mean

mean, in mathematics, a quantity that has a value intermediate between those of the extreme members of some set. Several kinds of means exist, and the method of calculating a mean depends upon the relationship known or assumed to govern the other members. The arithmetic mean, denoted x, of a set of n numbers x₁, x₂, …, x_n is defined as the sum of the numbers divided by n:

The arithmetic mean (usually synonymous with average) represents a point about which the numbers balance. For example, if unit masses are placed on a line at points with coordinates x₁, x₂, …, x_n, then the arithmetic mean is the coordinate of the centre of gravity of the system. In statistics, the arithmetic mean is commonly used as the single value typical of a set of data. For a system of particles having unequal masses, the centre of gravity is determined by a more general average, the weighted arithmetic mean. If each number (x) is assigned a corresponding positive weight (w), the weighted arithmetic mean is defined as the sum of their products (wx) divided by the sum of their weights. In this case,

The weighted arithmetic mean also is used in statistical analysis of grouped data: each number x_i is the midpoint of an interval, and each corresponding value of w_i is the number of data points within that interval.

For a given set of data, many possible means can be defined, depending on which features of the data are of interest. For example, suppose five squares are given, with sides 1, 1, 2, 5, and 7 cm. Their average area is (1² + 1² + 2² + 5² + 7²)/5, or 16 square cm, the area of a square of side 4 cm. The number 4 is the quadratic mean (or root mean square) of the numbers 1, 1, 2, 5, and 7 and differs from their arithmetic mean, which is 3 1/5. In general, the quadratic mean of n numbers x₁, x₂, …, x_n is the square root of the arithmetic mean of their squares, The arithmetic mean gives no indication of how widely the data are spread or dispersed about the mean. Measures of the dispersion are provided by the arithmetic and quadratic means of the n differences x₁ − x, x₂ − x, …, x_n − x. The quadratic mean gives the standard deviation of x₁, x₂, …, x_n.

The arithmetic and quadratic means are the special cases p = 1 and p = 2 of the pth-power mean, M_p, defined by the formulawhere p may be any real number except zero. The case p = −1 is also called the harmonic mean. Weighted pth-power means are defined by

If x is the arithmetic mean of x₁ and x₂, the three numbers x₁, x, x₂ are in arithmetic progression. If h is the harmonic mean of x₁ and x₂, the numbers x₁, h, x₂ are in harmonic progression. A number g such that x₁, g, x₂ are in geometric progression is defined by the condition that x₁/g = g/x₂, or g² = x₁x₂; hence This g is called the geometric mean of x₁ and x₂. The geometric mean of n numbers x₁, x₂, …, x_n is defined to be the nth root of their product:

All the means discussed are special cases of a more general mean. If f is a function having an inverse f ⁻¹ (a function that “undoes” the original function), the number is called the mean value of x₁, x₂, …, x_n associated with f. When f(x) = x^p, the inverse is f ⁻¹(x) = x^1/p, and the mean value is the pth-power mean, M_p. When f(x) = ln x (the natural logarithm), the inverse is f ⁻¹(x) = e^x (the exponential function), and the mean value is the geometric mean.

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For information on the development of various definitions of the mean, see probability and statistics. For further technical information, see statistics and probability theory.