mean value

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mean value - an average of n numbers computed by adding some function of the numbers and dividing by some function of nmeanstatistics - a branch of applied mathematics concerned with the collection and interpretation of quantitative data and the use of probability theory to estimate population parametersaverage, norm - a statistic describing the location of a distribution; "it set the norm for American homes"arithmetic mean, expected value, first moment, expectation - the sum of the values of a random variable divided by the number of valuesgeometric mean - the mean of n numbers expressed as the n-th root of their productharmonic mean - the mean of n numbers expressed as the reciprocal of the arithmetic mean of the reciprocals of the numbers

Mean Value

mean value

[′mēn ′val·yü] (mathematics) For a function ƒ(x) defined on an interval (a,b), the integral from a to b of ƒ(x) dx divided by b - a. (mathematics) mean

Mean Value

a number characterizing a group of numbers or functions.

(1) A mean value, or simply mean, of a given set of numbers x₁, x₂, . . . . , x_n 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 x_i, (i = 1,2,..., n) are positive, the pth-power mean M_p can be defined for any p ≠ 0:

The arithmetic, harmonic, and quadratic means are special cases of the pth-power mean: M_p 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 M_α ≤ M_β. 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 = . . . = w_n. 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 g_l are computed for the two positive numbers a and b, the arithmetic mean a₂ and geometric mean g₂are then found for the numbers a_x and g], and so on. The common limit of the two sequences a_n and g_n, 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.

mean value

Mean Value

mean value

Mean Value

mean value

Synonyms for mean value

noun an average of n numbers computed by adding some function of the numbers and dividing by some function of n

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