Harmonic mean

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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_i}}

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

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[edit] Examples

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 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 and the prices paid per share each month were $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 (which turns out to be $9.00) would be used. Note that in this example, the investor buying $1000 worth of the stock each month means buying 125 shares at $8 the first month, 111.11 shares at $9 the second month, and 100 shares at $10 in the third month. Fewer shares are purchased at higher prices while more shares are purchased at lower prices. Thus more weight is given to the lower prices than the higher prices in the calculation of the average cost per share ($8.926). If the investor had instead purchased 1000 shares each month then equal weight would be given to high and low purchase prices, leading to an average cost per share of $9.00. This explains why the harmonic mean is less than the arithmetic mean.

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.

This consequence arises in any work problem of n people. Shown here in simplified form,

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

[edit] Harmonic mean of two numbers

For just two numbers, the harmonic mean can be written as:

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.

[edit] 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). When the variables are set to equal each other, the harmonic mean is equal to the geometric mean, which equal to the arithmetic mean. For example, if you use 2 and 2 for your variables, you will find that the harmonic mean, geometric mean and arithmetic mean are all equal to each other (each of their values is 2).

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.[1] 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.[citation needed]

[edit] See also

[edit] References

  1. ^ *Statistical Analysis, Ya-lun Chou, Holt International, 1969, ISBN 0030730953

[edit] External links

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