Geometric Mean Calculator (Values, Growth Rates and Weights, with Steps)
Find the geometric mean of positive numbers or the average growth rate of percentage returns — with weights, the geometric standard deviation, a confidence interval, arithmetic and harmonic means for comparison, a table, chart and worked steps.
How it works
GM = (x₁ · x₂ · … · xₙ)^(1 ÷ n) = exp( Σ ln x ÷ n ) · weighted: exp( Σ wᵢ ln xᵢ ÷ Σ wᵢ ) · growth rate = (GM of (1 + r ÷ 100) − 1) × 100 · GSD = exp(SD of ln x) · CI = exp( mean ln x ± t × SD ÷ √n )
The geometric mean multiplies all n values together and takes the nth root, giving the central tendency of quantities that combine by multiplication rather than addition — growth rates, investment returns, index ratios, dilution titres and areas. To stay numerically stable this calculator computes it as the exponential of the average log, which is mathematically identical. Switch the data type to % changes to average returns directly: each percentage becomes a growth factor (+10% → 1.10), the geometric mean of the factors gives the compound average rate, and a period-by-period table and chart show that this constant rate reaches the same end value as the real sequence. Under More options you can attach weights, choose how zeros and negatives are handled, and pick the rounding. Alongside the geometric mean you get the arithmetic and harmonic means (always AM ≥ GM ≥ HM), the median, the geometric standard deviation — the multiplicative spread factor — and a confidence interval built on the log scale with the t distribution.
Worked example
For 2 and 8 the product is 16 and its square root is 4, so the geometric mean is 4, while the arithmetic mean is (2 + 8) ÷ 2 = 5 and the harmonic mean is 3.2. For returns, a fund that gains 10% one year and 50% the next has growth factors 1.10 and 1.50: √(1.10 × 1.50) = √1.65 = 1.2845, so the average return is 28.45% a year, not the 30% a simple average suggests — and 100 grows to 165 either way. A sequence of +50% then −50% averages 0% arithmetically but the geometric mean of 1.5 and 0.5 is 0.866, a real loss of 13.4% a year.
Sources: Geometric mean — Wikipedia · Geometric standard deviation — Wikipedia
Frequently asked questions
When should I use the geometric mean instead of the arithmetic mean?
Use it for things that compound or combine multiplicatively: average growth rates, investment returns over several years, ratios, or rates of change. Averaging yearly growth factors with the geometric mean gives the true equivalent constant rate, which the arithmetic mean overstates.
How do I find the geometric mean of percentage returns?
Choose “% changes / returns” and type the percentages (10, −5, 8). The calculator converts each to a factor (1.10, 0.95, 1.08), takes their geometric mean and subtracts 1. You do not need to convert anything by hand, and negative returns down to just above −100% work.
Why can’t the values be zero or negative?
The geometric mean multiplies the numbers and takes a root. A single zero makes the whole product zero, and negatives make the root undefined for real numbers. Under More options you can leave such values out or use the add-1-then-subtract-1 workaround that is common for count data — but say so if you report it.
Is the geometric mean always smaller than the arithmetic mean?
It is smaller whenever the values differ, and equal only when every value is identical. This is the AM–GM inequality; the harmonic mean is smaller still. The greater the spread in the data, the wider the gap.
What is the geometric standard deviation?
It is e raised to the standard deviation of the logged values, a multiplicative spread factor. A GSD of 2 means values typically lie between the geometric mean divided by 2 and multiplied by 2 — roughly two-thirds of the data for log-normal data.
How do I calculate the geometric mean in Excel?
Use =GEOMEAN(A1:A10) for positive values. For returns, put the growth factors (1 + rate) in the cells, use GEOMEAN and subtract 1. The results match this calculator.
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