harmonic mean

Harmonic mean

$${\bar{x}}_h=\frac{N}{{\sum }_{i=1}^N\frac{1}{x_i}}$$

Use it when values are rates with a fixed numerator (speeds over equal distances, prices when buying fixed € amounts).

On a 40km commute, driving 40km/h from A to B and 80km/h from B to A

The average speed is not 60km/h, but

$${\bar{x}}_h=\frac{2}{\frac{1}{40}+\frac{1}{80}}=53.33$$

Intuition

Each $x_i$ is of the form $\frac{\text{fixed quantity}}{\text{variable quantity}}$, so $\frac{1}{x_i}$ is of the form $\frac{\text{variable quantity}}{\text{fixed quantity}}$. The harmonic mean is the fixed quantity divided by the average variable quantity.