Harmonic Mean Calculator

Every value must be strictly positive: zero has no reciprocal, and a mix of positive and negative rates makes the reciprocal sum meaningless. The calculator reports any offending values instead of silently skipping them.

Because small values dominate the reciprocal sum, one tiny entry pulls the harmonic mean down hard — double-check suspiciously small inputs before trusting the result.

The harmonic mean flips every value, averages the flips, and flips back:

The structure explains the behavior. Because each value enters as a reciprocal, small values contribute large terms to the sum in the denominator and therefore dominate the result. An arithmetic mean is pulled up by one huge value; a harmonic mean is pulled down by one tiny value. Whichever quantity genuinely accumulates in your problem — hours per kilometer, dollars per share, errors per prediction — is the one that should be averaged directly, and the harmonic mean is what you get when that accumulating quantity is the reciprocal of what you are quoting.

Speed is distance over time. When the distances of two trip legs are equal, the slow leg takes more time than the fast leg, so the slow speed governs a larger share of the journey. Averaging the two speed numbers arithmetically weights each leg by count instead of by time, which overstates the true average. Dividing total distance by total time — the definition of average speed — collapses algebraically to exactly the harmonic mean of the leg speeds whenever the distances match.

The same logic covers any "per" quantity with a fixed numerator or denominator: pages per hour when each session covers the same number of pages, fuel consumption over identical routes, download speeds for files of the same size. When the legs are not equal, the right tool is a weighted harmonic mean with the distances as weights — the plain version this calculator computes assumes equal weighting.

Two famous applications live far from the highway. In finance, the average price-to-earnings ratio of a portfolio is computed harmonically: P/E has price on top, so a stock with enormous P/E (tiny earnings) would explode an arithmetic average while contributing almost nothing to real earning power. The harmonic mean weights each holding by the earnings a dollar of investment actually buys, matching the portfolio-level ratio of total price to total earnings.

In machine learning, the F1 score is the harmonic mean of precision and recall: F1 = 2PR / (P + R). The harmonic mean is chosen deliberately for its harshness — a classifier with precision 1.0 but recall 0.1 gets an F1 near 0.18, not the flattering 0.55 an arithmetic average would report. A model cannot buy a good F1 score by maximizing one metric and abandoning the other.

For any set of positive numbers that are not all identical, the three classical means line up in strict order:

The harmonic mean is always the most conservative of the three and the most sensitive to small values. A useful mnemonic ties each mean to its arithmetic: add values for the arithmetic mean, multiply them for the geometric mean, and add their reciprocals for the harmonic mean. For plain measurements rather than rates, the ordinary average from the mean, median, mode calculator remains the right default.

You drive to a city 120 km away at 60 km/h and return over the same road at 40 km/h. Intuition says the average speed is (60 + 40) ÷ 2 = 50 km/h. The clock disagrees:

1. Outbound time: 120 ÷ 60 = 2 hours.
2. Return time: 120 ÷ 40 = 3 hours.
3. Total: 240 km in 5 hours, so the true average speed is 240 ÷ 5 = 48 km/h.
4. Harmonic mean check: HM = 2 ÷ (1/60 + 1/40) = 2 × 60 × 40 ÷ (60 + 40) = 4800 ÷ 100 = 48.

The arithmetic mean of 50 km/h is wrong because the return leg at 40 km/h occupied 3 of the 5 hours — the slow speed deserves more weight, and the harmonic mean gives it exactly the weight the clock does. For contrast, the geometric mean of 60 and 40 is √2400 ≈ 48.99, landing between the other two just as the AM ≥ GM ≥ HM chain requires: 50 ≥ 48.99 ≥ 48.