When should I use the harmonic mean instead of the arithmetic mean?

Use the harmonic mean when you are averaging rates or ratios — for example, speeds (km/h), prices per unit, or rates per time. The arithmetic mean over-weights larger values, while the harmonic mean gives equal weight to each unit of the denominator.

Why can the harmonic mean not include zero or negative numbers?

The formula requires dividing 1 by each value (1/x). Division by zero is undefined, and negative values produce a result that has no meaningful interpretation as an average for most real-world rate problems.

Is the harmonic mean always less than or equal to the arithmetic mean?

Yes. By the AM–HM inequality, the arithmetic mean is always greater than or equal to the harmonic mean for any set of positive numbers, with equality only when all numbers are identical.

What is the formula for harmonic mean?

HM = n / (1/x₁ + 1/x₂ + … + 1/xₙ), where n is the count of values. Equivalently, it is the reciprocal of the arithmetic mean of the reciprocals.

How is the harmonic mean used in the F1 score?

The F1 score is the harmonic mean of precision and recall: F1 = 2 × (precision × recall) / (precision + recall). The harmonic mean penalizes extreme imbalances, so a high F1 requires both precision and recall to be high.

Harmonic Mean Calculator

Enter a list of numbers and instantly compute their harmonic mean — the reciprocal of the arithmetic mean of reciprocals.

Enter Numbers

Values (comma, space, or new line separated)

Quick Examples

Result

Enter numbers on the left and press Calculate.

Values & Reciprocals

#	Value (x)	Reciprocal (1/x)
Sum	—	—

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Summary

The harmonic mean is a type of average calculated as the reciprocal of the arithmetic mean of the reciprocals of the values. It is especially useful when averaging rates, speeds, or ratios where equal weight should be given to each unit. This calculator accepts any list of positive numbers and instantly returns the harmonic mean along with a step-by-step breakdown.

How it works

Enter your numbers in the input box, separated by commas, spaces, or new lines.
The calculator validates each value — all numbers must be positive (non-zero).
It computes the reciprocal (1/x) of every number in the list.
It averages those reciprocals: sums them and divides by the count.
Finally it takes the reciprocal of that average to produce the harmonic mean.
Results update automatically as you type.

Use cases

Find the average speed when the same distance is traveled at different speeds.
Average price-to-earnings (P/E) ratios across a portfolio of stocks.
Combine multiple rate measurements in physics or engineering.
Calculate the F1 score in machine learning (harmonic mean of precision and recall).
Average fuel-efficiency figures across different driving conditions.
Determine the effective resistance of parallel resistors.
Blend rate-based metrics in financial analysis.
Average any set of ratios where the denominator units must be equal.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu