Geometric Mean Calculator
The geometric mean is the n-th root of the product of n positive numbers — the right kind of average for growth rates, annualised returns and any data that compounds multiplicatively. A +10 % year followed by a −10 % year arithmetically averages to 0 %, but the geometric mean honestly returns −0.50 % — and your capital really has shrunk.
Geometric mean (GM)
—
Arithmetic mean (AM)
—
The ordinary "average" — sum divided by n.
Harmonic mean (HM)
—
n ÷ Σ(1/xᵢ). Best for averaging speeds or rates per unit.
Other statistics
- Count
- —
- Sum
- —
- Product Π xᵢ
- —
- Min
- —
- Max
- —
- Spread (max ÷ min)
- —
Formula
GM = (x₁ · x₂ · … · xₙ)^(1/n) = exp( (Σ ln xᵢ) / n ) Annualised return (CAGR) = GM − 1, when each xᵢ is (1 + period return). AM-GM-HM inequality: AM ≥ GM ≥ HM, with equality iff every xᵢ is the same.
- · All inputs must be positive; zero or negative values are stripped (the logarithm is undefined there).
- · The implementation sums logarithms instead of raw products, so the geometric mean stays finite even when the product itself overflows (e.g. many values around 10²⁰⁰).
- · The geometric mean is always ≤ the arithmetic mean (the AM-GM inequality); the more spread out the values, the larger the gap.
- · For percentage returns, enter the values as (1 + rate) — e.g. −5 % becomes 0.95 — then read the annualised compound rate as GM − 1.
- · For averaging speeds or rates per unit (e.g. "30 km/h uphill, 60 km/h downhill"), the harmonic mean is the right tool — this calculator shows all three side-by-side so you can compare.
- · All computation uses IEEE-754 double-precision floats; extreme values may lose a few digits of precision.
Frequently asked
How is the geometric mean different from the arithmetic mean — and when should I use which?
The arithmetic mean is "sum divided by n" — good for quantities that add (heights, test scores, monthly salaries). The geometric mean is "product to the 1/n" — good for ratios that compound (annual returns, population growth, CPI, anything multiplicative). Quick test: if your numbers come from "previous × factor," use geometric; if from "previous + quantity," use arithmetic. Three yearly returns of +10 %, +20 %, −30 % average arithmetically to 0 %, but their geometric mean (over 1.10, 1.20, 0.70) is ≈ −1.96 % — the real annualised return.
Can I use this to compute CAGR (annualised return)?
Yes — that is the textbook use case. Convert each year's return to a growth ratio (e.g. +8 % → 1.08, −3 % → 0.97) and enter the lot. The geometric mean is the equivalent per-year growth factor, so CAGR = (geometric mean) − 1. Compared with our ROI / CAGR calculators, this one lets you plug in an arbitrary list of yearly returns at once.
Why must every value be positive?
Because the geometric mean is defined through logarithms, and ln 0 / ln of a negative number are undefined. If you have signed percentage changes (e.g. −20 %), first convert them to growth ratios (1 + change = 0.80) so every input is positive, then take the geometric mean. The calculator flags zero and negative tokens and skips them automatically.
How does the geometric mean relate to the harmonic mean?
They sit in a chain of inequalities: AM ≥ GM ≥ HM for any list of positive numbers, with all three equal only when every value is identical. The harmonic mean is best for "rate-per-unit" data such as average driving speed when you spend equal distance (not time) at each speed. This calculator shows all three so you can pick the one that fits your problem.
Related tools
Percentage Calculator
Percent of, percent change, and percent add/subtract in one.
GCD & LCM Calculator
Enter 2–6 positive integers to get the greatest common divisor (HCF / GCD) and least common multiple (LCM), with the Euclidean step chain shown.
Average Calculator (Mean / Median / Mode)
Enter a list of numbers to get the mean, median, mode, range plus standard deviation, variance and total.
Quadratic Equation Solver
Enter the coefficients of ax² + bx + c = 0 to find the real or complex roots, discriminant and vertex.
Password Strength (Entropy) Calculator
Estimate a password's bit entropy, brute-force time and strength tier. All computation happens in your browser.
Scientific Notation Converter
Convert between standard decimal numbers and scientific notation, with significant figures and order of magnitude.
Permutations & Combinations (nPr / nCr) Calculator
Compute permutations P(n,r), combinations C(n,r) and factorial n! — useful for probability problems, lottery odds and combinatorics homework.
Standard Deviation Calculator
Paste a list of numbers to compute mean, median, sample and population variance and standard deviation — with the working shown.
Triangle Calculator (SSS / SAS / ASA)
Solve a triangle from 3 sides, 2 sides + 1 angle, or 2 angles + 1 side — area, perimeter and remaining parts via the law of sines / cosines.
Pythagorean Theorem Calculator
Given any two sides of a right triangle (two legs, or one leg plus the hypotenuse), instantly find the third side, area, perimeter and the two non-right angles.
Circle Calculator (radius / diameter / circumference / area)
Enter any one of radius, diameter, circumference or area to get the other three — useful for design, engineering and DIY.
Roman Numeral Converter
Two-way conversion between Arabic numbers (1–3999) and Roman numerals (I, V, X, L, C, D, M) — handy for typesetting, chapter titles and homework.
Slope & Line Equation Calculator (y = mx + b from Two Points)
Enter two points (x₁, y₁) and (x₂, y₂) to instantly get the slope, y-intercept, line equation y = mx + b, distance, and midpoint — a classroom staple for algebra and coordinate geometry.
Birthday Paradox Calculator
Enter group size n to see the probability that at least two people share a birthday — the classic birthday problem.
Logarithm Calculator (log / ln / log₂ / any base)
Compute logₐ(x) for any base — natural log (ln), common log (log₁₀), binary log (log₂) and a custom base, with the change-of-base steps shown.
Z-Score (Standard Score) Calculator
Enter a value, the mean and the standard deviation to compute the z-score and the corresponding normal-distribution percentile and probabilities.
Screen Pixel Density (PPI) Calculator
Enter the screen resolution and diagonal size to get pixel density (PPI), real width/height, dot pitch and total pixel count.
Hong Kong Mark Six Odds Calculator
Enter the number of tickets / selections and see the actual probability of hitting first, second … prizes in a Mark Six (6-of-49) draw.
Decimal to Fraction Converter
Convert any decimal (including repeating decimals) to a simplified fraction and a mixed number.
Sphere Volume & Surface Area Calculator
Give a sphere any one of radius, diameter, surface area or volume and instantly get the other three — plus the great-circle circumference and area.
Cylinder Volume & Surface Area Calculator
Enter the radius and height of a cylinder to get volume (π r²h), lateral surface, base area and total surface area.
Prime Factorization Calculator
Factor any integer from 2 up to 10¹² into primes, see the canonical exponent form, count and list all divisors.
Fibonacci Sequence Calculator
Enter any integer n from 0 to 1500 to instantly compute F(n) and F(n−1) with BigInt precision, the consecutive-term ratio (converging to the golden ratio φ) and the first 30 terms of the sequence.