# Geometric Mean Calculator

Free geometric mean calculator. Calculate the geometric mean of positive numbers for growth rates, financial returns, and ratios.

## What this calculates

Calculate the geometric mean of up to 6 positive values. The geometric mean is the appropriate average for growth rates, returns, and any multiplicative quantities.

## Inputs

- **Value 1** — min 0.0001
- **Value 2** — min 0.0001
- **Value 3** — min 0.0001
- **Value 4** — min 0 — Enter 0 to skip.
- **Value 5** — min 0 — Enter 0 to skip.
- **Value 6** — min 0 — Enter 0 to skip.

## Outputs

- **Geometric Mean** — The geometric mean of the values.
- **Arithmetic Mean** — The arithmetic mean for comparison.
- **Harmonic Mean** — The harmonic mean for comparison.
- **Number of Values** — How many values were used.
- **Mean Relationship** — formatted as text — Relationship between the three means.

## Details

The geometric mean is the nth root of the product of n values.

Formula:
GM = (x₁ × x₂ × ... × xₙ)^(1/n)

Or equivalently: GM = exp((1/n) × Σ ln(xᵢ))

When to Use

- Averaging growth rates or returns over time

- Averaging ratios or proportions

- Data spanning several orders of magnitude

Mean Inequality:
For positive values: Harmonic Mean ≤ Geometric Mean ≤ Arithmetic Mean

Equality holds only when all values are identical.

## Frequently Asked Questions

**Q: When should I use the geometric mean instead of the arithmetic mean?**

A: Use the geometric mean when: (1) averaging percentages, ratios, or rates of change, (2) data is multiplicative rather than additive (e.g., compound growth), (3) data spans several orders of magnitude. The arithmetic mean overestimates the average in these cases.

**Q: Why can't the geometric mean handle zero or negative values?**

A: The geometric mean requires taking the product of all values. If any value is zero, the product is zero. Negative values make the root undefined in the real numbers. For data with zeros, consider using the geometric mean of (values + 1) and then subtracting 1 from the result.

**Q: What is the relationship between geometric mean and CAGR?**

A: CAGR (Compound Annual Growth Rate) is essentially a geometric mean of growth factors. If an investment grows by factors of 1.10, 1.05, and 0.95 over three years, the CAGR = geometric mean = (1.10 × 1.05 × 0.95)^(1/3) - 1 ≈ 3.2% per year.

---

Source: https://vastcalc.com/calculators/math/geometric-mean
Category: Math
Last updated: 2026-04-21