Harmonic Mean Harmonic Meanz The harmonic mean is one of the three Pythagorean means. For all positive data sets containing at least one pair of nonequal 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 always in between. (If all values in a nonempty dataset are equal, the three means are always equal to one another; e.g. the harmonic, geometric, and arithmetic means of {2, 2, 2} are all 2.) 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. In the speed example below for instance the arithmetic mean 50 is incorrect, and too big. The harmonic mean is related to the other Pythagorean means, as seen in the third formula in the above equation. This is noticed if we interpret the denominator to be the arithmetic mean of the product of numbers n times but each time we omit the jth term.
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That is, for the first term we multiply all n numbers but omit the first, for the second we multiply all n numbers but omit 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 nth harmonic mean is related to the nth geometric and arithmetic means. 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. For a set of real variables it has been shown that A - H ≼ s2/(2 M) where A is the arithmetic mean, H is the harmonic mean, M is the maximum of the set and s2 is the variance of the set. In physics :- In certain situations, especially many situations involving rates and ratios, the harmonic mean provides the truest average. For instance, if a vehicle travels a certain distance at a speed x (e.g. 60 kilometres per hour) and then the same distance again at a speed y (e.g. 40 kilometres per hour), then its average speed is the harmonic mean of x and y (48 kilometres per hour), and its total travel time is the same as if it had traveled the whole distance at that average speed. However, if the vehicle travels for a certain amount of time at a speed x and then the same amount of time at a speed y, then its average speed is the arithmetic mean of x and y, which in the above example is 50 kilometres per hour. 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. In other sciences :- In computer science, specifically information retrieval and machine learning, the harmonic mean of the precision and the recall is often used as an aggregated performance score for the evaluation of algorithms and systems: the F-score (or F-measure).
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An interesting consequence arises from basic algebra in problems of working 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 (6 路 4)/(6 + 4), which is equal to 2.4 hours, to drain the pool together. Interestingly, this is one-half of the harmonic mean of 6 and 4. In finance :- The harmonic mean is the preferable method for averaging multiples, such as the price/earning ratio, in which price is in the numerator. If these ratios are averaged using an arithmetic mean (a common error), high data points are given greater weights than low data points. The harmonic mean, on the other hand, gives equal weight to each data point. In geometry :- In any triangle, the radius of the incircle is one-third 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. In trigonometry :- In the case of the double-angle tangent identity, if the tangent of an angle A is given as a/b, then the tangent of 2A is the product of (1) the harmonic mean of the numerator and denominator of tan A and (2) the reciprocal of (the denominator minus the numerator of tan A).
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