StatReason

Harmonic Mean Calculator

Find the correct average of rates measured over equal distances or quantities, such as speeds over the same distance.

All values must be strictly positive.

What this answers

The harmonic mean answers "what single rate, applied consistently, would produce the same overall result as this set of rates, when each rate applies over the same distance, quantity, or time?" The classic example is average speed: if you drive the same distance at 30 mph and then 60 mph, your true average speed for the whole trip is not 45 mph (the arithmetic mean), because you spent more time at the slower speed.

When to use it, and when not to

Use the harmonic mean specifically when rates apply to an equal distance, quantity, or unit each. If your rates instead apply to equal amounts of time (not equal distances), the ordinary arithmetic mean is actually correct, so it is worth checking carefully which situation you are in before choosing this calculator.

How it is calculated

The harmonic mean is n divided by the sum of the reciprocals of your values. This gives more weight to smaller values in the average, which is exactly why it correctly handles the equal-distance rate scenario where the arithmetic mean would not.

Worked example

For rates of 30 and 60 (say, miles per hour over the same distance in each direction): the harmonic mean is 2 divided by (1/30 plus 1/60), which works out to exactly 40. This is the true average speed for the round trip, correctly weighted by the fact that more time was spent traveling at the slower 30 mph rate.

Assumption audit

Calculated from your data: the count of values and whether every value is strictly positive, which the harmonic mean requires.
Evidence to review: whether your rates genuinely apply to equal distances or quantities, the specific condition that makes the harmonic mean the correct choice rather than the arithmetic mean.
You must verify: the actual real-world structure behind your rates (equal distance versus equal time), since this calculator cannot infer that from the numbers alone, and choosing the wrong average for the situation gives a plausible-looking but incorrect answer.

Limitations

The harmonic mean is undefined for zero or negative values and is rarely appropriate outside the specific equal-distance or equal-quantity rate scenario it is built for.