Geometric Mean Calculator

For n values, the geometric mean is the nth root of their product. With 2, 8 and 32: multiply to get 2 × 8 × 32 = 512, then take the cube root, which is 8. The arithmetic mean of the same three numbers is 14 — a big gap, and a deliberate one. The defining property is worth stating plainly: replacing every value with the geometric mean leaves the product unchanged. 8 × 8 × 8 = 512. The arithmetic mean does the same job for sums. Which average is "right" depends entirely on whether the quantity you care about is a sum or a product. Multiplying long lists directly overflows quickly, so this page computes it in logarithms instead: take the natural log of each value, average those, and exponentiate. exp(mean of ln x) is mathematically identical to the nth root of the product and stays numerically stable across hundreds of values.

Anything that multiplies period after period — growth rates, investment returns, population change, inflation — has to be averaged geometrically. Suppose a quantity grows 10% one year and 30% the next. The growth factors are 1.10 and 1.30, and their product is 1.43: a 43% total increase. The geometric mean is √1.43 ≈ 1.1958, an average of about 19.58% per year — and 1.1958² ≈ 1.43, which checks out. The arithmetic mean says 20%, and 1.20² = 1.44. It overstates, every time. The gap becomes dramatic with volatility. Gain 50%, then lose 50%: the factors are 1.5 and 0.5, product 0.75, so you finish 25% down. The arithmetic mean of +50% and −50% is zero, which describes nothing that happened. The geometric mean is √0.75 ≈ 0.866 — about −13.4% per period — and that number does reproduce the ending value. Compound annual growth rate is exactly this calculation in disguise: CAGR = (end ÷ start)^(1/years) − 1 is the geometric mean of the yearly growth factors.

The geometric mean is never larger than the arithmetic mean of the same positive numbers, and the two are equal only when every value is identical. This is the arithmetic mean–geometric mean inequality, one of the most-used results in all of mathematics. The practical reading is that the size of the gap measures spread. For 10, 10 and 10 both means are 10. For 2, 8 and 32 the arithmetic mean is 14 and the geometric mean is 8. Same count, same order of magnitude, wildly different dispersion — and the two averages diverge accordingly. This calculator shows both figures side by side for that reason. If they are close, your values are tightly clustered and the choice of average barely matters. If they are far apart, the choice matters a great deal, and picking the wrong one produces a number that does not reconstruct the thing you measured.

The second natural home for the geometric mean is ratios, where the arithmetic mean gives an answer that depends on which quantity you put on top. Average the ratios 2/1 and 1/2 arithmetically and you get 1.25. Flip both ratios, average again, and you get 1.25 once more — two answers that should be reciprocals but are not. The geometric mean gives 1 either way, because GM(a ÷ b) = GM(a) ÷ GM(b). That invariance is not a curiosity; it is the reason index numbers, price relatives and performance ratios are so often averaged geometrically. The same property explains why the geometric mean is used to combine indicators measured on different scales into one composite. The United Nations Human Development Index, for instance, combines its component indices geometrically, which keeps a very low score on one dimension from being papered over by a very high score on another — something an arithmetic average happily allows.

The geometric mean is defined only for positive numbers. A single zero drives the product to zero and the mean with it; negative values make the root ambiguous or complex. This calculator skips any value that is zero or below and tells you how many it dropped — read that note, because silently discarding data changes what the average means. That restriction rules out whole classes of data. Temperatures, profit-and-loss figures and anything that can legitimately be negative are not candidates. Working with returns is the standard workaround: a −20% return is a factor of 0.8, which is positive, and it is the factors you average. The geometric mean is also not a robust statistic in the everyday sense, though it is less sensitive to large outliers than the arithmetic mean and considerably more sensitive to small ones. A single value near zero pulls the result down hard. And it is simply the wrong tool for additive quantities. Heights, weights, test scores, order values — anything where a total is meaningful — belong to the arithmetic mean. Use the geometric mean when the meaningful aggregate is a product, not a sum.