The median of a set of numbers 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. The basic feature of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of the center. Median income , for example, may be a better way to describe the center of the income distribution because increases in the largest incomes alone have no effect on the median. For this reason, the median is of central importance in robust statistics .
73-408: The median of a finite list of numbers is the "middle" number, when those numbers are listed in order from smallest to greatest. If the data set has an odd number of observations, the middle one is selected (after arranging in ascending order). For example, the following list of seven numbers, has the median of 6 , which is the fourth value. If the data set has an even number of observations, there
146-444: A ↦ E ( ‖ X − a ‖ ) . {\displaystyle a\mapsto \operatorname {E} (\|X-a\|).\,} The spatial median is unique when the data-set's dimension is two or more. An alternative proof uses the one-sided Chebyshev inequality; it appears in an inequality on location and scale parameters . This formula also follows directly from Cantelli's inequality . For
219-425: A color wheel —there is no mean to the set of all colors. In these situations, you must decide which mean is most useful. You can do this by adjusting the values before averaging, or by using a specialized approach for the mean of circular quantities . The Fréchet mean gives a manner for determining the "center" of a mass distribution on a surface or, more generally, Riemannian manifold . Unlike many other means,
292-430: A convex space , not only a vector space. The arithmetic mean has several properties that make it interesting, especially as a measure of central tendency. These include: The arithmetic mean may be contrasted with the median . The median is defined such that no more than half the values are larger, and no more than half are smaller than it. If elements in the data increase arithmetically when placed in some order, then
365-763: A probability density function f ), nor does it require a discrete one . In the former case, the inequalities can be upgraded to equality: a median satisfies P ( X ≤ m ) = ∫ − ∞ m f ( x ) d x = 1 2 {\displaystyle \operatorname {P} (X\leq m)=\int _{-\infty }^{m}{f(x)\,dx}={\frac {1}{2}}} and P ( X ≥ m ) = ∫ m ∞ f ( x ) d x = 1 2 . {\displaystyle \operatorname {P} (X\geq m)=\int _{m}^{\infty }{f(x)\,dx}={\frac {1}{2}}\,.} Any probability distribution on
438-421: A probability distribution is the long-run arithmetic average value of a random variable having that distribution. If the random variable is denoted by X {\displaystyle X} , then the mean is also known as the expected value of X {\displaystyle X} (denoted E ( X ) {\displaystyle E(X)} ). For a discrete probability distribution ,
511-400: A truncated mean . It involves discarding given parts of the data at the top or the bottom end, typically an equal amount at each end and then taking the arithmetic mean of the remaining data. The number of values removed is indicated as a percentage of the total number of values. The interquartile mean is a specific example of a truncated mean. It is simply the arithmetic mean after removing
584-402: A comment on a subsequent proof by O'Cinneide, Mallows in 1991 presented a compact proof that uses Jensen's inequality twice, as follows. Using |·| for the absolute value , 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
657-424: A continuous variable, the probability of multiple sample values being exactly equal to the median is 0, so one can calculate the density of at the point v {\displaystyle v} directly from the trinomial distribution: Arithmetic mean In mathematics and statistics , the arithmetic mean ( / ˌ æ r ɪ θ ˈ m ɛ t ɪ k / arr-ith- MET -ik ), arithmetic average , or just
730-479: A data set X {\displaystyle X} is denoted as X ¯ {\displaystyle {\overline {X}}} ). The arithmetic mean can be similarly defined for vectors in multiple dimensions, not only scalar values; this is often referred to as a centroid . More generally, because the arithmetic mean is a convex combination (meaning its coefficients sum to 1 {\displaystyle 1} ), it can be defined on
803-417: A function f ( x ) {\displaystyle f(x)} . Intuitively, a mean of a function can be thought of as calculating the area under a section of a curve, and then dividing by the length of that section. This can be done crudely by counting squares on graph paper, or more precisely by integration . The integration formula is written as: In this case, care must be taken to make sure that
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#1732766039586876-518: A given group of data , illustrating the magnitude and sign of the data set . Which of these measures is most illuminating depends on what is being measured, and on context and purpose. The arithmetic mean , also known as "arithmetic average", is the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x 1 , x 2 , ..., x n is typically denoted using an overhead bar , x ¯ {\displaystyle {\bar {x}}} . If
949-399: A list of numbers, is the sum of all of the numbers divided by their count. Similarly, the mean of a sample x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} , usually denoted by x ¯ {\displaystyle {\bar {x}}} , is the sum of the sampled values divided by
1022-421: A measure of central tendency ). The mean of a set of observations is the arithmetic average of the values; however, for skewed distributions , the mean is not necessarily the same as the middle value (median), or the most likely value (mode). For example, mean income is typically skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast,
1095-426: A measure of location when one attaches reduced importance to extreme values, typically because a distribution is skewed , extreme values are not known, or outliers are untrustworthy, i.e., may be measurement or transcription errors. For example, consider the multiset The median is 2 in this case, as is the mode , and it might be seen as a better indication of the center than the arithmetic mean of 4, which
1168-419: 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
1241-416: A result of 180 ° . This is incorrect for two reasons: In general application, such an oversight will lead to the average value artificially moving towards the middle of the numerical range. A solution to this problem is to use the optimization formulation (that is, define the mean as the central point: the point about which one has the lowest dispersion) and redefine the difference as a modular distance (i.e.,
1314-421: A single pass over the sample. The distributions of both the sample mean and the sample median were determined by Laplace . The distribution of the sample median from a population with a density function f ( x ) {\displaystyle f(x)} is asymptotically normal with mean μ {\displaystyle \mu } and variance where m {\displaystyle m}
1387-411: A situation with n {\displaystyle n} numbers being averaged). If a numerical property, and any sample of data from it, can take on any value from a continuous range instead of, for example, just integers, then the probability of a number falling into some range of possible values can be described by integrating a continuous probability distribution across this range, even when
1460-404: A variable x as med( x ), x͂ , as μ 1/2 , or as M . 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 a special case of other ways of summarizing the typical values associated with a statistical distribution : it is the 2nd quartile , 5th decile , and 50th percentile . The median can be used as
1533-476: A well-defined mean, such as the Cauchy distribution : 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 . In particular, if m is a sample median, then it minimizes
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#17327660395861606-485: A wide range of other notions of mean are often used in geometry and mathematical analysis ; examples are given below. In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music. The arithmetic mean (or simply mean or average ) of
1679-427: Is 3 + 5 2 = 4 {\displaystyle {\frac {3+5}{2}}=4} , or equivalently 3 ⋅ 1 2 + 5 ⋅ 1 2 = 4 {\displaystyle 3\cdot {\frac {1}{2}}+5\cdot {\frac {1}{2}}=4} . In contrast, a weighted mean in which the first number receives, for example, twice as much weight as the second (perhaps because it
1752-468: Is a C function, but the reverse does not hold. If f is a C function, then If the medians are not unique, the statement holds for the corresponding suprema. 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,
1825-486: Is an average which is useful for sets of numbers which are defined in relation to some unit , as in the case of speed (i.e., distance per unit of time): For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is If we have five pumps that can empty a tank of a certain size in respectively 4, 36, 45, 50, and 75 minutes, then the harmonic mean of 15 {\displaystyle 15} tells us that these five different pumps working together will pump at
1898-399: Is an even number of classes. (For odd number classes, one specific class is determined as the median.) A geometric median , on the other hand, is defined in any number of dimensions. A related concept, in which the outcome is forced to correspond to a member of the sample, is the medoid . There is no widely accepted standard notation for the median, but some authors represent the median of
1971-525: Is assumed to appear twice as often in the general population from which these numbers were sampled) would be calculated as 3 ⋅ 2 3 + 5 ⋅ 1 3 = 11 3 {\displaystyle 3\cdot {\frac {2}{3}}+5\cdot {\frac {1}{3}}={\frac {11}{3}}} . Here the weights, which necessarily sum to one, are 2 3 {\displaystyle {\frac {2}{3}}} and 1 3 {\displaystyle {\frac {1}{3}}} ,
2044-408: Is larger than all but one of the values. However, the widely cited empirical relationship that the mean is shifted "further into the tail" of a distribution than the median is not generally true. At most, one can say that the two statistics cannot be "too far" apart; see § Inequality relating means and medians below. As a median is based on the middle data in a set, it is not necessary to know
2117-409: Is no distinct middle value and the median is usually defined to be the arithmetic mean of the two middle values. For example, this data set of 8 numbers has a median value of 4.5 , that is ( 4 + 5 ) / 2 {\displaystyle (4+5)/2} . (In more technical terms, this interprets the median as the fully trimmed mid-range ). In general, with this convention,
2190-417: 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 when—and only when— data is uncontaminated by data from heavy-tailed distributions or from mixtures of distributions. Even then, the median has a 64% efficiency compared to
2263-492: Is the probability density function . In all cases, including those in which the distribution is neither discrete nor continuous, the mean is the Lebesgue integral of the random variable with respect to its probability measure . The mean need not exist or be finite; for some probability distributions the mean is infinite ( +∞ or −∞ ), while for others the mean is undefined . The generalized mean , also known as
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2336-520: Is the median of f ( x ) {\displaystyle f(x)} and n {\displaystyle n} is the sample size: A modern proof follows below. Laplace's result is now understood as a special case of the asymptotic distribution of arbitrary quantiles . For normal samples, the density is f ( m ) = 1 / 2 π σ 2 {\displaystyle f(m)=1/{\sqrt {2\pi \sigma ^{2}}}} , thus for large samples
2409-473: 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 from an experiment , an observational study , or a survey . The term "arithmetic mean" is preferred in some mathematics and statistics contexts because it helps distinguish it from other types of means, such as geometric and harmonic . In addition to mathematics and statistics,
2482-409: The absolute deviation function a ↦ E ( | X − a | ) {\displaystyle a\mapsto \operatorname {E} (|X-a|)} . Mallows's proof can be generalized to obtain a multivariate version of the inequality simply by replacing the absolute value with a norm : where m is a spatial median , that is, a minimizer of the function
2555-421: 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
2628-407: The distribution of income for which a few people's incomes are substantially higher than most people's, the arithmetic mean may not coincide with one's notion of "middle". In that case, robust statistics, such as the median , may provide a better description of central tendency. The arithmetic mean of a set of observed data is equal to the sum of the numerical values of each observation, divided by
2701-626: The drawing in the definition of expected value for arbitrary real-valued random variables ). An equivalent phrasing uses a random variable X distributed according to F : P ( X ≤ m ) ≥ 1 2 and P ( X ≥ m ) ≥ 1 2 . {\displaystyle \operatorname {P} (X\leq m)\geq {\frac {1}{2}}{\text{ and }}\operatorname {P} (X\geq m)\geq {\frac {1}{2}}\,.} Note that this definition does not require X to have an absolutely continuous distribution (which has
2774-469: The 1980s, the median income in the United States has increased more slowly than the arithmetic average of income. A weighted average, or weighted mean, is an average in which some data points count more heavily than others in that they are given more weight in the calculation. For example, the arithmetic mean of 3 {\displaystyle 3} and 5 {\displaystyle 5}
2847-511: The Fréchet mean is defined on a space whose elements cannot necessarily be added together or multiplied by scalars. It is sometimes also known as the Karcher mean (named after Hermann Karcher). In geometry, there are thousands of different definitions for the center of a triangle that can all be interpreted as the mean of a triangular set of points in the plane. This is an approximation to
2920-452: The arithmetic mean is frequently used in economics , anthropology , history , and 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 : it is greatly influenced by outliers (values much larger or smaller than most others). For skewed distributions , such as
2993-475: The arithmetic mean is: If the data set is a statistical population (i.e., consists of every possible observation and not just a subset of them), then the mean of that population is called the population mean and denoted by the Greek letter μ {\displaystyle \mu } . If the data set is a statistical sample (a subset of the population), it is called the sample mean (which for
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3066-452: The arithmetic mean of the absolute deviations. Note, however, that in cases where the sample contains an even number of elements, this minimizer is not unique. 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 . If
3139-535: The base letter "x" plus a code for the line above ( ̄ or ¯). In some document formats (such as PDF ), the symbol may be replaced by a "¢" ( cent ) symbol when copied to a text processor such as Microsoft Word . Mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency ") in mathematics , especially in statistics . Each attempts to summarize or typify
3212-586: The case of unimodal distributions, one can achieve a sharper bound on the distance between the median and the mean: A similar relation holds between the median and the mode: A typical heuristic is that positively skewed distributions have mean > median. This is true for all members of the Pearson distribution family . However this is not always true. For example, the Weibull distribution family has members with positive mean, but mean < median. Violations of
3285-526: The data Amount of total numbers within the data {\displaystyle {\frac {\text{Total of all numbers within the data}}{\text{Amount of total numbers within the data}}}} For example, if the monthly salaries of 10 {\displaystyle 10} employees are { 2500 , 2700 , 2400 , 2300 , 2550 , 2650 , 2750 , 2450 , 2600 , 2400 } {\displaystyle \{2500,2700,2400,2300,2550,2650,2750,2450,2600,2400\}} , then
3358-453: The distance on the circle: so the modular distance between 1° and 359° is 2°, not 358°). The arithmetic mean is often denoted by a bar ( vinculum or macron ), as in x ¯ {\displaystyle {\bar {x}}} . Some software ( text processors , web browsers ) may not display the "x̄" symbol correctly. For example, the HTML symbol "x̄" combines two codes —
3431-410: The distribution has finite variance, then the distance between the median X ~ {\displaystyle {\tilde {X}}} and the mean X ¯ {\displaystyle {\bar {X}}} is bounded by one standard deviation . This bound was proved by Book and Sher in 1979 for discrete samples, and more generally by Page and Murty in 1982. In
3504-410: The former being twice the latter. The arithmetic mean (sometimes called the "unweighted average" or "equally weighted average") can be interpreted as a special case of a weighted average in which all weights are equal to the same number ( 1 2 {\displaystyle {\frac {1}{2}}} in the above example and 1 n {\displaystyle {\frac {1}{n}}} in
3577-470: The formula for the case of discrete variables is given below in § Empirical local density . The sample can be summarized as "below median", "at median", and "above median", which corresponds to a trinomial distribution with probabilities F ( v ) {\displaystyle F(v)} , f ( v ) {\displaystyle f(v)} and 1 − F ( v ) {\displaystyle 1-F(v)} . For
3650-443: The integral converges. But the mean may be finite even if the function itself tends to infinity at some points. Angles , times of day, and other cyclical quantities require modular arithmetic to add and otherwise combine numbers. In all these situations, there will not be a unique mean. For example, the times an hour before and after midnight are equidistant to both midnight and noon. It is also possible that no mean exists. Consider
3723-411: The lowest and the highest quarter of values. assuming the values have been ordered, so is simply a specific example of a weighted mean for a specific set of weights. In some circumstances, mathematicians may calculate a mean of an infinite (or even an uncountable ) set of values. This can happen when calculating the mean value y avg {\displaystyle y_{\text{avg}}} of
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#17327660395863796-422: The mean and size of sample i {\displaystyle i} respectively. In other applications, they represent a measure for the reliability of the influence upon the mean by the respective values. Sometimes, a set of numbers might contain outliers (i.e., data values which are much lower or much higher than the others). Often, outliers are erroneous data caused by artifacts . In this case, one can use
3869-598: The mean is given by ∑ x P ( x ) {\displaystyle \textstyle \sum xP(x)} , where the sum is taken over all possible values of the random variable and P ( x ) {\displaystyle P(x)} is the probability mass function . For a continuous distribution , the mean is ∫ − ∞ ∞ x f ( x ) d x {\displaystyle \textstyle \int _{-\infty }^{\infty }xf(x)\,dx} , where f ( x ) {\displaystyle f(x)}
3942-411: The mean, as shown in the figure. Jensen's inequality states that for any random variable X with a finite expectation E [ X ] and for any convex function f This inequality generalizes to the median as well. We say a function f : R → R is a C function if, for any t , is a closed interval (allowing the degenerate cases of a single point or an empty set ). Every convex function
4015-466: The median and arithmetic average are equal. For example, consider the data sample { 1 , 2 , 3 , 4 } {\displaystyle \{1,2,3,4\}} . The mean is 2.5 {\displaystyle 2.5} , as is the median. However, when we consider a sample that cannot be arranged to increase arithmetically, such as { 1 , 2 , 4 , 8 , 16 } {\displaystyle \{1,2,4,8,16\}} ,
4088-407: The median and arithmetic average can differ significantly. In this case, the arithmetic average is 6.2 {\displaystyle 6.2} , while the median is 4 {\displaystyle 4} . The average value can vary considerably from most values in the sample and can be larger or smaller than most. There are applications of this phenomenon in many fields. For example, since
4161-458: 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: if A is the sample laid out as an array , and then The remedian is an estimator for the median that requires linear time but sub-linear memory, operating in
4234-476: The median can be defined as follows: For a data set x {\displaystyle x} of n {\displaystyle n} elements, ordered from smallest to greatest, Formally, a median of a population is any value such that at least half of the population is less than or equal to the proposed median and at least half is greater than or equal to the proposed median. As seen above, medians may not be unique. If each set contains more than half
4307-409: The median income is the level at which half the population is below and half is above. The mode income is the most likely income and favors the larger number of people with lower incomes. While the median and mode are often more intuitive measures for such skewed data, many skewed distributions are in fact best described by their mean, including the exponential and Poisson distributions. The mean of
4380-555: 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. For any real -valued probability distribution with cumulative distribution function F , a median is defined as any real number m that satisfies the inequalities lim x → m − F ( x ) ≤ 1 2 ≤ F ( m ) {\displaystyle \lim _{x\to m-}F(x)\leq {\frac {1}{2}}\leq F(m)} (cf.
4453-402: The naive probability for a sample number taking one certain value from infinitely many is zero. In this context, the analog of a weighted average, in which there are infinitely many possibilities for the precise value of the variable in each range, is called the mean of the probability distribution . The most widely encountered probability distribution is called the normal distribution ; it has
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#17327660395864526-429: The number of items in the sample. For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is: The geometric mean is an average that is useful for sets of positive numbers, that are interpreted according to their product (as is the case with rates of growth) and not their sum (as is the case with the arithmetic mean): For example, the geometric mean of five values: 4, 36, 45, 50, 75 is: The harmonic mean
4599-475: The numbers are from observing a sample of a larger group , the arithmetic mean is termed the sample mean ( x ¯ {\displaystyle {\bar {x}}} ) to distinguish it from the group mean (or expected value ) of the underlying distribution, denoted μ {\displaystyle \mu } or μ x {\displaystyle \mu _{x}} . Outside probability and statistics,
4672-517: The parameter m , the following types of means are obtained: This can be generalized further as the generalized f -mean and again a suitable choice of an invertible f will give The weighted arithmetic mean (or weighted average) is used if one wants to combine average values from different sized samples of the same population: Where x i ¯ {\displaystyle {\bar {x_{i}}}} and w i {\displaystyle w_{i}} are
4745-426: The population, then some of the population is exactly equal to the unique median. The median is well-defined for any ordered (one-dimensional) data and is independent of any distance metric . The median can thus be applied to school classes which are ranked but not numerical (e.g. working out a median grade when student test scores are graded from F to A), although the result might be halfway between classes if there
4818-511: The power mean or Hölder mean, is an abstraction of the quadratic , arithmetic, geometric, and harmonic means. It is defined for a set of n positive numbers x i by x ¯ ( m ) = ( 1 n ∑ i = 1 n x i m ) 1 m {\displaystyle {\bar {x}}(m)=\left({\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{m}\right)^{\frac {1}{m}}} By choosing different values for
4891-407: The property that all measures of its central tendency, including not just the mean but also the median mentioned above and the mode (the three Ms ), are equal. This equality does not hold for other probability distributions, as illustrated for the log-normal distribution here. Particular care is needed when using cyclic data, such as phases or angles . Taking the arithmetic mean of 1° and 359° yields
4964-430: The real number set R {\displaystyle \mathbb {R} } has at least one median, but in pathological cases there may be more than one median: if F is constant 1/2 on an interval (so that f = 0 there), then any value of that interval is a median. The medians of certain types of distributions can be easily calculated from their parameters; furthermore, they exist even for some distributions lacking
5037-411: The rule are particularly common for discrete distributions. For example, any Poisson distribution has positive skew, but its mean < median whenever μ mod 1 > ln 2 {\displaystyle \mu {\bmod {1}}>\ln 2} . See for a proof sketch. When the distribution has a monotonically decreasing probability density, then the median is less than
5110-407: The same rate as much as five pumps that can each empty the tank in 15 {\displaystyle 15} minutes. AM, GM, and HM satisfy these inequalities: Equality holds if all the elements of the given sample are equal. In descriptive statistics , the mean may be confused with the median , mode or mid-range , as any of these may incorrectly be called an "average" (more formally,
5183-424: The total number of observations. Symbolically, for a data set consisting of the values x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} , the arithmetic mean is defined by the formula: (For an explanation of the summation operator, see summation .) In simpler terms, the formula for the arithmetic mean is: Total of all numbers within
5256-491: The value of extreme results in order to calculate it. 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. Because the median is simple to understand and easy to calculate, while also a robust approximation to the mean , the median is a popular summary statistic in descriptive statistics . In this context, there are several choices for
5329-419: The variance of the median equals ( π / 2 ) ⋅ ( σ 2 / n ) . {\displaystyle ({\pi }/{2})\cdot (\sigma ^{2}/n).} (See also section #Efficiency below.) We take the sample size to be an odd number N = 2 n + 1 {\displaystyle N=2n+1} and assume our variable continuous;
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