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Geometric Mean Calculator

Enter the values to calculate the geometric mean step-by-step.

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Geometric Mean Calculator

Use this Geometric Mean Calculator to find the geometric mean of a set of numbers or percentages. It works for small or large datasets, including integers, decimals, and even negative values when converted to growth rates, such as calculating return on investment.

What Is a Geometric Mean?

The geometric mean is a measure of central tendency for positive real numbers. It is calculated as the nth root of the product of n numbers.

Formula:

Geometric Mean Formula

n = total number of values
a₁, a₂, …, aₙ = the dataset values

How to Calculate Geometric Mean

Multiply all numbers together.
Take the nth root of the product.

Example:

Find the geometric mean of 12, 23, and 34.

Step 1: Multiply the values: 12 × 23 × 34 = 9384
Step 2: Take the cube root (∛9384) ≈ 21.0926

Geometric Mean for Negative Numbers

By definition, the geometric mean is for positive numbers, as negative products can lead to undefined or complex results. For negative values representing growth/decline:

Convert percentages to growth factors: decline → 1 - rate, growth → 1 + rate.
Multiply the factors and take the nth root.
Convert back to percentage if needed.

Example:

Calculate the geometric mean of -20%, 30%, and -15%:

-20% → 1 - 0.20 = -0.8
30% → 1 + 0.30 = 1.3
-15% → 1 - 0.15 = -0.85

Multiply factors: -0.8 × 1.3 × -0.85 = 0.884

Cube root: ∛0.884 ≈ 0.9597 → Convert back to %: (0.9597 - 1) × 100 ≈ -4.03%

✅ Interpretation: Even with declines, a meaningful geometric mean can be calculated.

Geometric Mean Using Logarithms

For large datasets or when handling negative numbers indirectly:

Take the logarithm of each value.
Compute the average of the logs.
Take the antilog of the result to obtain the geometric mean.

This simplifies multiplication into addition and is numerically stable.

Geometric Mean With Zero

If zero appears, the geometric mean becomes undefined. Options include:

Replace zero with 1 (or 100%) if it represents no change.
Remove zeros representing missing data.

⚠️ Note: Most calculators do not adjust for zeros automatically.

When To Use Geometric Mean

Average rate of return or growth rate over multiple periods.
Situations involving multiplication of values or exponential growth (e.g., population growth, investments).

References

Wikipedia: Geometric mean
Investopedia: Geometric Mean in Investing
LibreTexts: Geometric Mean

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