Harmonic Mean CalculatorFree online math calculator — instant results, no sign-up
Calculate the harmonic mean of positive values.
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About the Harmonic Mean Calculator
The Harmonic Mean Calculator finds the reciprocal-based average of a list of positive values. It is useful for averaging rates when each rate applies to an equal quantity, such as speeds over equal distances. For the default values 2, 4, 6, 8, 10, the harmonic mean is 4.3795620438.
- Values
- Enter the positive numbers to average, using consistent units such as kilometers per hour for speeds, or no units for dimensionless data.
Formula and method
H = n / (Σᵢ₌₁ⁿ 1/xᵢ); for the default inputs, H = 5 / (1/2 + 1/4 + 1/6 + 1/8 + 1/10) = 600/137 ≈ 4.3795620438.
- HThe harmonic mean, expressed in the same units as the input values.
- nThe number of input values; n = 5 for the default list.
- xᵢThe positive value at position i in the input list.
- iThe index identifying each value, from 1 through n.
- ΣThe summation symbol, instructing you to add the reciprocals of all input values.
Count the values, take the reciprocal of each one, and add those reciprocals. Divide the count by that sum. For 2, 4, 6, 8, 10, the reciprocal sum is 137/120, so dividing 5 by 137/120 gives 600/137, displayed as 4.3795620438.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Values2, 4, 6, 8, 10
Results
- Harmonic mean4.3795620438
How to use the Harmonic Mean Calculator
- 1Enter 2, 4, 6, 8, 10 in the Values field to reproduce the default example, or replace the list with your own data.
- 2Check that every entry is numeric and strictly greater than zero.
- 3Make sure all values use the same units and that an equally weighted harmonic mean suits your data.
- 4Run the calculation to divide the number of entries by the sum of their reciprocals.
- 5Read the Harmonic mean result; the default list produces 4.3795620438.
Reading your result
The Harmonic mean output row reports the reciprocal-based average in the same units as the inputs. For 2, 4, 6, 8, 10, the result 4.3795620438 lies between the smallest and largest values and is below the arithmetic mean of 6. Smaller values have a stronger influence on this average because their reciprocals contribute more to the denominator.
Limitations
This calculator requires a nonempty list of strictly positive values; zero cannot be used because its reciprocal is undefined, and negative values are outside the tool's scope. Each entry receives equal weight, so data requiring unequal weights need a weighted harmonic mean instead. Averaging rates harmonically is appropriate only when the relevant quantities are equal, such as equal distances for speeds, not equal travel times. Very small positive entries can strongly reduce the result, and the displayed decimal is rounded.
Common questions
Why does the default list give 4.3795620438 instead of 6?+
The arithmetic mean adds 2, 4, 6, 8, 10 and divides by 5, giving 6. The harmonic mean instead divides 5 by the sum of their reciprocals, giving 600/137 ≈ 4.3795620438.
Can I include zero or negative values?+
No. This calculator accepts only positive values. Zero makes the reciprocal calculation undefined, and negative values are excluded from this positive-data harmonic mean.
When should I use this calculator to average speeds?+
Use it when each speed applies to an equal distance. If speeds apply for equal amounts of time, use the arithmetic mean instead; for unequal distances, use total distance divided by total time or an appropriately weighted harmonic mean.
Is the harmonic mean always smaller than the arithmetic mean?+
For positive values with equal weights, the harmonic mean is no greater than the arithmetic mean. They are equal only when all input values are identical.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.