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mean value[′mēn ′val·yü]
a number characterizing a group of numbers or functions.
(1) A mean value, or simply mean, of a given set of numbers x1, x2, . . . . , xn is a number that is neither greater than the largest of the numbers nor smaller than the smallest of them. The most commonly used means are the arithmetic mean a, geometric mean g, harmonic mean h, and quadratic mean q:
If all the xi, (i = 1,2,..., n) are positive, the pth-power mean Mp can be defined for any p ≠ 0:
The arithmetic, harmonic, and quadratic means are special cases of the pth-power mean: Mp is equal to a, h, and q when p = 1, –1, and 2, respectively. As p → 0, the ρth-power mean approaches the geometric mean. Consequently, we can set M0 = g. If α ≤ β, then Mα ≤ Mβ. This inequality is of great importance; in particular,
h ≤ g ≤ a ≤ q
The arithmetic and quadratic means have numerous applications in, for example, probability theory, mathematical statistics, and calculations based on the method of least squares.
The means discussed above can all be derived from the formula
when a suitable choice for the function f(ξ) is made; here, f–1(ŋ) is the inverse of f(ξ) (seeINVERSE FUNCTION). Thus, the arithmetic mean is obtained if f(ξ) = ξ, the geometric mean if f(ξ) = In ξ, the harmonic mean if f(ξ) = 1/ξ, and the quadratic mean if f(ξ) = ξ2.
In addition to ordinary pth-power means, weighted pth-power means are also used:
In particular, when p = 1,
Weighted pth-power means become ordinary pth-power means when w1, = w2 = . . . = wn. Weighted means are particularly important in processing the results of observations (seeCALCULUS OF OBSERVATIONS) when the accuracy (weight) of different observations differs.
(2) The arithmetic-geometric mean is of some interest. Suppose the arithmetic mean a, and the geometric mean gl are computed for the two positive numbers a and b, the arithmetic mean a2 and geometric mean g2are then found for the numbers ax and g], and so on. The common limit of the two sequences an and gn, whose existence was proved by K. Gauss, is the arithmetic-geometric mean of a and b. This mean is of importance in the theory of elliptic functions.
(3) A mean value of a function is a number between the greatest and least values of the function. Several mean value theorems, which establish the existence of points at which a function or its derivative attains some mean value, are known in the differential and integral calculus.
The most important mean value theorem of the differential calculus is due to Lagrange. This theorem states that if f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that f(b) – f(a) = (b – a)f′(c).
The most important mean value theorem of the integral calculus states that if f(x) is continuous on the closed interval [a, b] and if φ(x) does not change sign, then there exists a point c in (a, b) such that
In particular, if φ(x) = l.then
As a consequence, the quantity
is usually understood as the mean value of the function f(x) on the closed interval [a, b]. The mean value of a function of several variables in some domain is defined in a similar manner.