Harmonic Mean Calculator
Calculate the harmonic mean of a set of positive numbers, ideal for rates and ratios.
About Harmonic Mean Calculator
The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. It is always ≤ the arithmetic mean and is the correct average for rates (like speed, efficiency, or price-to-earnings ratios).
Formula
Harmonic Mean
H = n / (1/x₁ + 1/x₂ + ... + 1/xₙ)
- H
- Harmonic mean
- n
- Number of values
- xᵢ
- Each positive value
Worked example
With the default values below, Harmonic Mean Calculator returns Harmonic Mean of 48.000000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Values (comma separated)
- 40, 60
Results
- Harmonic Mean
- 48.000000
- Count
- 2
- Sum of Reciprocals
- 0.04166667
Step by step
- 1Count of values= 2
- 2Sum of reciprocals= 0.04166667
- 3Harmonic mean = n / sum of reciprocals= 48.000000
Assumptions and limits
- The result is derived only from the single input listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
When is the harmonic mean useful?
Use it when averaging rates. For example, if you drive 40 km/h on the way out and 60 km/h on the return (same distance), the average speed is the harmonic mean: 2/(1/40+1/60) = 48 km/h, not the arithmetic mean of 50.
Why must all values be positive?
A zero value would make its reciprocal infinite, and negative values would cancel in ways that make the harmonic mean meaningless or undefined.