Harmonic Mean Calculator
Calculate n / sum(1/xi) for rates and ratios.
This tool is for informational and educational purposes only. It is not a substitute for professional financial, medical, legal, or engineering advice. See Terms of Service.
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Enter non-zero numbers separated by commas or spaces. The calculator computes n divided by the sum of reciprocals. If any value is zero, a warning appears because dividing by zero is undefined. The arithmetic mean is shown for comparison.
About Harmonic Mean
The harmonic mean is the reciprocal of the arithmetic mean of reciprocals. It is the correct average for rates: if you drive 40 mph one way and 60 mph back, the average speed is the harmonic mean (48 mph), not the arithmetic mean (50 mph). It is also used in finance for price-to-earnings ratios and in physics for parallel resistances.
Frequently Asked Questions
When is harmonic mean the right choice?
Use harmonic mean when averaging rates or ratios over equal intervals, such as speed over equal distances, prices per unit, or rates of work completion.
Why is harmonic mean always less than arithmetic mean?
By the AM-HM inequality, for positive numbers the harmonic mean is always less than or equal to the arithmetic mean, with equality only when all values are identical. The harmonic mean gives more weight to smaller values.
Can harmonic mean handle negative numbers?
Technically yes, but it rarely makes practical sense. Harmonic mean is designed for positive quantities. Negative values can cause the reciprocal sum to approach zero, giving misleading results.