How the Harmonic Mean Works
The harmonic mean is a type of average best suited to rates and ratios — quantities like speed, price-to-earnings ratios, or resistance in parallel circuits. For a set of n positive numbers x₁, x₂, …, xₙ, the harmonic mean is HM = n / Σ(1/xᵢ) — that is, n divided by the sum of the reciprocals of each value. Because it is based on reciprocals, the harmonic mean weights smaller values more heavily than larger ones, which makes it pull toward the low end of a data set rather than the middle.
Formula and method
To compute the harmonic mean by hand: (1) take the reciprocal of every value (1 divided by the value), (2) sum those reciprocals, and (3) divide the count of values, n, by that sum. For example, with 4, 8, and 16: the reciprocals are 0.25, 0.125, and 0.0625, which sum to 0.4375. Dividing n = 3 by 0.4375 gives HM ≈ 6.857. When some observations should count more than others, use the weighted harmonic mean: HM_w = Σwᵢ / Σ(wᵢ/xᵢ), where each value xᵢ is paired with a weight wᵢ. If no weights are supplied, this calculator treats every value as equally weighted, so the weighted result matches the plain harmonic mean.
Common sources of error
- Zero or negative values: the harmonic mean requires every value to be strictly positive — a zero makes 1/x undefined, and negative values make the result meaningless for rate-style data.
- Using the arithmetic mean for rates: averaging speeds, ratios, or rates with a simple arithmetic mean (sum ÷ count) typically overstates the true average — the harmonic mean corrects for this when the quantities share a common numerator (e.g., a fixed distance or fixed cost).
- Mismatched weight count: when using weights, make sure you enter exactly one weight per value, in the same order, or the weighted result will not correspond to the correct pairing.
Checking your result
The harmonic mean of a set of positive numbers is always less than or equal to the geometric mean, which is always less than or equal to the arithmetic mean — the relationship AM ≥ GM ≥ HM always holds, with equality only when all the values are identical. This calculator reports all three means, so you can use that inequality as a sanity check: if your harmonic mean comes out larger than the arithmetic mean, double-check your inputs.