Harmonic mean

It is the most appropriate average for ratios and rates such as speeds,[1][2] and is normally only used for positive arguments.

[citation needed] The harmonic mean is also concave for positive arguments, an even stronger property than Schur-concavity.

Since the harmonic mean of a list of numbers tends strongly toward the least elements of the list, it tends (compared to the arithmetic mean) to mitigate the impact of large outliers and aggravate the impact of small ones.

by the inequality of arithmetic and geometric means, this shows for the n = 2 case that H ≤ G (a property that in fact holds for all n).

, the harmonic mean can be written as:[4] Three positive numbers H, G, and A are respectively the harmonic, geometric, and arithmetic means of three positive numbers if and only if[8]: p.74, #1834  the following inequality holds If a set of weights

[10] In many situations involving rates and ratios, the harmonic mean provides the correct average.

As with the previous example, the same principle applies when more than two resistors, capacitors or inductors are connected, provided that all are in parallel or all are in series.

The weighted harmonic mean is the preferable method for averaging multiples, such as the price–earnings ratio (P/E).

[14] The simple weighted arithmetic mean when applied to non-price normalized ratios such as the P/E is biased upwards and cannot be numerically justified, since it is based on equalized earnings; just as vehicles speeds cannot be averaged for a roundtrip journey (see above).

[15] In any triangle, the radius of the incircle is one-third of the harmonic mean of the altitudes.

For any point P on the minor arc BC of the circumcircle of an equilateral triangle ABC, with distances q and t from B and C respectively, and with the intersection of PA and BC being at a distance y from point P, we have that y is half the harmonic mean of q and t.[16] In a right triangle with legs a and b and altitude h from the hypotenuse to the right angle, h2 is half the harmonic mean of a2 and b2.

[17][18] Let t and s (t > s) be the sides of the two inscribed squares in a right triangle with hypotenuse c. Then s2 equals half the harmonic mean of c2 and t2.

Let a trapezoid have vertices A, B, C, and D in sequence and have parallel sides AB and CD.

One application of this trapezoid result is in the crossed ladders problem, where two ladders lie oppositely across an alley, each with feet at the base of one sidewall, with one leaning against a wall at height A and the other leaning against the opposite wall at height B, as shown.

This is used in information retrieval because only the positive class is of relevance, while number of negatives, in general, is large and unknown.

A consequence arises from basic algebra in problems where people or systems work together.

In hydrology, the harmonic mean is similarly used to average hydraulic conductivity values for a flow that is perpendicular to layers (e.g., geologic or soil) - flow parallel to layers uses the arithmetic mean.

This apparent difference in averaging is explained by the fact that hydrology uses conductivity, which is the inverse of resistivity.

In sabermetrics, a baseball player's Power–speed number is the harmonic mean of their home run and stolen base totals.

The harmonic mean takes into account the fact that events such as population bottleneck increase the rate genetic drift and reduce the amount of genetic variation in the population.

This is a result of the fact that following a bottleneck very few individuals contribute to the gene pool limiting the genetic variation present in the population for many generations to come.

When considering fuel economy in automobiles two measures are commonly used – miles per gallon (mpg), and litres per 100 km.

As the dimensions of these quantities are the inverse of each other (one is distance per volume, the other volume per distance) when taking the mean value of the fuel economy of a range of cars one measure will produce the harmonic mean of the other – i.e., converting the mean value of fuel economy expressed in litres per 100 km to miles per gallon will produce the harmonic mean of the fuel economy expressed in miles per gallon.

The geometric (G), arithmetic and harmonic means of the distribution are related by[21] The harmonic mean of type 1 Pareto distribution is[22] where k is the scale parameter and α is the shape parameter.

The expectation of the harmonic mean is the same as the non-length biased version E( x ) The problem of length biased sampling arises in a number of areas including textile manufacture[25] pedigree analysis[26] and survival analysis[27] Akman et al. have developed a test for the detection of length based bias in samples.

[28] If X is a positive random variable and q > 0 then for all ε > 0[29] Assuming that X and E(X) are > 0 then[29] This follows from Jensen's inequality.

Gurland has shown that[30] for a distribution that takes only positive values, for any n > 0 Under some conditions[31] where ~ means approximately equal to.

Similarly a first order approximation to the bias and variance of H3 are[33] In numerical experiments H3 is generally a superior estimator of the harmonic mean than H1.

The Environmental Protection Agency recommends the use of the harmonic mean in setting maximum toxin levels in water.

[34] In geophysical reservoir engineering studies, the harmonic mean is widely used.