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The harmonic mean is denoted by H in purple, while the arithmetic mean is A in red and the geometric mean is G in blue. Q denotes a fourth mean, the quadratic mean . Since a hypotenuse is always longer than a leg of a right triangle , the diagram shows that H ≤ G ≤ A ≤ Q {\displaystyle H\leq G\leq A\leq Q} .
The geometric mean is one of the three classical Pythagorean means, together with the arithmetic mean and the harmonic mean. For all positive data sets containing at least one pair of unequal values, the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is ...
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–harmonic mean is equivalent to the geometric mean. The arithmetic–geometric mean can be used to compute – among others – logarithms , complete and incomplete elliptic integrals of the first and second kind , [ 13 ] and Jacobi elliptic functions .
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:
In mathematics, the geometric–harmonic mean M(x, y) of two positive real numbers x and y is defined as follows: we form the geometric mean of g 0 = x and h 0 = y and call it g 1, i.e. g 1 is the square root of xy. We also form the harmonic mean of x and y and call it h 1, i.e. h 1 is the reciprocal of the arithmetic mean of the reciprocals of ...
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
In mathematics, generalised 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 ).