Foundations · Averages
Geometric Mean Calculator
The geometric mean is the right way to average things that multiply — growth rates, investment returns, ratios — where the familiar arithmetic mean quietly lies. Average a +50% year and a −40% year the usual way and you get +5%; compound them and you have actually lost money. This computes the geometric mean of any positive values, or the true average growth rate of a series of returns, and shows it against the arithmetic mean so the gap is impossible to miss.
Geometric mean = the nth root of the product. Use it for ratios, indices and anything that compounds; all values must be greater than zero.
Enter each period’s percentage change. The calculator finds the constant rate that would compound to the same result — the true average — and contrasts it with the (overstated) arithmetic average.
Result
In plain English
The arithmetic mean adds the numbers and divides; the geometric mean multiplies them and takes a root. That difference matters whenever the quantities combine by multiplication rather than addition — a 10% rise followed by a 10% fall does not return you to where you started, because the fall acts on a bigger base. For growth, returns and ratios, the geometric mean is the honest average; the arithmetic mean systematically overstates, and the more the numbers bounce around, the worse it overstates.
- geometric mean
- The nth root of the product of n positive numbers: (x₁ · x₂ · … · xₙ)^(1∕n). Equivalently, the arithmetic mean done in log-space and converted back.
- arithmetic mean (AM)
- The everyday average: add and divide. Correct for quantities that add (heights, test scores), wrong for quantities that multiply.
- harmonic mean (HM)
- The reciprocal of the average of the reciprocals. The right average for rates defined per unit of something — average speed over equal distances, for instance.
- AM ≥ GM ≥ HM
- For any set of positive numbers the three means line up in this order, with equality only when every value is identical. The more spread out the values, the wider the gaps.
- average growth rate (CAGR)
- The single constant rate that compounds to the same final result — the geometric mean of the growth factors, minus one. The honest “per-period” figure for a series of returns.
- volatility drag
- The gap between the arithmetic average return and the geometric one. Swings cost you: the bigger the ups and downs, the more the compounded result falls short of the simple average.
Frequently asked
How do you calculate the geometric mean?
Multiply all n values together and take the nth root: GM = (x₁ · x₂ · … · xₙ)^(1∕n). For 2, 4 and 8 that is (2 · 4 · 8)^(1∕3) = 64^(1∕3) = 4. In practice it is computed through logarithms — average the logs of the values and exponentiate — which avoids overflow and makes clear that the geometric mean is just the arithmetic mean on a multiplicative (log) scale. Every value must be positive, since the log of zero or a negative number is undefined.
When should I use the geometric mean instead of the arithmetic mean?
Use the geometric mean whenever the quantities multiply rather than add: rates of growth, investment returns, ratios, index numbers, things measured as “× per period.” Use the arithmetic mean for quantities that add — heights, weights, test scores. A quick test: if averaging the numbers and then “undoing” it should reproduce a product (a final balance, a cumulative factor), you want the geometric mean. Averaging year-on-year returns arithmetically is the classic mistake this page is built to expose.
Why is the average of +50% and −40% not +5%?
Because returns compound. A +50% year multiplies your money by 1.5; a −40% year multiplies by 0.6. Together that is 1.5 × 0.6 = 0.9 — you have lost 10% over the two years, not gained. The arithmetic average, (50 − 40) ∕ 2 = +5%, ignores that the −40% hits a larger balance. The geometric mean of the factors, √(1.5 × 0.6) = 0.9487, gives the true average of about −5.1% per year, which compounds correctly back to the 0.9 you actually ended with.
Can the geometric mean handle negative numbers or zero?
Not directly — the product of values including a zero is zero, and an odd number of negatives makes the product negative, so the root is undefined or meaningless. For returns that go negative, you do not feed in the percentages; you convert each to a growth factor (1 + return), which stays positive as long as no single loss is −100% or worse. A −100% return (total wipeout) sends the geometric mean to zero, correctly: once you are at zero, no later gain brings you back.