Harmonic Mean Calculator
The right average for speeds and rates.
Work out Harmonic Mean. The right average for speeds and rates. Says where the rule comes from.
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Harmonic mean
40
Arithmetic mean is 45
The harmonic mean averages rates over a fixed quantity, which is why it is the correct one for speed. Driving one leg at 60 and an equal leg at 30 averages 40, not 45 — you spend twice as long on the slow leg, and the harmonic mean is what accounts for that automatically.
How the Harmonic Mean Calculator works
The harmonic mean averages rates over a fixed quantity, which is why it is the correct one for speed. Driving one leg at 60 and an equal leg at 30 averages 40, not 45 — you spend twice as long on the slow leg, and the harmonic mean accounts for that automatically.
Also known as: average speed calculator · harmonic mean formula · harmonic average calculator · how to calculate harmonic mean
Why average speed is not the average of the speeds
Drive one leg at 60 and an equal leg at 30, and the average speed is 40, not 45.
The reason is that you spend twice as long on the slow leg. Over 60 miles each way, the fast leg takes one hour and the slow leg two — so 120 miles in three hours, which is 40 mph.
The formula and its family
The harmonic mean is the count divided by the sum of the reciprocals: 2 ÷ (1/60 + 1/30) = 40. Taking reciprocals is what converts a rate into a time, which is exactly the weighting the problem needs.
The three Pythagorean means always order the same way: harmonic ≤ geometric ≤ arithmetic, with equality only when every value is identical. For 1, 2 and 4 they are 1.714, 2 and 2.333.
Each answers a different question. Arithmetic for quantities that add, geometric for those that multiply, harmonic for rates over a fixed quantity. Using the wrong one is not a rounding difference — it is the wrong number.
Frequently asked questions
What is the harmonic mean?
The count divided by the sum of the reciprocals. For 60 and 30 it is 2 ÷ (1/60 + 1/30) = 40.
Why is average speed not the simple average?
Because you spend longer at the slower speed, so it carries more of the journey. Equal distances at 60 and 30 average 40, not 45.
When should I use the harmonic mean?
For rates measured against a fixed quantity — speed over fixed distance, price-earnings ratios across a portfolio, throughput across stages.
How does it compare to the other means?
It is always the smallest: harmonic ≤ geometric ≤ arithmetic, with equality only when every value is identical.
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