Enter a set of positive numbers in the Data Values field (comma-separated, space-separated, or one per line) and this Harmonic Mean Calculator computes the harmonic mean of your dataset. You'll see the harmonic mean, along with the count of values and their sum of reciprocals — all the key figures for your statistical analysis.
Harmonic Mean
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Count (n)
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Sum of Reciprocals
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Arithmetic Mean
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Geometric Mean
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Minimum Value
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Maximum Value
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The harmonic mean is a type of average, or measure of central tendency, calculated as the reciprocal of the arithmetic mean of the reciprocals of a set of values. It is defined as n divided by the sum of (1/x) for each value x. It is most appropriate when averaging rates, ratios, or speeds.
The arithmetic mean sums all values and divides by the count, while the harmonic mean takes the reciprocal of the average of the reciprocals. For a given dataset, the harmonic mean is always less than or equal to the arithmetic mean. The harmonic mean gives more weight to smaller values, making it useful for rates and ratios.
Use the harmonic mean when you are averaging rates, speeds, or ratios — for example, average speed over a fixed distance, or price-to-earnings ratios in finance. When the quantities you are averaging are defined as a ratio of two values, the harmonic mean typically gives the most meaningful result.
No. All values must be strictly positive (greater than zero). Including zero would result in division by zero, and negative numbers do not have a meaningful harmonic mean. This calculator will ignore any zero or negative values and notify you accordingly.
For any set of positive numbers, the following inequality always holds: Harmonic Mean ≤ Geometric Mean ≤ Arithmetic Mean. Equality holds only when all values in the dataset are identical. This relationship is known as the AM-GM-HM inequality.
Type or paste your numbers into the Data Values field. You can separate them with commas, spaces, or by placing each value on its own line. All entries must be positive numbers greater than zero. The calculator will parse them automatically and compute the harmonic mean.
Suppose a car travels 60 km at 40 km/h and then another 60 km at 60 km/h. The average speed is not (40+60)/2 = 50 km/h, but the harmonic mean: 2 / (1/40 + 1/60) = 48 km/h. This correctly reflects the total distance divided by the total time taken.
Yes, but differently from the arithmetic mean. Because the harmonic mean uses reciprocals, it is particularly sensitive to very small values (values close to zero pull the harmonic mean down significantly). Very large values have comparatively less influence on the result.