mean

There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
Pythagorean means consist of arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM). The AM is the sum of numbers divided by the number of numbers, GM is an average for sets of positive numbers based on their product, and HM is an average for sets of numbers defined in relation to a unit of measurement. The relationship between AM, GM, and HM is represented by the inequality AM ≥ GM ≥ HM.
Statistical location covers mean, median, and mode, where mean may not always be the same as the median or mode for skewed distributions. The mean of a probability distribution is the long-run average value of a random variable with that distribution.
Generalized means include power mean and f-mean. The power mean is an abstraction of quadratic, arithmetic, geometric, and harmonic means, while the f-mean generalizes the concept further.
Other specialized means discussed are weighted arithmetic mean, truncated mean, interquartile mean, mean of a function, mean of angles and cyclical quantities, Fréchet mean, triangular sets, and Swanson's rule.
For a data set, the arithmetic mean, also known as "arithmetic average", is a measure of central tendency of a finite 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 using an overhead bar,






x
¯





{\displaystyle {\bar {x}}}
. 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 (






x
¯





{\displaystyle {\bar {x}}}
) to distinguish it from the mean, or expected value, of the underlying distribution, the population mean (denoted



μ


{\displaystyle \mu }
or




μ

x




{\displaystyle \mu _{x}}
).Outside probability and statistics, a wide range of other notions of mean are often used in geometry and mathematical analysis; examples are given below.

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