Description:
The harmonic mean is the number of variables divided by the sum of the reciprocals of the variables. Typically, it is appropriate for situations when the average of rates is desired.
Example:
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.
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.
Sources:
http://en.wikipedia.org/wiki/Harmonic_mean
