Geometric Mean Calculator

Statistics & The Pythagorean Means Hierarchy

In classical mathematics, the three Pythagorean means (Arithmetic, Geometric, and Harmonic) describe different aspects of central tendency:

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• Geometric Mean (GM) Formula**:

GM = √[n]x_1 · x_2 · · x_n =

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• Arithmetic Mean (AM)**:

AM = (1) / (n) Σ_i=1^n x_i

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• Harmonic Mean (HM)**:

HM = (n) / (Σ_i=1)^n (1) / (x_i)

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• The Universal AM-GM-HM Inequality**:

For any collection of positive real numbers:

AM GM HM

Equality holds if and only if all numbers in the set are identical.

How to Use This Calculator

Enter your Enter Positive Numbers (Comma or Space-Separated, e.g. 4, 9, 16, 25) (e.g. 4, 9, 16, 25) and Pythagorean Benchmark Benchmark — choose from Custom Numbers Set, 4, 9, 16, 25 (GM: 10.9545 / AM: 13.50), 4 & 16 (GM: 8.00 / √(4 × 16) = 8). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Formula Reference: Geometric Mean Calculator

Input variables:

numbersInputString — Enter Positive Numbers (Comma or Space-Separated, e.g. 4, 9, 16, 25)

Computed outputs:

geometricMeanResult — Geometric Mean (GM)

arithmeticMeanResult — Arithmetic Mean (AM)

harmonicMeanResult — Harmonic Mean (HM)

totalCountElements — Sample Size (n)

pythagoreanInequalityProof — AM ≥ GM ≥ HM Inequality Check

Mathematical relationships extracted from calculation logic:

sum = nums.reduce((a, b) => a + b, 0)

logSum = nums.reduce((acc, val) => acc + Math.log(val), 0)

invSum = nums.reduce((acc, val) => acc + safeDiv(1.0, val), 0)

Standard: NIST / ISO mathematical definitions

Worked Example

An investment portfolio growing by +10%, +25%, -15%, and +15% over 4 years has an arithmetic average of +8.75%, but its true compounded geometric return (CAGR) is +7.74%.

Understanding Your Result

Statistical results require context: a mean alone is misleading without knowing the standard deviation (spread). Normal distribution reference: 68% of values fall within ±1 SD of the mean, 95% within ±2 SD, 99.7% within ±3 SD. For small sample sizes (n < 30), use t-distributions rather than z-scores. Statistical significance (p < 0.05) does not imply practical significance — effect size (Cohen's d, R², odds ratio) measures real-world importance. Correlation coefficient r: |r| < 0.3 weak, 0.3–0.7 moderate, > 0.7 strong.