What happens if one value is zero?

The mean is undefined, because a reciprocal of zero has no value. That is not a limitation of the formula but a correct description of the situation it models: a leg of a journey travelled at zero speed never finishes, so the average speed of the whole journey is not a number. The same holds for the F1 score of a classifier that recalls nothing. Where zeros are legitimate data rather than a signal that something did not happen, the harmonic mean is the wrong summary and an arithmetic or a median one is the right one.

How do I know which mean a problem wants?

Ask what has to stay true after averaging. If the total is a sum — of costs, of masses, of counts — the arithmetic mean preserves it. If the total is a product, as with successive growth rates, the geometric mean does. If the quantities are rates and what is fixed is the thing in their denominator, the harmonic mean is the one that makes the overall rate come out right. Writing down the quantity that must be preserved takes a moment and picks the mean without any memorised rule.

Forty Out and Sixty Back Averages Forty-Eight

Published 10/9/2026 · 3 min read · Everyday calculators

Lena Hoffmann — Science & education writer at OneKitly

Mathematics · Physics

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In short

The harmonic mean is the count divided by the sum of the reciprocals: for 1, 2 and 4 that is 3 ÷ (1 + 0.5 + 0.25) = 1.714, well below the arithmetic mean of 2.33. Its natural home is averaging rates over a fixed quantity of something else. Drive out at 40 km/h and back at 60 over the same road: the outward leg takes longer, so more of the journey is spent at the slower speed, and the average for the whole trip is the harmonic mean, 48 km/h. Averaging the two speeds directly gives 50 and would be correct only if equal time were spent at each — which would mean different distances. The rule is short: average the rate itself when the time is fixed, and take the harmonic mean when the distance is.

The harmonic mean of 40 and 60 is 48, not 50, because the slower leg takes longer and therefore counts for more. For 1, 2 and 4 it is 1.714 against an arithmetic mean of 2.33.

The three means are always in that order

Harmonic is never larger than geometric, and geometric is never larger than arithmetic, for any set of positive numbers. They coincide only when every value is identical, and the gap between them widens with the spread — which makes the difference between two of them a crude measure of how uneven the data is. For 40 and 60 the three sit within two units of each other; for 1 and 100 the harmonic mean is 1.98 and the arithmetic 50.5, a factor of twenty-five. Seeing an arithmetic mean far above a harmonic mean of the same data is a reliable sign that a few large values are carrying the average.

It also appears where nobody calls it that

The F1 score used to summarise a classifier is the harmonic mean of precision and recall, chosen precisely because it refuses to be flattered by one of them: a system with precision 0.6 and recall 0.8 scores 0.686, closer to the weaker number than an arithmetic mean would put it. The same reasoning gives the average price-to-earnings ratio of a portfolio, which has to be computed harmonically because each ratio has earnings in its denominator. Wherever the quantity being averaged is a ratio and the denominators are what stay fixed, the harmonic mean is the one that keeps the total correct.

Three means of the same numbers
Set	Harmonic	Geometric	Arithmetic
1, 2, 4	1.714	2.000	2.333
40, 60	48.00	48.99	50.00

Harmonic mean calculatorCompute the harmonic mean of a list of positive numbers.Try the tool →

Frequently asked questions

What happens if one value is zero?: The mean is undefined, because a reciprocal of zero has no value. That is not a limitation of the formula but a correct description of the situation it models: a leg of a journey travelled at zero speed never finishes, so the average speed of the whole journey is not a number. The same holds for the F1 score of a classifier that recalls nothing. Where zeros are legitimate data rather than a signal that something did not happen, the harmonic mean is the wrong summary and an arithmetic or a median one is the right one.
How do I know which mean a problem wants?: Ask what has to stay true after averaging. If the total is a sum — of costs, of masses, of counts — the arithmetic mean preserves it. If the total is a product, as with successive growth rates, the geometric mean does. If the quantities are rates and what is fixed is the thing in their denominator, the harmonic mean is the one that makes the overall rate come out right. Writing down the quantity that must be preserved takes a moment and picks the mean without any memorised rule.

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