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The geometric mean has from time to time been used to calculate financial indices (the averaging is over the components of the index). For example, in the past the FT 30 index used a geometric mean. [8] It is also used in the CPI calculation [9] and recently introduced "RPIJ" measure of inflation in the United Kingdom and in the European Union.
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 geometric standard deviation is used as a measure of log-normal dispersion analogously to the geometric mean. [3] As the log-transform of a log-normal distribution results in a normal distribution, we see that the geometric standard deviation is the exponentiated value of the standard deviation of the log-transformed values, i.e. = ( ()).
Nomograms to graphically calculate arithmetic (1), geometric (2) and harmonic (3) means, z of x=40 and y=10 (red), and x=45 and y=5 (blue) Of all pairs of different natural numbers of the form ( a , b ) such that a < b , the smallest (as defined by least value of a + b ) for which the arithmetic, geometric and harmonic means are all also ...
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:
The number of digits in which a n and g n agree (underlined) approximately doubles with each iteration. The arithmetic–geometric mean of 24 and 6 is the common limit of these two sequences, which is approximately 13.458 171 481 725 615 420 766 813 156 974 399 243 053 838 8544.
Proof of = (geometric mean). For the purpose of the proof, we will assume without loss of generality that [,] and = = We can rewrite the definition of using the exponential function as
area of grey square = area of grey rectangle: = = In Euclidean geometry, the right triangle altitude theorem or geometric mean theorem is a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse.