Geometric Mean Calculator

What Is the Geometric Mean?

The geometric mean is a type of average that is calculated by multiplying all values together and then taking the nth root, where n is the number of values. It is especially useful when values are multiplicative — such as growth rates or ratios.

The Formula

For n values:

Geometric Mean = (x₁ × x₂ × x₃ × … × xₙ)^(1/n)

Or equivalently using logarithms (better for large datasets):

Geometric Mean = exp[(ln x₁ + ln x₂ + … + ln xₙ) ÷ n]

When to Use the Geometric Mean vs Arithmetic Mean

Practical Example — Investment Returns

An investment returns: +50% in year 1, −20% in year 2, +30% in year 3.

As multipliers: 1.50, 0.80, 1.30

Arithmetic average = (1.50 + 0.80 + 1.30) ÷ 3 = 1.20 (suggests 20% average return — misleading)

Geometric mean = (1.50 × 0.80 × 1.30)^(1/3) = (1.56)^(1/3) = 1.1597 (actual 15.97% average — correct)

The arithmetic mean overstates the actual compound growth rate. The geometric mean gives the true compound annual growth rate (CAGR).

Another Example — Aspect Ratios

What is the geometric mean screen size between a 9-inch and 16-inch dimension?

Geometric mean = √(9 × 16) = √144 = 12 inches

This is why 12-inch screens fit naturally between standard 9:16 content.

Limitations

• All values must be positive (cannot take the log of zero or a negative number)
• For investment returns entered as percentages (e.g. +50%, −20%), convert to multipliers first: add 1 to each percentage value