Harmonic Mean Calculator
Paste positive rates to get their harmonic mean.
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Harmonic Mean Calculator by TheCalcTools
How the Harmonic Mean Calculator works
The harmonic mean is the number of values divided by the sum of their reciprocals. It is the right average for rates when the same amount of something is done at each rate.
Drive 60 km at 60 km/h and 60 km back at 40 km/h and your average speed is the harmonic mean, 48 km/h, not 50 km/h, because you spend longer at the slower speed.
The formula
Worked example
Using the default values shown in the calculator:
| Positive numbers (rates) | 60, 40 |
|---|
- 2 ÷ (1/60 + 1/40) = 2 ÷ 0.041667 = 48.0.
- The arithmetic mean, 50, overstates the average speed.
Result:
| Harmonic mean | 48.0000 |
|---|---|
| Arithmetic mean | 50.0000 |
| Geometric mean | 48.9898 |
Frequently asked questions
When do I use the harmonic mean?
When averaging rates over equal amounts of the thing in the numerator’s denominator: speeds over equal distances, prices per unit when spending equal money, or P/E ratios with equal investment.
Is the harmonic mean always the smallest?
For positive values that are not all equal, harmonic ≤ geometric ≤ arithmetic mean.
Sources and further reading
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