Harmonic mean

The harmonic mean is the number of values divided by the sum of their reciprocals. It is the correct average for rates and ratios, such as average speed or portfolio price-earnings ratios.

What it is

See it move

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A van covers the same 60 km route on four days at 60, 40, 30 and 60 km/h. The arithmetic mean, (60+40+30+60) ÷ 4 = 47.5 km/h, overstates the true average. The harmonic mean, 4 ÷ (1/60+1/40+1/30+1/60) = 480/11 ≈ 43.64 km/h, correctly reflects the extra time spent at the slower speeds.

The formula

LaTeX

HM = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}

Variables

$n$: Number of observed values (count)
$x_{i}$: Each individual observed value (a rate or ratio)

The correct average for rates and ratios computed over a fixed base, such as speed over a fixed distance.

Check yourself

PracticeCORE

A fund holds three stocks in equal euro amounts, with price-earnings ratios of 10, 15 and 30. Using the harmonic mean, what is the portfolio's aggregate P/E ratio?

Select an answer to check your understanding.