Subcontrary mean

Harmonic mean

In mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired.

The harmonic mean H of the positive real numbers x1, x2, ..., xn is defined to be

H = frac{n}{frac{1}{x_1} + frac{1}{x_2} + cdots + frac{1}{x_n}} = frac{n}{sum_{i=1}^n 1/x_i}, qquad x_i > 0 text{ for all } i.

Equivalently, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals.

Relationship with other means

The harmonic mean is one of the three Pythagorean means. For a given data set, the harmonic mean is always the least of the three, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between. When the data values to be averaged are equal to each other, the harmonic mean is equal to both the geometric mean and the arithmetic mean. For example, if the values are 2 and 2, the harmonic mean, geometric mean, and arithmetic mean are all equal (in this case, 2).

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 incorrectly used in places calling for the harmonic mean. In the speed example below for instance the arithmetic mean 50 is incorrect, and too big.

Weighted harmonic mean

If a set of weights w_1, ..., w_n is associated to the dataset x_1, ..., x_n, the weighted harmonic mean is defined by
sum_{i=1}^n w_i bigg/ sum_{i=1}^n frac{w_i}{x_i}.
The harmonic mean is the special case where all weights are equal to 1.


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. (If neither is the case, then a weighted harmonic mean or weighted arithmetic mean is needed.)

Similarly, if one connects two electrical resistors in parallel, one having resistance x (e.g. 60Ω) and one having resistance y (e.g. 40Ω), 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 (48Ω): the equivalent resistance in either case is 24Ω (one-half of the harmonic mean). However, if one connects the resistors in series, then the average resistance is the arithmetic mean of x and y (with total resistance equal to the sum of x and y). And, as with previous example, the same principle applies when more than two resistors are connected, provided that all are in parallel or all are in series.

In other sciences

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 frac {{6} cdot {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.

Harmonic mean of two numbers

For the special case of just two numbers x_1 and x_2, the harmonic mean can be written
H = frac{2 x_1 x_2}{x_1 + x_2}.

In this special case, the harmonic mean is related to the arithmetic mean A = (x_1 + x_2)/2 and the geometric mean G = sqrt{x_1 x_2}, by

H = frac {G^2} {A}.

So G = sqrt{A H}, which means the geometric mean, for two numbers, is the geometric mean of the arithmetic mean and the harmonic mean.

See also


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