Сандар жиынының кері шамаларының орташасының керісі

Inverse of the average of the inverses of a set of numbers
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired. The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observations. As a simple example, the harmonic mean of 1, 4, and 4 is

If a set of weights , , is associated to the dataset , , , the weighted harmonic mean is defined by

The unweighted harmonic mean can be regarded as the special case where all of the weights are equal.

Similarly, if one wishes to estimate the density of an alloy given the densities of its constituent elements and their mass fractions (or, equivalently, percentages by mass), then the predicted density of the alloy (exclusive of typically minor volume changes due to atom packing effects) is the weighted harmonic mean of the individual densities, weighted by mass, rather than the weighted arithmetic mean as one might at first expect. To use the weighted arithmetic mean, the densities would have to be weighted by volume. Applying dimensional analysis to the problem while labeling the mass units by element and making sure that only like element masses cancel makes this clear.

If one connects two electrical resistors in parallel, one having resistance x (e. g., 60 Ω) and one having resistance y (e. g., 40 Ω), then the effect is the same as if one had used two resistors with the same resistance, both equal to the harmonic mean of x and y (48 Ω): the equivalent resistance, in either case, is 24 Ω (one half of the harmonic mean). This same principle applies to capacitors in series or to inductors in parallel. However, if one connects the resistors in series, then the average resistance is the arithmetic mean of x and y (50 Ω), with total resistance equal to twice this, the sum of x and y (100 Ω). This principle applies to capacitors in parallel or to inductors in series. 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 "conductivity effective mass" of a semiconductor is also defined as the harmonic mean of the effective masses along the three crystallographic directions.

As for other optic equations, the thin lens equation = + can be rewritten such that the focal length f is one half of the harmonic mean of the distances of the subject u and object v from the lens. Two thin lenses of focal length f1 and f2 in series is equivalent to two thin lenses of focal length fhm, their harmonic mean, in series. Expressed as optical power, two thin lenses of optical powers P1 and P2 in series is equivalent to two thin lenses of optical power Pam, their arithmetic mean, in series.

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.

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. 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. Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD. Then FG is the harmonic mean of AB and DC. (This is provable using similar triangles.) 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. The ladders cross at a height of h above the alley floor. Then h is half the harmonic mean of A and B. This result still holds if the walls are slanted but still parallel and the "heights" A, B, and h are measured as distances from the floor along lines parallel to the walls. This can be proved easily using the area formula of a trapezoid and area addition formula. In an ellipse, the semi latus rectum (the distance from a focus to the ellipse along a line parallel to the minor axis) is the harmonic mean of the maximum and minimum distances of the ellipse from a focus.

In computer science, specifically information retrieval and machine learning, the harmonic mean of the precision (true positives per predicted positive) and the recall (true positives per real positive) is often used as an aggregated performance score for the evaluation of algorithms and systems: the F score (or F measure). This is used in information retrieval because only the positive class is of relevance, while number of negatives, in general, is large and unknown. It is thus a trade off as to whether the correct positive predictions should be measured in relation to the number of predicted positives or the number of real positives, so it is measured versus a putative number of positives that is an arithmetic mean of the two possible denominators. A consequence arises from basic algebra in problems where people or systems work together. As an example, if a gas powered pump can drain a pool in 4 hours and a battery powered pump can drain the same pool in 6 hours, then it will take both pumps , which is equal to 2.4 hours, to drain the pool together. This is one half of the harmonic mean of 6 and 4: That is, the appropriate average for the two types of pump is the harmonic mean, and with one pair of pumps (two pumps), it takes half this harmonic mean time, while with two pairs of pumps (four pumps) it would take a quarter of this harmonic mean time. 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. In population genetics, the harmonic mean is used when calculating the effects of fluctuations in the census population size on the effective population size. 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. For calculating the average fuel consumption of a fleet of vehicles from the individual fuel consumptions, the harmonic mean should be used if the fleet uses miles per gallon, whereas the arithmetic mean should be used if the fleet uses litres per 100 km. In the USA the CAFE standards (the federal automobile fuel consumption standards) make use of the harmonic mean. In chemistry and nuclear physics the average mass per particle of a mixture consisting of different species (e. g., molecules or isotopes) is given by the harmonic mean of the individual species' masses weighted by their respective mass fraction.

For a random sample, the harmonic mean is calculated as above. Both the mean and the variance may be infinite (if it includes at least one term of the form 1/0).