Quasi-arithmetic mean

In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean or Kolmogorov-Nagumo-de Finetti mean is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function . It is also called Kolmogorov mean after Soviet mathematician Andrey Kolmogorov. It is a broader generalization than the regular generalized mean. If f is a function which maps an interval of the real line to the real numbers, and is both continuous and injective, the f-mean of numbers is defined as , which can also be written We require f to be injective in order for the inverse function to exist. Since is defined over an interval, lies within the domain of . Since f is injective and continuous, it follows that f is a strictly monotonic function, and therefore that the f-mean is neither larger than the largest number of the tuple nor smaller than the smallest number in . If = R, the real line, and , (or indeed any linear function , not equal to 0) then the f-mean corresponds to the arithmetic mean. If = R+, the positive real numbers and , then the f-mean corresponds to the geometric mean. According to the f-mean properties, the result does not depend on the base of the logarithm as long as it is positive and not 1. If = R+ and , then the f-mean corresponds to the harmonic mean. If = R+ and , then the f-mean corresponds to the power mean with exponent . If = R and , then the f-mean is the mean in the log semiring, which is a constant shifted version of the LogSumExp (LSE) function (which is the logarithmic sum), . The corresponds to dividing by n, since logarithmic division is linear subtraction. The LogSumExp function is a smooth maximum: a smooth approximation to the maximum function. The following properties hold for for any single function : Symmetry: The value of is unchanged if its arguments are permuted. Idempotency: for all x, . Monotonicity: is monotonic in each of its arguments (since is monotonic). Continuity: is continuous in each of its arguments (since is continuous).

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Quasi-arithmetic mean

Generalized mean

In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means). If p is a non-zero real number, and are positive real numbers, then the generalized mean or power mean with exponent p of these positive real numbers is (See p-norm).

Average

In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, and 9 (summing to 25) is 5. Depending on the context, an average might be another statistic such as the median, or mode. For example, the average personal income is often given as the median—the number below which are 50% of personal incomes and above which are 50% of personal incomes—because the mean would be higher by including personal incomes from a few billionaires.

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