Geometric Mean Calculator

Geometric mean: the right average for growth rates

For n positive values, the geometric mean is GM = ⁿ√(x₁ · x₂ · … · xₙ). In practice you compute it as exp(mean of logs), which is more stable numerically. Use it when you're dealing with compound growth rates (CAGR) or any multiplicative quantity. Take returns of 2×, 8× and 32×: GM = ∛(2·8·32) = ∛512 = 8. The AM-GM inequality tells us GM is never larger than the arithmetic mean, and the two only match when every value is identical. That gap is exactly why averaging returns with the arithmetic mean flatters your compound performance. One caveat: the values have to be strictly positive, so for negatives you'll need a shift or a log-return transformation.

Applications

It shows up in financial returns (the annualized equivalent rate, fund performance), in population and economic growth (CAGR), in photometry and astronomy where you want a central value on a log scale, in antibody titers in biology, and in the UN Human Development Index (HDI). The HDI runs its three dimensions through the geometric mean precisely so that imbalance gets penalized.

FAQ

When use GM instead of arithmetic mean? Reach for it whenever the values multiply together: rates, ratios, percentage changes. Across several periods of returns, the GM hands you the equivalent constant rate. The arithmetic mean won't.

What if there are zeros or negatives? A single zero drags GM down to 0. Negatives leave the n-th root undefined when n is even. The fix for returns is to work with 1 + r rather than r, which keeps everything positive.

Why does HDI use GM? Because the geometric mean punishes imbalance. A country that's rich but poorly educated ends up scoring below one whose three dimensions are roughly level, which nudges the whole measure toward balanced development.