Harmonic Mean of a List Calculator
Compute the harmonic mean of a list of positive numbers — the right average for rates, ratios and speeds.
How it works
Harmonic mean = n ÷ Σ(1 ÷ xᵢ)
The harmonic mean is the reciprocal of the average of the reciprocals, and it is the correct average whenever the quantities are rates defined per unit of something fixed — average speed over equal distances, average price-earnings across equal dollar amounts, or parallel resistances. It always sits at or below the geometric and arithmetic means and is pulled hard toward the smallest values, so a single tiny number drags it down sharply.
Worked example
For 1, 2, 4: Σ(1/xᵢ) = 1 + 0.5 + 0.25 = 1.75 and n = 3, so the harmonic mean = 3 ÷ 1.75 = 1.714. Compare the arithmetic mean of 2.333 — the harmonic mean is smaller, as it always is for unequal positive numbers.
Sources: Harmonic mean — Wikipedia
Frequently asked questions
When should I use the harmonic mean instead of the ordinary average?
Use it for rates over a fixed base — for example the average speed of a round trip that covers equal distances at different speeds, where the arithmetic mean would overstate the true average.
Why can there be no zeros or negatives?
The formula divides by each value. A zero makes 1 ÷ x infinite and a negative makes the mean meaningless, so the harmonic mean is defined only for positive numbers.
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This calculator is for educational purposes. Double-check important results before acting on them.