Harmonic Mean

Tax Glossary Definition

The Harmonic Mean is a type of average calculated by dividing the number of observations by the reciprocal of each number in the series. It is particularly useful in situations where the rates of change are important, such as in finance or economics.

Example 1: Average Speed

Scenario:

A car travels 60 km at 30 km/h and then another 60 km at 60 km/h.
Total distance = 120 km

Average speed using HM:

$\frac{2}{\frac{1}{30} + \frac{1}{60}} = \frac{2}{\frac{2 + 1}{60}} = \frac{2}{\frac{3}{60}} = \frac{2 \times 60}{3} = 40 \text{ km/h}$

Arithmetic mean (45 km/h) would overstate the true average because the car spends more time at the slower speed.
Example 2: Finance – Price-to-Earnings Ratio (P/E)

Scenario:
- Two companies have P/E ratios of 10 and 20.
Harmonic mean P/E:
$\frac{2}{\frac{1}{10} + \frac{1}{20}} = \frac{2}{0.1 + 0.05} = \frac{2}{0.15} \approx 13.33$
- The harmonic mean gives a more accurate average P/E when considering investment portfolios