In mathematics, mean has several different definitions depending on the context. In probability and statistics, population mean and expected value are used synonymously to refer to one measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. In the case of a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability P(x), and then adding all these products together, giving μ = ∑ x P ( x ) {\displaystyle \mu =\sum xP(x)} . An analogous formula applies to the case of a continuous probability distribution. Not every probability distribution has a defined mean; see the Cauchy distribution for an example. Moreover, for some distributions the mean is infinite: for example, when the probability of the value 2 n {\displaystyle 2^{n}} is 1 2 n {\displaystyle {\tfrac {1}{2^{n}}}} for n = 1, 2, 3, .... For a data set, the terms arithmetic mean, mathematical expectation, and sometimes average are used synonymously to refer to a central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x1, x2, ..., xn is typically denoted by x ¯ {\displaystyle {\bar {x}}} , pronounced "x bar". If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is termed the sample mean (denoted x ¯ {\displaystyle {\bar {x}}} ) to distinguish it from the population mean (denoted μ {\displaystyle \mu } or μ x {\displaystyle \mu _{x}} ). For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual divided by the total number of individuals. The sample mean may differ from the population mean, especially for small samples. The law of large numbers dictates that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean. Outside probability and statistics, a wide range of other notions of "mean" are often used in geometry and analysis; examples are given below.
From Middle English menen (“to intend; remember; lament; comfort”), from Old English mǣnan (“to mean, signify; lament”), from Proto-Germanic *mainijaną (“to mean, think; lament”), from Proto-Indo-European *meyn- (“to think”). Germanic cognates include West Frisian miene (“to deem, think”) (Old Frisian mēna (“signify”)), Dutch menen (“to believe, think, mean”) (Middle Dutch menen (“think, intend”)), German meinen (“to think, mean, believe”), Old Saxon mēnian. Indo-European cognates include Old Irish mían (“wish, desire”) and Polish mienić (“signify, believe”). Related to moan.
mean (third-person singular simple present means, present participle meaning, simple past and past participle meant)
mean (third-person singular simple present means, present participle meaning, simple past and past participle meaned)
From Middle English mene, imene, from Old English mǣne, ġemǣne (“common, public, general, universal”), from Proto-Germanic *gamainiz (“common”), from Proto-Indo-European *mey- (“to change, exchange, share”). Cognate with West Frisian mien (“general, universal”), Dutch gemeen (“common, mean”), German gemein (“common, mean, nasty”), Gothic 𐌲𐌰𐌼𐌰𐌹𐌽𐍃 (gamains, “common, unclean”), Latin commūnis (“shared, common, general”) (Old Latin comoinem).
mean (comparative meaner, superlative meanest)
From Middle English meene, from Old French meien (French moyen), Late Latin mediānus (“that is in the middle, middle”), from Latin medius (“middle”). Cognate with mid. For the musical sense, compare the cognate Italian mezzano.
mean (not comparable)
mean (plural means)
meen