon (comp. Jer 51:11, 28). Daniel gives an account of the reign of Darius the Mede, who was made viceroy by Cyrus (Dan 6:1-28). The decree of Cyrus, Ezra informs us (6:2-5), was found in "the palace that is in the province of the Medes," Achmetha or Ecbatana of the Greeks, which is the only Median city mentioned in Scripture.

This entry includes text from Easton's Bible Dictionary, 1897.

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Gaming

Up to date as of February 01, 2010

From Wikia Gaming, your source for walkthroughs, games, guides, and more!

Median

Developer(s) Midway
Publisher(s) Midway
Release date Bally Astrocade:
1977 (NA)
Genre Edutainment
Mode(s) Single player
Age rating(s) N/A
Bally Astrocade
Platform(s) Bally Astrocade
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Simple English

In probability theory and statistics, the median is a number. This number has the property that it divides a set of observed values in two equal halves, so that half of the values are below it, and half are above.

If there are a finite number of elements, the median is easy to find. The values need to be arranged in a list, lowest to highest. If there is an odd number of values, the median is the one at position (n+1)/2. For example, if there are 13 values, they can be arranged into two groups of 6, with the median in between, at position 7. With an even number of values, as there is no single number which divides all of the numbers to two halves, the median is defined as the mean of the two central elements. With 14 observations, this would be the mean of elements 7 and 8, which is their sum divided by 2.

Median and mean

Median and mean are different in several ways. Mean is a better statistical measure in many cases, because many of the statistical tests can use mean and standard deviation of two observations to compare them, while the same comparison cannot be performed using the medians.

On the opposite, median is a better statistical measure in some cases where the variance of the values is not important and we only need a central measure of the values. If the maximum value of a set of numbers changes while the other numbers of this set are kept the same, the mean of this set of numbers changes, but the median does not.

One of the other advantages of median is that, it can be calculated sooner when we are studying survival data. For example, a researcher can calculate the median survival of patients with kidney transplant, when half the patients participated in his study die; in contrast, if he wants to calculate the mean survival, he must continue the study and follow all of the patients until their death.[1]

References

  1. Dawson, B; Trapp, R (2001). Basic & Clinical Biostatistics (4th Edition ed.). McGrawHill. http://www.accessmedicine.com/resourceTOC.aspx?resourceID=62. 


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