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 is the number of variables divided by the sum of the reciprocals of the variables.

The harmonic mean H of the positive real numbers a1, a2, ..., an is defined to be


 * $$H = \frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}} = \frac{n}{\sum_{i=1}^n \frac{1}{a_n}}$$

That is, the harmonic mean of a group of terms is the reciprocal of the arithmetic mean of the terms' reciprocals.

Examples
In certain situations, the harmonic mean provides the truest average. For instance, if for half the distance of a trip you travel at 40 kilometres per hour and for the other half of the distance you travel at 60 kilometres per hour, then your average speed for the trip is given by the harmonic mean of 40 and 60, which is 48; that is, the total amount of time for the trip is the same as if you travelled the entire trip at 48 kilometres per hour. If you had travelled for half the time at one speed and the other half at another, the arithmetic mean, in this case 50 kilometres per hour, would provide the correct average.

Similarly, if an electrical circuit contains two resistors connected in parallel, one with a resistance of 40Ω and the other with 60Ω, then the average resistance of the two resistors is 48Ω; that is, the resistance of the circuit is the same as two 48Ω resistors connected in parallel. This is not to be confused with their equivalent resistance, 24Ω, which is the resistance needed for a single resistor to replace the two parallel resistors. The equivalent resistance is equal to one half the value of the harmonic mean of the two parallel resistances.

In finance, the harmonic mean is used to calculate the average cost of shares purchased over a period of time. For example, an investor purchases $1000 worth of stock every month for three months. If the spot prices at execution time are $8, $9, and $10, then the average price the investor paid is $8.926 per share. However, if the investor purchased 1000 shares per month, the arithmetic mean would be used.

Harmonic mean of two numbers
When dealing with just two numbers, an equivalent, sometimes more convenient, formula of their harmonic mean is given by:


 * $$H = \frac {{2} {a_1} {a_2}} {{a_1} + {a_2}}.$$

In this case, their harmonic mean is related to their arithmetic mean,


 * $$A = \frac {{a_1} + {a_2}} {2},$$

and their geometric mean,


 * $$G = \sqrt {{a_1} \cdot {a_2}},$$

by


 * $$H = \frac {G^2} {A}.$$

so


 * $$G = \sqrt {{A} {H}}$$, i. e. the geometric mean is the geometric mean of the arithmetic mean and the harmonic mean.

Note that this result holds only in the case of just two numbers.

Relationship with other means
The harmonic mean is one of the 3 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 (as the geometric mean is actually the geometric mean applied to the other two means as shown above).

It is the special case $$M_{- 1}$$ of the power mean.

It is equivalent to a weighted arithmetic mean with each value's weight being the reciprocal of the value.

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 above for instance the arithmetic mean 50 is incorrect, and too big. Such an error was apparently made in a calculation of transport capacity of American ships during World War I. The arithmetic mean of the various ships' speed was used, resulting in a total capacity estimate which proved unattainable.