How to Calculate the Harmonic Mean

Takes the reciprocal of the mean of the reciprocals, n ÷ Σ(1/x). It is the right average for rates: driving out at 40 km/h and back at 60 km/h gives an average speed of 48 km/h, not 50.

Explanation

The harmonic mean averages the reciprocals of the values and then takes the reciprocal of that result.

H = \dfrac{n}{\dfrac{1}{x_1} + \dfrac{1}{x_2} + \cdots + \dfrac{1}{x_n}}

It is the right average for rates — quantities of the form "so much per unit", such as speed or fuel economy.

Example

You drive out at 40 km/h and return along the same road at 60 km/h. The average speed is not 50 km/h.

H = \dfrac{2}{\dfrac{1}{40} + \dfrac{1}{60}} = \dfrac{2}{\dfrac{3}{120} + \dfrac{2}{120}} = \dfrac{2}{\dfrac{5}{120}} = 48

The average speed is 48 km/h. Check it with a 120 km leg: the trip out takes 3 hours, the return 2 hours, so 240 km in 5 hours, and $240 \div 5 = 48$ .

You spend more time at the slower speed, which is why the answer falls below the naive 50.

Watch out

All values must be positive, since the formula takes reciprocals
Equal distances call for the harmonic mean, but equal times call for the arithmetic mean. What is held constant decides which average applies
The arithmetic mean is always at least the geometric mean, which is always at least the harmonic mean