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The generalized mean, also known as the power mean or Hölder mean, is an abstraction of the quadratic, arithmetic, geometric, and harmonic means. It is defined for a set of n positive numbers xi by. [1] By choosing different values for the parameter m, the following types of means are obtained: maximum of. quadratic mean.
The arithmetic mean of a set of observed data is equal to the sum of the numerical values of each observation, divided by the total number of observations. Symbolically, for a data set consisting of the values , the arithmetic mean is defined by the formula: (For an explanation of the summation operator, see summation.)
Average. In ordinary language, an average is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the arithmetic mean – the sum of the numbers divided by how many numbers are in the list.
Harmonic Means for Beta distribution Purple=H (X), Yellow=H (1-X), larger values alpha and beta in front. The harmonic mean of a beta distribution with shape parameters α and β is: The harmonic mean with α < 1 is undefined because its defining expression is not bounded in [0, 1]. Letting α = β.
Logarithmic mean. Three-dimensional plot showing the values of the logarithmic mean. In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient. This calculation is applicable in engineering problems involving heat and mass transfer.
The Catalan numbers can be interpreted as a special case of the Bertrand's ballot theorem. Specifically, is the number of ways for a candidate A with n + 1 votes to lead candidate B with n votes. The two-parameter sequence of non-negative integers is a generalization of the Catalan numbers.
Weighted arithmetic mean. The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also ...
Harmonic number. The harmonic number with (red line) with its asymptotic limit (blue line) where is the Euler–Mascheroni constant. In mathematics, the n -th harmonic number is the sum of the reciprocals of the first n natural numbers: [1] Starting from n = 1, the sequence of harmonic numbers begins: Harmonic numbers are related to the ...