In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.
It is sometimes used for ratios and rates such as speeds,Course and is normally used for positive arguments only.
The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the numbers, that is, the generalized f-mean with . For example, the harmonic mean of 1, 4, and 4 is
Definition The harmonic mean H of the positive real numbers isIt is the reciprocal of the arithmetic mean of the reciprocals, and vice versa:where the arithmetic mean is
The harmonic mean is a Schur-concave function. It is greater than or equal to the minimum of its arguments: for positive arguments, . Indeed, if , then , while , which gives the two inequalities from the definition above. Thus, the harmonic mean cannot be made arbitrarily large by changing some values to bigger ones while having at least one value unchanged.
The harmonic mean is also concave for positive arguments, an even stronger property than Schur-concavity.
Relationship with other means
For all positive data sets containing at least one pair of nonequal values, the harmonic mean is always the least of the three Pythagorean means,Da-Feng Xia, Sen-Lin Xu, and Feng Qi, "A proof of the arithmetic mean-geometric mean-harmonic mean inequalities", RGMIA Research Report Collection, vol. 2, no. 1, 1999, http://ajmaa.org/RGMIA/papers/v2n1/v2n1-10.pdf while the arithmetic mean is always the greatest of the three and the geometric mean is always in between. (If all values in a nonempty data set are equal, the three means are always equal.)
It is the special case M−1 of the power mean: 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.
The arithmetic mean is often mistakenly used in places calling for the harmonic mean.*Statistical Analysis, Ya-lun Chou, Holt International, 1969, In the speed example below, for instance, finding the average speed based on equal amounts of distance, the arithmetic mean of 40 is incorrect. Since the arithmetic mean is always greater than or equal to the harmonic mean, this necessarily means 40 is larger than the correct answer.
The harmonic mean is related to the other Pythagorean means, as seen in the equation below. This can be seen by interpreting the denominator to be the arithmetic mean of the product of numbers n times but each time omitting the j-th term. That is, for the first term, we multiply all n numbers except the first; for the second, we multiply all n numbers except the second; and so on. The numerator, excluding the n, which goes with the arithmetic mean, is the geometric mean to the power n. Thus the n-th harmonic mean is related to the n-th geometric and arithmetic means. The general formula is If a set of non-identical numbers is subjected to a mean-preserving spread — that is, two or more elements of the set are "spread apart" from each other while leaving the arithmetic mean unchanged — then the harmonic mean always decreases.Mitchell, Douglas W., "More on spreads and non-arithmetic means," The Mathematical Gazette 88, March 2004, 142–144.
Harmonic mean of two or three numbers
Two numbers
For the special case of just two numbers, and , the harmonic mean can be written as: or (Note that the harmonic mean is undefined if , i.e. .)
In this special case, the harmonic mean is related to the arithmetic mean and the geometric mean by
Since 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). It also follows that , meaning the two numbers' geometric mean equals the geometric mean of their arithmetic and harmonic means.
Three numbers For the special case of three numbers, , and , the harmonic mean can be written as:
Three positive numbers H, G, and A are respectively the harmonic, geometric, and arithmetic means of three positive numbers if and only ifInequalities proposed in "Crux Mathematicorum", . the following inequality holds
Weighted harmonic mean If a set of weights , ..., is associated to the data set , ..., , the weighted harmonic mean is defined by Ferger F (1931) The nature and use of the harmonic mean. Journal of the American Statistical Association 26(173) 36-40
The unweighted harmonic mean can be regarded as the special case where all of the weights are equal.
Examples In analytic number theory Prime number theory The prime number theorem states that the number of primes less than or equal to is asymptotically equal to the harmonic mean of the first natural numbers.
In physics Average speed In many situations involving rates and ratios, the harmonic mean provides the correct average. For instance, if a vehicle travels a certain distance s outbound at a speed (e.g. 60 km/h) and returns the same distance at a speed (e.g. 20 km/h), then its average speed is the harmonic mean of and (30 km/h), not the arithmetic mean (40 km/h). The total travel time is the same as if it had traveled the whole distance at that average speed. This can be proven as follows:
However, if the vehicle travels for a certain amount of time at a speed and then the same amount of time at a speed , then its average speed is the arithmetic mean of and , which in the above example is 40 km/h.
The same principle applies to more than two segments: given a series of sub-trips at different speeds, if each sub-trip covers the same distance, then the average speed is the harmonic mean of all the sub-trip speeds, and if each sub-trip takes the same amount of time, then the average speed is the arithmetic mean of all the sub-trip speeds. If neither is the case, then a weighted harmonic mean or weighted arithmetic mean is needed. For the arithmetic mean, the speed of each portion of the trip is weighted by the duration of that portion, while for the harmonic mean, the corresponding weight is the distance. In both cases, the resulting formula reduces to dividing the total distance by the total time.
However, one may avoid the use of the harmonic mean for the case of "weighting by distance". Pose the problem as finding the pace of the trip where pace (in hours per kilometer) is the inverse of speed. When trip pace is found, invert it so as to find the "true" average trip speed. For each trip segment , the pace . Then take the weighted arithmetic mean of the weighted by their respective distances (optionally with the weights normalized so they sum to 1 by dividing them by trip length). This gives the true average pace (in time per kilometer). It turns out that this procedure, which can be done with no knowledge of the harmonic mean, amounts to the same mathematical operations as one would use in solving this problem by using the harmonic mean. Thus it illustrates why the harmonic mean works in this case.
Density
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.
Electricity
If one connects two electrical resistors in parallel, one having resistance R1 (e.g., 60 Ω) and one having resistance R2 (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 R1 and R2 (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.
Average resistance for both resistors in parallel
= = = 48
However, if one connects the resistors in series, then the average resistance is the arithmetic mean of R1 and R2 (50 Ω), with total resistance equal to twice this, the sum of R1 and R2 (100 Ω). This principle applies to capacitors in parallel or to inductors in series.
Average resistance for both resistors in series
= 50
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.
Optics 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 finance The weighted harmonic mean is the preferable method for averaging multiples, such as the price–earnings ratio (P/E). If these ratios are averaged using a weighted arithmetic mean, high data points are given greater weights than low data points. The weighted harmonic mean, on the other hand, correctly weights each data point. 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).
In geometry 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, is half the harmonic mean of and .Voles, Roger, "Integer solutions of ," Mathematical Gazette 83, July 1999, 269–271.Richinick, Jennifer, "The upside-down Pythagorean Theorem," Mathematical Gazette 92, July 2008, 313–;317.
Let t and s (t > s) be the sides of the two inscribed squares in a right triangle with hypotenuse c. Then equals half the harmonic mean of and .
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 other sciences