Harmonic mean

The harmonic mean is one of the types of mean we typically use when we are working with rates.

Average speed

For example, imagine we took a train on a day trip, and the average speed of the train was 120 km/h. On the way back we went by train again, along the same route, but this time the train crawled along at just 40 km/h. What was the average speed of the train for the whole journey there and back? It is rather tempting to work out the average speed using the arithmetic mean, like this

$$\frac{120\text{km}/\text{h}+40\text{km}/\text{h}}{2}=80\text{km}/\text{h}$$

and to say that the average speed was 80 km/h. But that is not true. We can easily show this: suppose the train travelled for one hour on the way there. Since its average speed was 120 km/h, it covered exactly 120 km in that hour. If it travelled at an average speed of 40 km/h on the way back, it needed 3 hours to cover 120 km. So altogether the train covered 240 km in 4 hours. From this we can work out the average speed

$$\frac{240\text{km}}{4\text{h}}=60\text{km}/\text{h}$$

So the arithmetic mean let us down. How would we have to work out the mean of the two speeds, 120 and 40, to get the correct result?

The harmonic mean

Instead of the arithmetic mean we can use the harmonic mean. We work it out by building a fraction whose numerator is the number of values we are averaging – in our case 2, because we have two speeds:

$$\frac{2}{?}$$

In the denominator we put the sum of the reciprocals of both speeds. If the speed is 120, its reciprocal is 120⁻¹, which is $\frac{1}{120}$. We get the fraction

$$\frac{2}{\frac{1}{120}+\frac{1}{40}}$$

We rewrite the fractions in the denominator over a common denominator:

$$\frac{2}{\frac{1}{120}+\frac{3}{120}}$$

Now we can add the fractions easily:

$$\frac{2}{\frac{4}{120}}$$

This fraction looks odd, but don't worry, it works out well. We rewrite the fraction as a product like this:

$$\frac{2}{\frac{4}{120}}=2\cdot \frac{120}{4}$$

If this step confused you, remember that dividing by one half is the same as multiplying by two. So dividing 2 by the fraction $\frac{4}{120}$ is the same as multiplying 2 by the reciprocal fraction $\frac{120}{4}$. Now we just simplify the fraction a little and we get our result:

$$2\cdot \frac{120}{4}=\frac{240}{4}=60$$

The harmonic mean came out as 60, which is the correct result.

General formula for the harmonic mean

Take n numbers x₁, x₂, …, x_n. Their harmonic mean H is then

$$H=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\dots +\frac{1}{x_n}}$$

If you like sigma notation, we can also write the harmonic mean like this:

$$H=\frac{n}{\sum _{i=1}^n\frac{1}{x_i}}$$

For example, the harmonic mean of the four values 1, 2, 3, 4 is

$$\begin{array}{rcl}H& =& \frac{4}{\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}}\\ & =& \frac{4}{\frac{12}{12}+\frac{6}{12}+\frac{4}{12}+\frac{3}{12}}\\ & =& \frac{4}{\frac{25}{12}}\\ & =& 4\cdot \frac{12}{25}=\frac{48}{25}=1.92\end{array}$$