Harmonic Mean Calculator

What the harmonic mean calculator does

The harmonic mean is the average built for rates and ratios. This calculator finds it from your data, along with how many numbers you entered.

Its natural home is averaging speeds, rates, and prices per unit, where adding and dividing the usual way gives a figure that is quietly too high. Below is how it works and the kind of question it answers.

How to use it

1. Enter your numbers in the box, separated by commas, spaces, or new lines. They should all be positive.
2. Press Calculate to see the harmonic mean and the count, or Reset to clear it.

How the harmonic mean is worked out

The harmonic mean works through reciprocals, which is just one divided by each number. You flip every value, average those, and flip the result back:

Harmonic mean = how many numbers there are ÷ the sum of their reciprocals

Working in reciprocals is what makes it right for rates, because a rate already has a quantity tucked into its bottom, like miles per hour or dollars per item. Flipping the numbers lines those up correctly before averaging, which the ordinary mean does not do.

When the harmonic mean is the right one

The classic case is average speed over the same distance. If you drive somewhere at one speed and back at another, you spend more time at the slower speed, so your true average speed is dragged toward the slow end. The harmonic mean captures that automatically, while the ordinary average would flatter you with too high a number.

The same goes for any rate where the thing on top is held fixed: prices per unit when you spend the same amount at each price, or rates of work when the job size is the same. The harmonic mean always leans toward the smaller values, which is exactly the correction these situations need.

A worked example: there and back

You drive to a town at 60 miles per hour and return along the same road at 30. What was your average speed for the trip?

The tempting answer is 45, the ordinary average, but that is wrong, because you spent twice as long on the slower return. The harmonic mean gets it right: 2 ÷ (1/60 + 1/30) = 2 ÷ 0.05 = 40 miles per hour. That lower figure is the honest one, since more of your time was spent going slowly.

Positive numbers only, and the rounding

The harmonic mean is meant for positive numbers. A zero cannot be flipped, since one divided by zero has no value, and mixing in negatives makes the reciprocals fight each other and the result stops meaning anything. So keep the data to positive rates, which is what the harmonic mean is for in the first place.

The result is shown to four decimal places with trailing zeros trimmed.

Questions people ask

What is the harmonic mean?

The count of your numbers divided by the sum of their reciprocals. It is the average designed for rates and ratios, and it always comes out lower than the ordinary average.

When should I use the harmonic mean?

When averaging rates where the top of the rate is fixed, the classic case being speeds over equal distances. It correctly accounts for the extra time spent at the slower rate.

Why is average speed not just the average of the two speeds?

Because you spend more time at the slower speed over the same distance, so it should count for more. The harmonic mean weights the time correctly and gives the true average.

Can I use negative numbers or zero?

No. A zero cannot be flipped into a reciprocal, and negatives make the result meaningless. The harmonic mean only works for positive values.

References

A quick note on where the methods here come from. The harmonic mean and the rate problems it solves are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook that covers the harmonic mean alongside the other averages.

1. NIST/SEMATECH e-Handbook of Statistical Methods (measures of location). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (measures of the center of the data). https://openstax.org/details/books/introductory-statistics-2e