How To Calculate A Harmonic Mean

The harmonic mean of n positive numbers is n divided by the sum of their reciprocals. It is the third of the classical Pythagorean means, sitting alongside the arithmetic mean you get by adding and the geometric mean you get by multiplying.

It is the average you need when the numbers are rates. A rate has something in the denominator, and when that denominator is what actually adds up, the plain average gives the wrong answer. Speed is the textbook case: kilometres per hour, where hours are what accumulate.

Drive 60 miles out at 30 miles per hour and 60 miles back at 60 miles per hour. The arithmetic mean of the two speeds is 45, but the trip takes two hours out and one hour back, so the real average is 120 miles over 3 hours, which is 40. The harmonic mean of 30 and 60 is 40.

The harmonic mean is never larger than the geometric mean, which is never larger than the arithmetic mean. For 30, 60 and 90 the three averages line up as 49.0909, 54.5136 and 60, and the gap grows as the values spread apart.

It always lands between the smallest and the largest value, and it leans towards the small end. One slow leg of a journey drags the average down far more than one fast leg lifts it, because slow legs eat the time.

For two numbers there is a shortcut: the harmonic mean is twice the product divided by the sum. That is the same shape as the formula for two resistors in parallel, which is why electricians meet this average without calling it that.

The same pattern shows up whenever two things work together. If one painter finishes a wall in 3 hours and another in 6, the combined rate is the harmonic mean of their rates, and the wall takes 2 hours rather than the 4.5 the arithmetic mean suggests.

Investors meet it through dollar cost averaging. Spend a fixed amount each month and the average price you pay is the harmonic mean of the prices, not the arithmetic mean. Buying at 10, 20 and 5 gives an average cost of 8.5714 rather than 11.6667, which is exactly why the strategy works.

Machine learning uses it for the F1 score. Precision and recall are both rates over the same set of predictions, so a single number has to be combined harmonically; the F1 score is the harmonic mean of the two and collapses to zero if either one does.

It demands positive numbers. A zero in the list forces a division by zero and the whole calculation dies, and a negative value can produce a denominator that cancels to something meaningless or a mean that falls outside the data.

The weighted version handles unequal legs. Divide the total distance by the total time rather than averaging the speeds, which is the honest way to report an average speed over a trip with legs of different lengths.

Do not reach for it with ordinary measurements. Averaging the heights of a class, the price of a basket of goods or a set of test scores is additive, and the harmonic mean would understate every one of them.

A single tiny value dominates the result. Because the smallest number contributes the biggest reciprocal, one value near zero pulls the harmonic mean down towards it, which is a feature for rates and a bug for anything else.

Compute it in three moves: flip every value, add the flips, and divide the count by that sum. On a calculator the reciprocal key does the flipping, and one more press flips the answer back.

A useful sanity check is the ordering. If your harmonic mean comes out above the arithmetic mean of the same numbers, either a value was mistyped or the reciprocals were averaged instead of summed.