Geometric mean

The geometric mean is the n-th root of the product of n positive values. It is the correct average for multiplicative processes such as investment returns over multiple periods, and is always at or below the arithmetic mean.

ByHoang TruongUpdated July 2, 2026

See it move

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An investment gains 50% in year one and loses 33% in year two. The arithmetic mean of these returns, 8.5%, looks like healthy growth, but €100 invested actually becomes €150 after year one and only €100.50 after year two, essentially flat. The geometric mean, about 0.25% a year, is the rate that truly reproduces this outcome.

Where it fits

TopicDescriptive StatisticsAdvancedSubjectData Analysis & StatisticsAdvanced

The formula

LaTeX

GM = \left(\prod_{i=1}^{n} x_i\right)^{1/n}

Variables

$x_{1}, x_{2}, \dots, x_{n}$: Positive values in the series
$n$: Number of values

The geometric mean equals the arithmetic mean only when all values are identical; otherwise it is always lower. The gap widens as variability in the series increases.

LaTeX

r_{GM} = \left(\prod_{i=1}^{n}(1+r_i)\right)^{1/n} - 1

Variables

$r_{i}$: Return in period i expressed as a decimal (e.g. 0.10 for 10%)
$n$: Number of periods

This is the single constant rate that, compounded over n periods, reproduces the actual cumulative return. The arithmetic average of period returns always overstates it when returns vary.

Check yourself

PracticeCORE

An investment portfolio achieves a return of +60% in year 1 and −40% in year 2. What is the approximate geometric mean annual return over the two years?

Select an answer to check your understanding.