Geometric Mean Calculator & Average Growth Rate

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Formula at a glance

Geometric Mean Calculator: Geometric mean — GM = (x₁ × x₂ × … × xₙ)^(1/n). Example: 4, 9, 16, 25 → Geometric mean: 10.95445115 — How the Geometric Mean Calculator works: Geometric mean — GM = (x₁ × x₂ × … × xₙ)^(1/n)

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📘 About the Geometric Mean Calculator

The geometric mean is the n-th root of the product of n numbers. It is the correct average for quantities that multiply rather than add: investment returns, population growth, inflation, ratios and index numbers. If an investment rises 10%, falls 5% and rises 20%, its average yearly growth is the geometric mean of 1.10, 0.95 and 1.20, minus 1.

Enter positive numbers, or percentages with the growth option. The calculator gives the geometric mean (or average growth rate), compares it with the arithmetic and harmonic means — always AM ≥ GM ≥ HM — and shows the working.

🧮 Formula

Geometric mean

GM = (x₁ × x₂ × … × xₙ)^(1/n)

Average growth rate

g = [(1 + r₁)(1 + r₂)…(1 + rₙ)]^(1/n) − 1

Inequality

AM ≥ GM ≥ HM

✏️ Worked examples

4, 9, 16, 25

You enter
Numbers	4, 9, 16, 25
Are these growth rates in %?	No — ordinary numbers

You get
Geometric mean	10.95445115
Arithmetic mean	13.5
Harmonic mean	8.627920911
AM ≥ GM ≥ HM	13.5 ≥ 10.9545 ≥ 8.62792 ✓
Product of the values	14400
Count	4

GM = 10.95445115.

Returns of +10%, −5%, +20%

You enter
Numbers	10, -5, 20
Are these growth rates in %?	Yes — e.g. 10, −5, 20 (% per year)

You get
Average growth rate (geometric)	7.83652% per period
Simple average of the rates	8.33333% (overstates real growth)
Total growth over 3 periods	25.4%
Check	(1 + 7.83652%)^3 = 1.254

7.83652% per period.

🧭 How to use this calculator

What you enter

NumbersSeparate numbers with commas, spaces or new lines.
Are these growth rates in %?

What you get

Geometric mean
Arithmetic mean
Harmonic mean
AM ≥ GM ≥ HM
Product of the values
Count

❓ Frequently asked questions

When should I use the geometric mean?

For rates of change and ratios — returns, growth, inflation — and for data that span several orders of magnitude.

Why is the geometric mean lower than the average?

Because ups and downs do not cancel: +50% then −50% leaves you at 75%, an average growth of −13.4% a year, not 0%.

Can the geometric mean use negative numbers?

No. Convert returns to growth factors (1 + r) first, which is what the growth option does.

√Π Geometric Mean Calculator (and Average Growth Rate)