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The geometric mean of a data set {,, …,} is given by: (=) =. [3]The above figure uses capital pi notation to show a series of multiplications. Each side of the equal sign shows that a set of values is multiplied in succession (the number of values is represented by "n") to give a total product of the set, and then the nth root of the total product is taken to give the geometric mean of the ...
In Euclidean geometry, the geometric mean theorem or right triangle altitude theorem is a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states that the geometric mean of the two segments equals the altitude. Expressed as a mathematical formula, if h denotes the ...
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 mathematics, the arithmetic–geometric mean(AGM or agM[1]) of two positive real numbersxand yis the mutual limit of a sequence of arithmetic meansand a sequence of geometric means. The arithmetic–geometric mean is used in fast algorithmsfor exponential, trigonometric functions, and other special functions, as well as some mathematical ...
The geometric distribution is the only memoryless discrete probability distribution.[4] It is the discrete version of the same property found in the exponential distribution. [1]: 228 The property asserts that the number of previously failed trials does not affect the number of future trials needed for a success.
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.
Geometric standard deviation. In probability theory and statistics, the geometric standard deviation (GSD) describes how spread out are a set of numbers whose preferred average is the geometric mean. For such data, it may be preferred to the more usual standard deviation. Note that unlike the usual arithmetic standard deviation, the geometric ...
A price index aggregates various combinations of base period prices ( ), later period prices ( ), base period quantities ( ), and later period quantities ( ). Price index numbers are usually defined either in terms of (actual or hypothetical) expenditures (expenditure = price * quantity) or as different weighted averages of price relatives ( ).