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Geometric Mean Calculator

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Geometric Mean Calculator

The geometric mean is the nth root of the product of n positive numbers. It is the right average for quantities that multiply rather than add: growth rates, ratios, index numbers and data spread over several orders of magnitude. Enter your values and the calculator returns the geometric mean with its working, alongside the arithmetic and harmonic means for comparison.

To avoid overflow with long lists, the calculation averages the natural logarithms and then exponentiates: GM = exp[(Σ ln x) ÷ n]. For 2, 8, 4, 16 and 32 the geometric mean is 8, while the arithmetic mean is 12.4, pulled up by the large values. The three means always satisfy harmonic ≤ geometric ≤ arithmetic, with equality only when all values are the same.

All values must be greater than zero. For average investment returns, convert percentages to growth factors first (a 10% gain is 1.10) and subtract 1 from the result; the geometric mean return formula page does this for you. The geometric standard deviation, shown below the result, describes multiplicative spread. For ordinary averages use the mean, median and mode calculator.

Separate values with commas, spaces or new lines.

Result and step-by-step solution

Geometric mean
8

Arithmetic mean = 12.4, harmonic mean = 5.16129 (always HM ≤ GM ≤ AM).

  1. Formula
    GM = (x₁ × x₂ × … × xₙ)1/n = exp[(Σ ln x) ÷ n]
  2. Sum of logs
    Σ ln x = ln 2 + ln 8 + ln 4 + ln 16 + ln 32 = 10.397208
  3. Average log
    (Σ ln x) ÷ n = 10.397208 ÷ 5 = 2.079442
  4. Back-transform
    GM = e2.079442 = 8
Geometric standard deviation2.992059

Frequently asked questions

How do you calculate the geometric mean?
Multiply the n values and take the nth root. For 4 and 9: √(4 × 9) = 6. For long lists, average the logarithms and exponentiate.
When should I use the geometric mean instead of the average?
For rates of growth, ratios and percentages that compound, and for skewed data spanning several orders of magnitude, such as bacterial counts.
Can the geometric mean be calculated with zero or negative numbers?
No. Every value must be positive. A zero makes the product zero, and logarithms of negative numbers are undefined.
How do I calculate the geometric mean in Excel?
Use =GEOMEAN(A1:A10). =HARMEAN gives the harmonic mean.

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Last reviewed: October 7, 2026. Calculations run in your browser and were validated against SciPy.

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