Harmonic Mean

The harmonic mean gives the correct average in situations where values are rates, such as speeds, prices per unit or ratios, and the quantity being held constant is the one in the numerator.

A simple example is travel speed. If you drive 120 kilometres to a city at 60 kilometres per hour and return the same distance at 40 kilometres per hour, your average speed is not 50. The trip there takes 2 hours and the trip back 3 hours, so you travel 240 kilometres in 5 hours, an average of 48 kilometres per hour. That is the harmonic mean of 60 and 40.

In finance, the harmonic mean is used when averaging valuation multiples, such as price-to-earnings ratios across a group of companies, because it reduces the influence of very high multiples. It also applies to pound-cost or dollar-cost averaging: if you invest the same amount of money at regular intervals, the average price you pay per share is the harmonic mean of the prices, which is lower than their arithmetic mean.

For any set of positive values that are not all equal, the harmonic mean is lower than the geometric mean, which is lower than the arithmetic mean.