Statistics/Summary/Averages/Harmonic Mean

The arithmetic mean cannot be used when we want to average quantities such as speed.

Consider the example below:

Example 1: X went from Town A to Town B at a speed of 40 km per hour and returned from Town B to Town A at a speed of 80 km per hour. The distance between the two towns is 40 km. Required: What was the average speed of X.

Solution: The answer to the question above is not 60, which would be the case if we took the arithmetic mean of the speeds. Rather, we must find the harmonic mean.

For two quantities A and B, the harmonic mean is given by: $\frac{2}{\frac{1}{A}+\frac{1}{B}}$

This can be simplified by adding in the denominator and multiplying by the reciprocal: $\frac{2}{\frac{1}{A}+\frac{1}{B}}=\frac{2}{\frac{B+A}{AB}}=\frac{2AB}{A+B}$

For N quantities: A, B, C......

Harmonic mean = $\frac{N}{\frac{1}{A}+\frac{1}{B}+\frac{1}{C}+\dots }$

Let us try out the formula above on our example:

Harmonic mean = $\frac{2AB}{A+B}$

Our values are A = 40, B = 80. Therefore, harmonic mean $=\frac{2\times 40\times 80}{40+80}=\frac{6400}{120}\approx 53.333$

Is this result correct? We can verify it. In the example above, the distance between the two towns is 40 km. So the trip from A to B at a speed of 40 km will take 1 hour. The trip from B to A at a speed to 80 km will take 0.5 hours. The total time taken for the round distance (80 km) will be 1.5 hours. The average speed will then be $\frac{80}{1.5}\approx$ 53.33 km/hour.

The harmonic mean also has physical significance.