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Statistics

Harmonic Mean Calculator

Calculate the harmonic mean of positive numbers, see the reciprocal calculation, and compare it with the arithmetic mean.

Data stays on this deviceConvention explained below
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Check the dataset and sample convention before interpreting results.

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Review the method below for assumptions and conventions.

How to use this tool

  1. Enter positive numbers separated by spaces or commas. Do not use commas as thousands separators.
  2. Calculate to see the reciprocal sum, average reciprocal and harmonic mean.
  3. For rates, check that the quantities being averaged have equal numerators, such as equal distances when averaging speeds.

Input reference and sample values

Use the units and choices shown for each field. Sample values are provided for practice, not as recommendations.

InputFormat and unitsSample
NumbersKeep the structure and separators shown in the sample. Line breaks are retained in this input.30, 60

Result for these sample inputs

40
Number of observations
2
Sum of reciprocals
0.05
Mean of reciprocals
0.025
Arithmetic mean (for comparison)
45

For equal-distance journeys use the harmonic mean of speeds. For equal time intervals, use the arithmetic mean instead.

Select Try the example in the form to load these values and calculate them again. This replaces the current form inputs. Compare the result with the method and worked explanation below.

The method, explained

Harmonic mean = n/sum(1/xi).

A WORKED EXAMPLE

For 30 and 60, the reciprocals add to 1/30 + 1/60 = 0.05. Divide the count, 2, by 0.05 to get 40. The arithmetic mean is 45. Travelling equal distances at 30 and 60 km/h gives an average speed of 40 km/h; travelling equal times gives 45 km/h.

Understanding your result

The harmonic mean gives greater influence to small positive observations. It is no larger than the arithmetic mean for the same positive values, with equality when all values are identical. The comparison does not mean one mean is universally better: the appropriate choice depends on what is held equal.

What to keep in mind

Accepts 1–1,000 strictly positive finite values within the input limits. Zero has no reciprocal and negative values are outside this tool’s convention. Unequal distances or other unequal weights require a weighted harmonic mean.

Reference: Harmonic mean — Math Is Fun

Common questions

Can I enter zero?

No. The calculation uses 1/x for every observation, so zero is undefined.

When should I use the arithmetic mean instead?

For average speed over equal time intervals, use the arithmetic mean. For equal distances, use the harmonic mean.

Does the order of numbers matter?

No. Reordering the same positive observations does not change the harmonic mean.

Methodology maintained by ClarityKit. How these tools are built and checked.