Geometric Mean Calculator

GM = (x₁ · x₂ · ... · xₙ)^(1/n). Right "average" for ratios, growth rates, multiplicative quantities.

Inputs

Result

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How to use this calculator

  • Enter positive numbers (the calculator rejects zero or negative).
  • Read GM and compare to AM.

About this calculator

The geometric mean is the right average for multiplicative quantities — investment returns, growth rates, or ratios. For a sequence of growth multipliers, GM gives the constant rate that would produce the same total growth. AM-GM inequality: GM ≤ AM, with equality only when all values are equal.

Frequently asked

When should I use geometric mean over arithmetic mean?+
For ratios, growth multipliers, percentages of change. If you grew 50% one year and 50% the next, the average growth isn't 50% (=AM); it's ~22.5% (=GM of 1.5 and 1.5 minus 1, sort of).
Why are negative numbers not allowed?+
Geometric mean involves taking products and roots; negative values produce nonsense answers (or complex numbers). For data that mixes signs, use arithmetic mean.
Compound annual growth rate (CAGR)?+
CAGR is a geometric mean of growth ratios minus 1. Investment returns are conventionally averaged geometrically.
Why is GM always ≤ AM?+
AM-GM inequality. Equality only when all values are identical. This is a fundamental fact of analysis.
How does this differ from harmonic mean?+
HM is for averaging rates (mph, throughput). GM is for averaging ratios (growth, returns). AM is for averaging absolute quantities. Different "averages" for different problems.

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