Arithmetic Geometric Harmonic Mean Calculator

Arithmetic, geometric and harmonic means

These are the three classical averages of n positive numbers x₁, …, xₙ. The arithmetic mean adds everything up and divides: AM = (x₁ + x₂ + … + xₙ) / n. The geometric mean multiplies the values and takes the n-th root: GM = ⁿ√(x₁ · x₂ · … · xₙ). The harmonic mean inverts each value, averages those, then inverts the result: HM = n / (1/x₁ + 1/x₂ + … + 1/xₙ). The classical AM–GM–HM inequality says AM ≥ GM ≥ HM, and the three coincide only when every xᵢ is identical. Run the numbers on 2, 4, 8, 16, 32: AM = 62/5 = 12.4, GM = ⁵√32768 = 8, and HM = 5 / (1/2 + 1/4 + 1/8 + 1/16 + 1/32) = 5 / (31/32) ≈ 5.161, which confirms 12.4 ≥ 8 ≥ 5.161.

Applications

Reach for the arithmetic mean with additive quantities, like grades, daily temperatures, or sample means in statistics. The geometric mean suits multiplicative growth — the Compound Annual Growth Rate (CAGR) of an investment, average inflation, and ratios or indices, where GM = (final/initial)^(1/n) − 1 + 1. The harmonic mean fits rates and ratios measured against a fixed quantity: average speed when the distances are equal, the F1-score in machine learning that combines precision and recall, or resistors wired in parallel in electronics.

FAQ

Which mean for a round-trip at 60 km/h going and 40 km/h coming back? The harmonic one. HM = 2 / (1/60 + 1/40) = 48 km/h. It's not the arithmetic 50, because you spend longer at the slower speed.

Why does GM require positive numbers? If the product includes a zero, its real n-th root is zero and the mean collapses. A negative value makes the root complex when n is even. So by convention the GM is defined only for strictly positive data.

Why is the geometric mean the right choice for average returns? Returns compound rather than add. Gain 10%, then lose 10%, and you land at 0.99, not 1. The GM captures that compounding faithfully, whereas the AM overstates what you actually earned.