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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 occurs in a more general ...
hide. Proof without words of the AM–GM inequality: PR is the diameter of a circle centered on O; its radius AO is the arithmetic mean of a and b. Using the geometric mean theorem, triangle PGR's altitude GQ is the geometric mean. For any ratio a:b,AO ≥ GQ. Visual proof that (x + y)2 ≥ 4xy. Taking square roots and dividing by two gives the ...
In statistics, the weighted geometric mean is a generalization of the geometric mean using the weighted arithmetic mean . Given a sample and weights , it is calculated as: [ 1] The second form above illustrates that the logarithm of the geometric mean is the weighted arithmetic mean of the logarithms of the individual values. If all the weights ...
Generalized mean. Plot of several generalized means . 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 ).
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. These inequalities often appear in mathematical competitions and have ...
Definition. 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: [ 2] (For an explanation of the summation operator, see summation .)
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is a weighted sum or weighted average. Weight functions occur frequently in statistics and analysis ...
In statistics, a moving average ( rolling average or running average or moving mean[ 1] or rolling mean) is a calculation to analyze data points by creating a series of averages of different selections of the full data set. Variations include: simple, cumulative, or weighted forms. Mathematically, a moving average is a type of convolution.
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