How this instrument works
The harmonic mean of a and b is 2ab ⁄ (a+b) — equivalently, it is the number whose reciprocal equals the average of the two reciprocals: 1 ⁄ HM = (1⁄a + 1⁄b) ⁄ 2. That reciprocal-of-reciprocals shape is not a curiosity; it is exactly what you need whenever the quantity you are combining is itself a rate — kilometres per hour, ohms of parallel resistance, price per share — because rates add correctly only after you flip them into 'per unit' form, average that, and flip back.
The classic illustration is a round trip. Drive out at 30 km/h and return over the identical distance at 60 km/h, and the average speed for the whole trip is not the arithmetic mean of 45; it is the harmonic mean, 2(30)(60) ⁄ 90 = 40 km/h. The arithmetic mean silently assumes equal time spent at each speed, but the slow leg eats more time to cover the same ground, so it must weigh more in the average — which is precisely what dividing by a sum of reciprocals achieves.
Ancient Greek mathematicians named it 'harmonic' for a musical reason: a string, half that string, and two-thirds of that string sound the octave, the fifth, and the fourth together, and the lengths 1, 1/2, 2/3 are in harmonic proportion — each term's reciprocal steps evenly from the one before. The formula later resurfaced as one of three classical means (arithmetic, geometric, harmonic), and it always ranks lowest of the three unless a and b are equal, when all three collapse to the same value.
- Enter the first quantity into a — any positive rate, ratio, or resistance value.
- Enter the second quantity into b, measured in the same unit as a.
- Read Harmonic mean for 2ab ⁄ (a+b), the correctly weighted average of the two.
- Change either a or b to recompute instantly; the result always sits below the plain average of the two inputs unless they are equal.
Worked example — averaging 4 and 6
Set a = 4 and b = 6. Harmonic mean returns HM = 2(4)(6) ⁄ (4+6) = 48 ⁄ 10 = 4.8. Compare that against the arithmetic mean of the same pair, (4+6) ⁄ 2 = 5, and the geometric mean, √(4×6) = √24 ≈ 4.899: the harmonic mean of 4.8 lands lowest of the three, exactly as the ranking AM ≥ GM ≥ HM predicts whenever the two inputs differ.
That ordering is not a coincidence of this pair; it holds for every unequal a and b, with equality across all three means only when a equals b. If 4 and 6 were speeds in km/h covering equal distances, 4.8 is the true average speed a single-number summary owes you — not 5, which would only be correct if equal time, rather than equal distance, had been spent at each speed.
Questions
When should I use the harmonic mean instead of the arithmetic mean?
Use it whenever you are averaging a rate — speed, price per unit, resistance in parallel — over quantities that are themselves fixed in the denominator, such as equal distances or equal purchase amounts. Averaging two speeds over equal distances, for example, needs 2ab ⁄ (a+b); the plain sum (a+b)/2 only applies when equal time, not equal distance, was spent at each rate.
Why is the harmonic mean always the smallest of the three classical means?
Because it weights small values more heavily. Reciprocals of small numbers are large, so averaging reciprocals and flipping back drags the result toward whichever input is smaller. The inequality AM ≥ GM ≥ HM holds for any positive a and b, collapsing to a single equal value only when a = b, as shown for a = b = 5 where all three means equal 5.
Where does the name 'harmonic' come from?
From music, not mathematics. Ancient Greek theorists observed that string lengths of 1, 1/2, and 2/3 — which sound an octave, a fifth, and a fourth in combination — have reciprocals in even steps, a pattern they called harmonic proportion. The middle term of such a triple is the harmonic mean of the outer two, and the name stuck long after the musical origin was forgotten by most users of the formula.
Does the harmonic mean work with more than two numbers?
Yes, though this sheet handles the two-number case. For n values the general form is HM = n ⁄ (1⁄x₁ + 1⁄x₂ + … + 1⁄xₙ), the count divided by the sum of reciprocals. Setting n = 2 and expanding that sum of reciprocals algebraically is exactly what reduces to the 2ab ⁄ (a+b) shown here.
What happens if a and b are very different in size?
The result pulls hard toward the smaller value rather than sitting near the midpoint. Averaging 1 and 100 gives a harmonic mean of only about 1.98, far below the arithmetic mean of 50.5 or even the geometric mean of 10, because the small number's large reciprocal dominates the sum being averaged.
Can the harmonic mean be used with zero or negative inputs?
Not meaningfully — a zero input makes a reciprocal undefined, and mixing positive and negative values produces a result that no longer sits between the two numbers the way a mean should. The formula 2ab ⁄ (a+b) is defined for positive a and b, which is the case this sheet is built for.