Harmonic Mean Calculator

Calculate the harmonic mean for positive rates, ratios or other reciprocal quantities

Your result will appear here.

About the Harmonic Mean Calculator

The harmonic mean is built for averaging rates when the quantity of work or distance is held constant and time varies inversely with the rate. It is often the right mean for average speed over equal distances, price-per-unit style rates and other reciprocal relationships.

It gives more influence to smaller values than the arithmetic mean. That behaviour is useful for rates but can be misleading when the observations are ordinary additive measurements.

How to Use the Harmonic Mean Calculator

  1. Enter the positive rates or ratios you want to average.
  2. Confirm that the scenario truly calls for reciprocal averaging, such as equal-distance travel at different speeds.
  3. Calculate the harmonic mean.
  4. Compare with the arithmetic mean if you are unsure which averaging model matches the problem.

How the Calculation Works

For n non-zero values x₁…xₙ, harmonic mean H = n ÷ (1/x₁ + 1/x₂ + … + 1/xₙ).

For two positive rates a and b, the formula simplifies to H = 2ab/(a+b). The denominator uses reciprocals because equal quantities completed at different rates consume time in inverse proportion to rate.

Worked example

Harmonic Mean Calculator Worked Example

Equal distances at 60 km/h and 90 km/h

H = 2×60×90 ÷ (60+90) = 10,800÷150 = 72 km/h. The arithmetic mean would be 75 km/h, which overstates the true average speed for equal distances.

Result guide

Understanding Your Harmonic Mean Calculator Result

The harmonic mean is especially useful when low rates create disproportionately long durations. A very small input can pull the mean down sharply because its reciprocal is large.

Before you rely on the answer

Assumptions and Limitations

  • Inputs are non-zero and normally positive.
  • The averaging problem is genuinely rate-based or reciprocal in structure.
  • Each rate carries the intended equal quantity weighting unless another weighting scheme is used.

Practical tip

Using This Result in Practice

Before choosing the harmonic mean, ask what is being held equal. Equal distances at different speeds call for harmonic averaging; equal time spent at different speeds calls for arithmetic averaging of the speeds.

Related calculations

Useful Calculations to Continue With

For multiplicative growth or ratio chains, use the Geometric Mean Calculator instead.

For ordinary equally weighted values that combine additively, use the Average Calculator.

Common questions

Harmonic Mean Calculator FAQ

Why is harmonic mean used for average speed?

It is appropriate when equal distances are travelled at different speeds because travel time is inversely related to speed.

Can harmonic mean be used with zero?

No in the ordinary formula because it requires division by each input value.

Why is harmonic mean sensitive to small values?

Small values have large reciprocals, so they exert stronger influence on the denominator.