Mean

Mean

There are several kinds of means in various branches of mathematics (especially statistics). For a data set, the arithmetic mean, also called the mathematical expectation or average, is the 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 the sample mean (denoted x ¯ {\displaystyle {\bar {x}}} ) to distinguish it from the mean of the underlying distribution, the population mean (denoted μ {\displaystyle \mu } or μ x {\displaystyle \mu _{x}} ). Pronounced "mew" /'mjuː/. In probability and statistics, the population mean, or expected value, are a 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 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.

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