Median

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 median is a commonly used measure of the properties of a data set in statistics and probability theory. 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.

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

Finite set of numbers

The median of a finite list of numbers can be found by arranging all the numbers from smallest to greatest.

If there is an odd number of numbers, the middle one is picked. For example, consider the list of numbers

1, 3, 3, 6, 7, 8, 9

This list contains seven numbers. The median is the fourth of them, which is 6.

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] For example, in the data set

1, 2, 3, 4, 5, 6, 8, 9

the median is the mean of the middle two numbers: this is , which is . (In more technical terms, this interprets the median as the fully trimmed mid-range).

The formula used to find the index of the middle number of a data set of n numerically ordered numbers is . This either gives the middle number (for an odd number of values) or the halfway point between the two middle values. For example, with 14 values, the formula will give an index of 7.5, and the median will be taken by averaging the seventh (the floor of this index) and eighth (the ceiling of this index) values. So the median can be represented by the following formula:

Comparison of common averages of values { 1, 2, 2, 3, 4, 7, 9 }
Type Description Example Result
Arithmetic mean Sum of values of a data set divided by number of values: (1+2+2+3+4+7+9) / 7 4
Median Middle value separating the greater and lesser halves of a data set 1, 2, 2, 3, 4, 7, 9 3
Mode Most frequent value in a data set 1, 2, 2, 3, 4, 7, 9 2

One can find the median using the Stem-and-Leaf Plot.

There is no widely accepted standard notation for the median, but some authors represent the median of a variable x either as or as μ1/2[1] sometimes also 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 used primarily for skewed distributions, which it summarizes differently from the arithmetic mean. Consider the multiset { 1, 2, 2, 2, 3, 14 }. The median is 2 in this case, (as is the mode), and it might be seen as a better indication of central tendency (less susceptible to the exceptionally large value in data) than the arithmetic mean of 4.

The median is a popular summary statistic used in descriptive statistics, 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. The widely cited empirical relationship between the relative locations of the mean and the median for skewed distributions is, however, not generally true.[5] There are, however, various relationships for the absolute difference between them; see below.

With an even number of observations (as shown above) no value need be exactly at the value of the median. 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.

In a population, at most half have values strictly less than the median and at most half have values strictly greater than it. 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. Indeed, as it is based on the middle data in a group, it is not necessary to even know the value of extreme results in order to calculate a median. For example, in a psychology test investigating the time needed to solve a problem, if a small number of people failed to solve the problem at all in the given time a median can still be calculated.[6]

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 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.

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. The median is the 2nd quartile, 5th decile, and 50th percentile. Since the median is the same as the second quartile, its calculation is illustrated in the article on quartiles. A median can be worked out for ranked but not numerical classes (e.g. working out a median grade when students are graded from A to F), although the result might be halfway between grades if there is an even number of cases.

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 asymptotic efficiency and references therein.

Probability distributions

Visualisation mode median mean
Geometric visualisation of the mode, median and mean of an arbitrary probability density function.[7]

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

or, equivalently, the inequalities

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

Any probability distribution on R has at least one median, but in specific cases there may be more than one median. Specifically, if a probability density is zero on an interval [ab], and the cumulative distribution function at a is 1/2, any value between a and b will also be a median.

Medians of particular distributions

The medians of certain types of distributions can be easily calculated from their parameters; furthermore, they exist even for some distributions lacking a well-defined mean, such as the Cauchy distribution:

Populations

Optimality property

The mean absolute error of a real variable c with respect to the random variable X is

Provided that the probability distribution of X is such that the above expectation exists, then m is a median of X if and only if m is a minimizer of the mean absolute error with respect to X.[9] In particular, m is a sample median if and only if m minimizes the arithmetic mean of the absolute deviations.

More generally, a median is defined as a minimum of

as discussed below in the section on multivariate medians (specifically, the spatial median).

This optimization-based definition of the median is useful in statistical data-analysis, for example, in k-medians clustering.

Unimodal distributions

Comparison mean median mode
Comparison of mean, median and mode of two log-normal distributions with different skewness.

It can be shown for a unimodal distribution that the median and the mean lie within (3/5)1/2 ≈ 0.7746 standard deviations of each other.[10] In symbols,

where |·| is the absolute value.

A similar relation holds between the median and the mode: they lie within 31/2 ≈ 1.732 standard deviations of each other:

Inequality relating means and medians

If the distribution has finite variance, then the distance between the median and the mean is bounded by one standard deviation.

This bound was proved by Mallows,[11] who used Jensen's inequality twice, as follows. We have

The first and third inequalities come from Jensen's inequality applied to the absolute-value function and the square function, which are each convex. The second inequality comes from the fact that a median minimizes the absolute deviation function

This proof also follows directly from Cantelli's inequality.[12] The result can be generalized to obtain a multivariate version of the inequality,[13] as follows:

where m is a spatial median, that is, a minimizer of the function The spatial median is unique when the data-set's dimension is two or more.[14][15] An alternative proof uses the one-sided Chebyshev inequality; it appears in an inequality on location and scale parameters.

Jensen's inequality for medians

Jensen's inequality states that for any random variable x with a finite expectation E(x) and for any convex function f

It has been shown[16] that if x is a real variable with a unique median m and f is a C function then

A C function is a real valued function, defined on the set of real numbers R, with the property that for any real t

is a closed interval, a singleton or an empty set.

Medians for samples

The sample median

Efficient computation of the sample median

Even though comparison-sorting n items requires Ω(n log n) operations, selection algorithms can compute the k'th-smallest of n items with only Θ(n) operations. This includes the median, which is the n/2'th order statistic (or for an even number of samples, the arithmetic mean of the two middle order statistics).

Selection algorithms still have the downside of requiring Ω(n) memory, that is, they need to have the full sample (or a linear-sized portion of it) in memory. Because this, as well as the linear time requirement, can be prohibitive, several estimation procedures for the median have been developed. A simple one is the median of three rule, which estimates the median as the median of a three-element subsample; this is commonly used as a subroutine in the quicksort sorting algorithm, which uses an estimate of its input's median. A more robust estimator is Tukey's ninther, which is the median of three rule applied with limited recursion:[17] if A is the sample laid out as an array, and

med3(A) = median(A[1], A[n/2], A[n]),

then

ninther(A) = med3(med3(A[1 ... 1/3n]), med3(A[1/3n ... 2/3n]), med3(A[2/3n ... n]))

The remedian is an estimator for the median that requires linear time but sub-linear memory, operating in a single pass over the sample.[18]

Easy explanation of the sample median

In individual series (if number of observation is very low) first one must arrange all the observations in order. Then count(n) is the total number of observation in given data.

If n is odd then Median (M) = value of ((n + 1)/2)th item term.

If n is even then Median (M) = value of [(n/2)th item term + (n/2 + 1)th item term]/2

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.

The median is the ((n + 1)/2)th item, where n is the number of values. For example, for the list {1, 2, 5, 7, 8}, we have n = 5, so the median is the ((5 + 1)/2)th item.

median = (6/2)th item
median = 3rd item
median = 5
For an even number of values

As an example, we will calculate the sample median for the following set of observations: 1, 6, 2, 8, 7, 2.

Start by sorting the values: 1, 2, 2, 6, 7, 8.

In this case, the arithmetic mean of the two middlemost terms is (2 + 6)/2 = 4. Therefore, the median is 4 since it is the arithmetic mean of the middle observations in the ordered list.

Sampling distribution

The distributions of both the sample mean and the sample median were determined by Laplace.[19] The distribution of the sample median from a population with a density function is asymptotically normal with mean and variance[20]

where is the median of and is the sample size.

These results have also been extended.[21] It is now known for the -th quantile that the distribution of the sample -th quantile is asymptotically normal around the -th quantile with variance equal to

where is the value of the distribution density at the -th quantile.

In the case of a discrete variable, the sampling distribution of the median for small-samples can be investigated as follows. We take the sample size to be an odd number . If a given value is to be the median of the sample then two conditions must be satisfied. The first is that at most observations can have a value of or less. The second is that at most observations can have a value of or more. Let be the number of observations which have a value of or less and let be the number of observations which have a value of or more. Then and both have a minimum value of 0 and a maximum of . If an observation has a value below , it is not relevant how far below it is and conversely, if an observation has a value above , it is not relevant how far above it is. We can therefore represent the observations as following a trinomial distribution with probabilities , and . The probability that the median will have a value is then given by

Summing this over all values of defines a proper distribution and gives a unit sum. In practice, the function will often not be known but it can be estimated from an observed frequency distribution. An example is given in the following table where the actual distribution is not known but a sample of 3,800 observations allows a sufficiently accurate assessment of .

v 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
f(v) 0.000 0.008 0.010 0.013 0.083 0.108 0.328 0.220 0.202 0.023 0.005
F(v) 0.000 0.008 0.018 0.031 0.114 0.222 0.550 0.770 0.972 0.995 1.000

Using these data it is possible to investigate the effect of sample size on the standard errors of the mean and median. The observed mean is 3.16, the observed raw median is 3 and the observed interpolated median is 3.174. The following table gives some comparison statistics. The standard error of the median is given both from the above expression for and from the asymptotic approximation given earlier.

Sample size
Statistic
3 9 15 21
Expected value of median 3.198 3.191 3.174 3.161
Standard error of median (above formula) 0.482 0.305 0.257 0.239
Standard error of median (asymptotic approximation) 0.879 0.508 0.393 0.332
Standard error of mean 0.421 0.243 0.188 0.159

The expected value of the median falls slightly as sample size increases while, as would be expected, the standard errors of both the median and the mean are proportionate to the inverse square root of the sample size. The asymptotic approximation errs on the side of caution by overestimating the standard error.

In the case of a continuous variable, the following argument can be used. If a given value is to be the median, then one observation must take the value . The elemental probability of this is . Then, of the remaining observations, exactly of them must be above and the remaining below. The probability of this is the th term of a binomial distribution with parameters and . Finally we multiply by since any of the observations in the sample can be the median observation. Hence the elemental probability of the median at the point is given by

Now we introduce the beta function. For integer arguments and , this can be expressed as . Also, we note that . Using these relationships and setting both and equal to allows the last expression to be written as

Hence the density function of the median is a symmetric beta distribution over the unit interval which supports . Its mean, as we would expect, is 0.5 and its variance is . The corresponding variance of the sample median is

However this finding can only be used if the density function is known or can be assumed. As this will not always be the case, the median variance has to be estimated sometimes from the sample data.

Estimation of variance from sample data

The value of —the asymptotic value of where is the population median—has been studied by several authors. The standard "delete one" jackknife method produces inconsistent results.[22] An alternative—the "delete k" method—where grows with the sample size has been shown to be asymptotically consistent.[23] This method may be computationally expensive for large data sets. A bootstrap estimate is known to be consistent,[24] but converges very slowly (order of ).[25] Other methods have been proposed but their behavior may differ between large and small samples.[26]

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 from the normal distribution, the efficiency for large N is

The efficiency tends to as tends to infinity.

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.[27]

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.[28] It is a statistical measure used by the states of Iowa, New York and South Dakota in estimating dues taxes.[29][30][31] In symbols

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 in which the sample sizes are large have been derived by Bonett and Seier[28] 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

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

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

and

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 univariate median, when the sample or population had one-dimension. 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.[27][32][33][34]

Medoid

Let be a set of points in a space with a distance function . Medoid is defined as

The medoid is often used in clustering using the k-medoid algorithm.

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.[27][35]

Spatial median

For N vectors in a normed vector space, a spatial median minimizes the average distance

where xn and a are vectors. The spatial median is unique when the data-set's dimension is two or more and the norm is the Euclidean norm (or another strictly convex norm).[14][15][27] The spatial median is also called the L1 median, even when the norm is Euclidean. Other names are used especially for finite sets of points: geometric median, Fermat point (in mechanics), or Weber or Fermat-Weber point (in geographical location theory).[36] In the special case where the norm is an L1-norm, then the spatial median and the marginal median are the same.

More generally, a spatial median is defined as a minimizer of

[27][37]

this general definition is convenient for defining a spatial median of a population in a finite-dimensional normed space, for example, for distributions without a finite mean.[14][27] Spatial medians are defined for random vectors with values in a Banach space.[14]

The spatial median is a robust and highly efficient estimator of a central tendency of a population.[27][37][38][39]

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

Interpolated median

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

Alternatively, if in an observed sample there are scores above the median category, scores in it and scores below it then the interpolated median is given by

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.[40] The Hodges–Lehmann estimator has been generalized to multivariate distributions.[37]

Variants of regression

The Theil–Sen estimator is a method for robust linear regression based on finding medians of slopes.[41]

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 : a left half with values less than the median and a right half with values greater than the median.[42] He suggested taking the means of the dependent and independent 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.[43] Brown and Mood in 1951 proposed the idea of using the medians of two subsamples rather the means.[44] 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.[45]

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:[46]

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.[47][48][49][50] Median-unbiased estimators are invariant under one-to-one transformations.

There are methods of construction median-unbiased estimators that are optimal (in a sense analogous to minimum-variance property considered for mean-unbiased estimators). Such constructions exist for probability distributions having monotone likelihood-functions.[51][52] One such procedure is an analogue of the Rao–Blackwell procedure for mean-unbiased estimators: The procedure holds for a smaller class of probability distributions than does the Rao—Blackwell procedure but for a larger class of loss functions.[53]

History

The idea of the median appeared in the 13th century in the Talmud [54][55] (further for possible older mentions)

The idea of the median also appeared later 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.[56]

In 1774, Laplace 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. Laplaces's criterion was generally rejected for 150 years in favor of the least squares method of Gauss and Legendre which minimizes to obtain the mean.[57] The distribution of both the sample mean and the sample median were determined by Laplace in the early 1800s.[19][58]

Antoine Augustin Cournot in 1843 was the first[59] 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.[60] 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.[60]

Francis Galton used the English term median in 1881,[61] having earlier used the terms middle-most value in 1869, and the medium in 1880.[62][63]

See also

References

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External links

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

Arithmetic mean

In mathematics and statistics, the arithmetic mean ( , stress on third syllable of "arithmetic"), or simply the mean or average when the context is clear, is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics because it helps distinguish it from other means, such as the geometric mean and the harmonic mean.

In addition to mathematics and statistics, the arithmetic mean is used frequently in many diverse fields such as economics, anthropology, and history, and it is used in almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation's population.

While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers (values that are very much larger or smaller than most of the values). Notably, for skewed distributions, such as the distribution of income for which a few people's incomes are substantially greater than most people's, the arithmetic mean may not coincide with one's notion of "middle", and robust statistics, such as the median, may be a better description of central tendency.

Average

In colloquial language, an average is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. In statistics, mean, median, and mode are all known as measures of central tendency, and in colloquial usage any of these might be called an average value.

Carpal tunnel syndrome

Carpal tunnel syndrome (CTS) is a medical condition due to compression of the median nerve as it travels through the wrist at the carpal tunnel. The main symptoms are pain, numbness and tingling in the thumb, index finger, middle finger and the thumb side of the ring fingers. Symptoms typically start gradually and during the night. Pain may extend up the arm. Weak grip strength may occur, and after a long period of time the muscles at the base of the thumb may waste away. In more than half of cases, both sides are affected.Risk factors include obesity, repetitive wrist work, pregnancy, genetics, and rheumatoid arthritis. There is tentative evidence that hypothyroidism increases the risk. Diabetes mellitus is weakly associated with CTS. The use of birth control pills does not affect the risk. Types of work that are associated include computer work, work with vibrating tools and work that requires a strong grip. Diagnosis is suspected based on signs, symptoms and specific physical tests and may be confirmed with electrodiagnostic tests. If muscle wasting at the base of the thumb is present, the diagnosis is likely.Being physically active can decrease the risk of developing CTS. Symptoms can be improved by wearing a wrist splint or with corticosteroid injections. Taking NSAIDs or gabapentin does not appear to be useful. Surgery to cut the transverse carpal ligament is effective with better results at a year compared to non surgical options. Further splinting after surgery is not needed. Evidence does not support magnet therapy.About 5% of people in the United States have carpal tunnel syndrome. It usually begins in adulthood, and women are more commonly affected than men. Up to 33% of people may improve without specific treatment over approximately a year. Carpal tunnel syndrome was first fully described after World War II.

Central tendency

In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution. It may also be called a center or location of the distribution. Colloquially, measures of central tendency are often called averages. The term central tendency dates from the late 1920s.The most common measures of central tendency are the arithmetic mean, the median and the mode. A central tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the normal distribution. Occasionally authors use central tendency to denote "the tendency of quantitative data to cluster around some central value."The central tendency of a distribution is typically contrasted with its dispersion or variability; dispersion and central tendency are the often characterized properties of distributions. Analysis may judge whether data has a strong or a weak central tendency based on its dispersion.

Household income in the United States

Household income is an economic measure that can be applied to one household, or aggregated across a large group such as a county, city, or the whole country. It is commonly used by the United States government and private institutions to describe a household's economic status or to track economic trends in the US.

One key measure is the real median level, meaning half of households have income above that level and half below, adjusted for inflation. According to the Census, this measure was $61,372 in 2017, an increase of $1,063 or 1.8% versus 2016, the second consecutive record level year. This measure was $60,309 in 2016 (up $1,833 or 3.1% vs. 2015) and $58,476 in 2015 (up $2,863 or 5.1% vs. 2014).The distribution of U.S. household income has become more unequal since around 1980, with the income share received by the top 1% trending upward from around 10% or less over the 1953–1981 period to over 20% by 2007. After falling somewhat due to the Great Recession in 2008 and 2009, inequality rose again during the economic recovery, a typical pattern historically.

Interquartile range

In descriptive statistics, the interquartile range (IQR), also called the midspread or middle 50%, or technically H-spread, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles, IQR = Q3 − Q1. In other words, the IQR is the first quartile subtracted from the third quartile; these quartiles can be clearly seen on a box plot on the data. It is a trimmed estimator, defined as the 25% trimmed range, and is a commonly used robust measure of scale.

The IQR is a measure of variability, based on dividing a data set into quartiles. Quartiles divide a rank-ordered data set into four equal parts. The values that separate parts are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.

Lateral consonant

A lateral is consonant in which the airstream proceeds along the sides of the tongue, but it is blocked by the tongue from going through the middle of the mouth. An example of a lateral consonant is the English l, as in Larry.

For the most common laterals, the tip of the tongue makes contact with the upper teeth (see dental consonant) or the upper gum (see alveolar consonant), but there are many other possible places for laterals to be made. The most common laterals are approximants and belong to the class of liquids, but lateral fricatives and affricates are also common in some parts of the world. Some languages, such as the Iwaidja and Ilgar languages of Australia, have lateral flaps, and others, such as the Xhosa and Zulu languages of Africa, have lateral clicks.

When pronouncing the labiodental fricatives [f] and [v], the lip blocks the airflow in the centre of the vocal tract, so the airstream proceeds along the sides instead. Nevertheless, they are not considered lateral consonants because the airflow never goes over the tongue. No known language makes a distinction between lateral and non-lateral labiodentals. Plosives are never lateral, but they may have lateral release. Nasals are never lateral either, but some languages have lateral nasal clicks. For consonants articulated in the throat (laryngeals), the lateral distinction is not made by any language, although pharyngeal and epiglottal laterals are reportedly possible.

Medes

The Medes (, Old Persian Māda-, Ancient Greek: Μῆδοι, Hebrew: מָדַי Madai) were an ancient Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Under the Neo-Assyrian Empire, late 9th to early 7th centuries BC, the region of Media was bounded by the Zagros Mountains to its west, to its south by the Garrin Mountain in Lorestan Province, to its northwest by the Qaflankuh Mountains in Zanjan Province, and to its east by the Dasht-e Kavir desert. Its neighbors were the kingdoms of Gizilbunda and Mannea in the northwest, and Ellipi and Elam in the south.In the 7th century BC, Media's tribes came together to form the Median Kingdom, which remained a Neo-Assyrian vassal. Between 616 and 609 BC, King Cyaxares (624–585 BC), allied with King Nabopolassar of the Neo-Babylonian Empire against the Neo-Assyrian Empire, after which the Median Empire stretched across the Iranian Plateau as far as Anatolia. Its precise geographical extent remains unknown.A few archaeological sites (discovered in the "Median triangle" in western Iran) and textual sources (from contemporary Assyrians and also ancient Greeks in later centuries) provide a brief documentation of the history and culture of the Median state. Apart from a few personal names, the language of the Medes is unknown. The Medes had an ancient Iranian religion (a form of pre-Zoroastrian Mazdaism or Mithra worshipping) with a priesthood named as "Magi". Later, during the reigns of the last Median kings, the reforms of Zoroaster spread into western Iran.

Median household income in Australia and New Zealand

Median household income is commonly used to measure the relative prosperity of populations in different geographical locations. It divides households into two equal segments with the first half of households earning less than the median household income and the other half earning more.

New Zealand and Australia are gradually being economically integrated through a process known as “Closer Economic Relations (CER)”. Their citizens are free to travel, live and work in either country. Information about their relative median household incomes is of interest, especially for those considering migration.

The latest release shows that the median gross household income in 2013–14 was $80,704, and the average of all households was $107,276.

Median income

Median income is the amount that divides the income distribution into two equal groups, half having income above that amount, and half having income below that amount. Mean income (average) is the amount obtained by dividing the total aggregate income of a group by the number of units in that group. Mode income is the most frequently occurring income in a given income distribution.

Median income can be calculated by household income, by personal income, or for specific demographic groups.

Median lethal dose

In toxicology, the median lethal dose, LD50 (abbreviation for "lethal dose, 50%"), LC50 (lethal concentration, 50%) or LCt50 is a measure of the lethal dose of a toxin, radiation, or pathogen. The value of LD50 for a substance is the dose required to kill half the members of a tested population after a specified test duration. LD50 figures are frequently used as a general indicator of a substance's acute toxicity. A lower LD50 is indicative of increased toxicity.

The test was created by J.W. Trevan in 1927. The term semilethal dose is occasionally used in the same sense, in particular with translations of foreign language text, but can also refer to a sublethal dose. LD50 is usually determined by tests on animals such as laboratory mice.

In 2011, the U.S. Food and Drug Administration approved alternative methods to LD50 for testing the cosmetic drug Botox without animal tests.

Median nerve

The median nerve is a nerve in humans and other animals in the upper limb. It is one of the five main nerves originating from the brachial plexus.

The median nerve originates from the lateral and medial cords of the brachial plexus, and has contributions from ventral roots of C5-C7 (lateral cord) and C8 and T1 (medial cord).

The median nerve is the only nerve that passes through the carpal tunnel. Carpal tunnel syndrome is the disability that results from the median nerve being pressed in the carpal tunnel.

Median strip

The median strip or central reservation is the reserved area that separates opposing lanes of traffic on divided roadways, such as divided highways, dual carriageways, freeways, and motorways. The term also applies to divided roadways other than highways, such as some major streets in urban or suburban areas. The reserved area may simply be paved, but commonly it is adapted to other functions; for example, it may accommodate decorative landscaping, trees, a median barrier or railway, rapid transit, light rail or streetcar lines.

Metres above sea level

"Feet above sea level" redirects here.Metres above mean sea level (MAMSL) or simply metres above sea level (MASL or m a.s.l.) is a standard metric measurement in metres of vertical distance (height, elevation or altitude) of a location in reference to a historic mean sea level taken as a vertical datum. Mean sea levels are affected by climate change and other factors and change over time. For this and other reasons, recorded measurements of elevation above sea level might differ from the actual elevation of a given location over sea level at a given moment.

Mode (statistics)

The mode of a set of data values is the value that appears most often. If X is a discrete random variable, the mode is the value x (i.e, X = x) at which the probability mass function takes its maximum value. In other words, it is the value that is most likely to be sampled.

Like the statistical mean and median, the mode is a way of expressing, in a (usually) single number, important information about a random variable or a population. The numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions.

The mode is not necessarily unique to a given discrete distribution, since the probability mass function may take the same maximum value at several points x1, x2, etc. The most extreme case occurs in uniform distributions, where all values occur equally frequently.

When the probability density function of a continuous distribution has multiple local maxima it is common to refer to all of the local maxima as modes of the distribution. Such a continuous distribution is called multimodal (as opposed to unimodal). A mode of a continuous probability distribution is often considered to be any value x at which its probability density function has a locally maximum value, so any peak is a mode.In symmetric unimodal distributions, such as the normal distribution, the mean (if defined), median and mode all coincide. For samples, if it is known that they are drawn from a symmetric unimodal distribution, the sample mean can be used as an estimate of the population mode.

Personal income in the United States

Personal income is an individual's total earnings from wages, investment interest, and other sources. The Bureau of Labor Statistics reported a median personal income of $865 weekly for all full-time workers in 2017. The U.S Bureau of the Census has the annual median personal income at $31,099 in 2016. Inflation-adjusted ("real") per-capita disposable personal income rose steadily in the U.S. from 1945 to 2008, but has since remained generally level.Income patterns are evident on the basis of age, sex, ethnicity and educational characteristics. In 2005 roughly half of all those with graduate degrees were among the nation's top 15% of income earners. Among different demographics (gender, marital status, ethnicity) for those over the age of 18, median personal income ranged from $3,317 for an unemployed, married Asian American female to $55,935 for a full-time, year-round employed Asian American male. According to the US Census, men tended to have higher income than women, while Asians and Whites earned more than African Americans and Hispanics.

Sea level

Mean sea level (MSL) (often shortened to sea level) is an average level of the surface of one or more of Earth's oceans from which heights such as elevation may be measured. MSL is a type of vertical datum – a standardised geodetic datum – that is used, for example, as a chart datum in cartography and marine navigation, or, in aviation, as the standard sea level at which atmospheric pressure is measured to calibrate altitude and, consequently, aircraft flight levels. A common and relatively straightforward mean sea-level standard is the midpoint between a mean low and mean high tide at a particular location.Sea levels can be affected by many factors and are known to have varied greatly over geological time scales. However 20th century and current millennium sea level rise is caused by global warming, and careful measurement of variations in MSL can offer insights into ongoing climate change.The term above sea level generally refers to above mean sea level (AMSL).

Skewness

In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive or negative, or undefined.

For a unimodal distribution, negative skew commonly indicates that the tail is on the left side of the distribution, and positive skew indicates that the tail is on the right. In cases where one tail is long but the other tail is fat, skewness does not obey a simple rule. For example, a zero value means that the tails on both sides of the mean balance out overall; this is the case for a symmetric distribution, but can also be true for an asymmetric distribution where one tail is long and thin, and the other is short but fat.

Tongue

The tongue is a muscular organ in the mouth of most vertebrates that manipulates food for mastication, and is used in the act of swallowing. It is of importance in the digestive system and is the primary organ of taste in the gustatory system. The tongue's upper surface (dorsum) is covered by taste buds housed in numerous lingual papillae. It is sensitive and kept moist by saliva, and is richly supplied with nerves and blood vessels. The tongue also serves as a natural means of cleaning the teeth. A major function of the tongue is the enabling of speech in humans and vocalization in other animals.

The human tongue is divided into two parts, an oral part at the front and a pharyngeal part at the back. The left and right sides are also separated along most of its length by a vertical section of fibrous tissue (the lingual septum) that results in a groove, the median sulcus on the tongue's surface.

There are two groups of muscles of the tongue. The four intrinsic muscles alter the shape of the tongue and are not attached to bone. The four paired extrinsic muscles change the position of the tongue and are anchored to bone.

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