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In mathematics, the QM-AM-GM-HM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean, geometric mean, arithmetic mean, and quadratic mean (also known as root mean square). Suppose that ,, …, are positive real numbers. Then
A geometric construction of the quadratic mean and the Pythagorean means (of two numbers a and b). Harmonic mean denoted by H, geometric by G, arithmetic by A and quadratic mean (also known as root mean square) denoted by Q. Comparison of the arithmetic, geometric and harmonic means of a pair of numbers.
In forestry, quadratic mean diameter or QMD is a measure of central tendency which is considered more appropriate than arithmetic mean for characterizing the group of trees which have been measured. For n trees, QMD is calculated using the quadratic mean formula:
The arithmetic mean, or less precisely the average, of a list of n numbers x 1, x 2, . . . , x n is the sum of the numbers divided by n: + + +. The geometric mean is similar, except that it is only defined for a list of nonnegative real numbers, and uses multiplication and a root in place of addition and division:
In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) [1] are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means ( arithmetic , geometric , and harmonic means ).
The RMS is also known as the quadratic mean (denoted ), [2] [3] a special case of the generalized mean. The RMS of a continuous function is denoted f R M S {\displaystyle f_{\mathrm {RMS} }} and can be defined in terms of an integral of the square of the function.
In the speed example below for instance, the arithmetic mean of 40 is incorrect, and too big. The harmonic mean is related to the other Pythagorean means, as seen in the equation below. This can be seen by interpreting the denominator to be the arithmetic mean of the product of numbers n times but each time omitting the j-th term.
The arithmetic mean can be similarly defined for vectors in multiple dimensions, not only scalar values; this is often referred to as a centroid. More generally, because the arithmetic mean is a convex combination (meaning its coefficients sum to ), it can be defined on a convex space, not only a vector space.