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📊 Geometric Mean Calculator
Calculate the geometric mean of any set of numbers. Supports raw values and percentage returns with step-by-step workings.
📌 Common Examples
Investment Returns
Years: +10%, +20%, -15%, +5%
Factors: 1.10, 1.20, 0.85, 1.05
Geo Mean: 1.0427 → +4.27% avg/yr
Population Growth
Growth rates: 2.5%, 3.1%, 1.8%
Factors: 1.025, 1.031, 1.018
Geo Mean: 1.0246 → 2.46% avg
Simple Values
Values: 2, 8
√(2 × 8) = √16
Geo Mean: 4
AM vs GM Gap
Values: 1, 100
Arithmetic Mean: 50.5
Geo Mean: 10 (much lower!)
How This Tool Works
The geometric mean is a type of average that multiplies all values together and then takes the nth root, where n is the count of values. Unlike the arithmetic mean (regular average), the geometric mean accounts for compounding and is the correct way to average rates of change, growth rates, and ratios. For example, if an investment returns +50% one year and −33% the next, the arithmetic mean suggests +8.5% average growth — but your money is actually flat (1.50 × 0.67 = 1.0). The geometric mean correctly reports 0% average growth.
Formula / Methodology
GM = (x₁ × x₂ × x₃ × … × xₙ)^(1/n) — multiply all n values together, then take the nth root. Equivalently: GM = exp((ln(x₁) + ln(x₂) + … + ln(xₙ)) / n) — average the natural logs, then exponentiate.💡 Tips & Best Practices
- 1Use the geometric mean for investment returns — it accounts for compounding and gives the true average annual return.
- 2All values must be positive. If you have percentage changes, convert them to growth factors first (e.g., +20% → 1.20, −10% → 0.90).
- 3The geometric mean is always less than or equal to the arithmetic mean — this is known as the AM-GM inequality.
- 4Common use cases: averaging financial returns, biological growth rates, aspect ratios, and any data where values are multiplied rather than added.
Frequently Asked Questions
When should I use geometric mean instead of arithmetic mean?
Use the geometric mean whenever you are averaging rates of change, growth rates, or ratios. If values compound (like investment returns), geometric mean gives the correct average. Use arithmetic mean for additive data like heights, weights, or test scores.
Why must all values be positive?
The geometric mean involves multiplying all values and taking roots. A zero makes the product zero regardless of other values, and negative numbers create problems with even roots (square root of a negative is undefined in real numbers). Convert percentage changes to growth factors to keep values positive.
How is geometric mean related to CAGR?
CAGR (Compound Annual Growth Rate) is essentially the geometric mean of annual growth factors minus 1. If you have yearly returns of 10%, 15%, and −5%, convert to factors (1.10, 1.15, 0.95), compute their geometric mean, and subtract 1 to get the CAGR.
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