Arithmetic Geometric Harmonic Mean Calculator
Computes arithmetic, geometric and harmonic means simultaneously for a list of positive numbers separated by comma.
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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.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.