Harmonic Mean Calculator
Calculate the harmonic mean of a set of numbers — the right average for rates like speed over equal distances.
Numbers (comma, space, or line separated, all positive)
Harmonic Mean
55.3846
Arithmetic Mean (for comparison)
60.0000
3 values
How to Calculate the Harmonic Mean
The harmonic mean of n numbers is n divided by the sum of the reciprocals of each number. It gives more weight to smaller values than the arithmetic mean does, which makes it the right choice for averaging rates and ratios rather than plain quantities.
Example
If you drive the same distance three times at 60, 40, and 80 mph, your true average speed for the whole trip is the harmonic mean: 3 ÷ (1/60 + 1/40 + 1/80) ≈ 54.5 mph — not the arithmetic mean of 60, which would overstate your actual average speed.
Common Use Cases
- Averaging speeds traveled over equal distances (e.g. multiple legs of a trip).
- Averaging rates like price-to-earnings ratios in finance, where the arithmetic mean is misleading.
- Computing average resistance of parallel electrical resistors or similar rate-based quantities.
FAQs
When should I use harmonic mean instead of arithmetic mean?
Use harmonic mean when averaging rates defined as a ratio (like distance per time) over equal amounts of the denominator — such as speed over equal distances, not equal times. If you traveled for equal durations instead, the arithmetic mean would be the correct average.
Why is the harmonic mean always the smallest of the three Pythagorean means?
For any set of positive numbers, harmonic mean ≤ geometric mean ≤ arithmetic mean, with equality only when all the numbers are identical. The harmonic mean is pulled down more strongly by small values since it works with reciprocals.
Why must all the numbers be positive?
The harmonic mean relies on taking the reciprocal of each value. A zero value would make its reciprocal undefined (division by zero), and negative values can produce misleading or undefined results, so this calculator requires strictly positive inputs.