The geometric mean is a multiplicative center. It replaces positive factors with one equal factor while preserving their product.
Use it for repeated growth factors, proportional changes, normalized ratios, and other quantities that combine by multiplication rather than addition.
The relationship
The defining equation
G = (x₁x₂⋯xₙ)^(1/n)
x₁ through xₙ are positive values, n is their count, and G is their geometric mean.
See the structure
Equal factors with the same product
1same product: 1×4 = 2×24equal factors: 2 and 2
Worked example
Geometric mean of 1 and 4
G = √(1×4) = √4 = 2
Interpret with care
What the result does—and does not—mean
Do not use it for ordinary additive totals. This calculator accepts zero by the conventional zero-product rule but rejects negative values in its real-number model.
Quick guide
How to use this calculator
Add one non-negative value per labelled row.
If any value is zero, the geometric mean is exactly zero.
Otherwise, read whether the result is exact or explicitly approximate.
Calculation method
Average logarithms, then exponentiate
For positive values, the geometric mean is the nth root of their product. Computing the mean of logarithms avoids constructing an enormous or tiny intermediate product.
Worked example
Geometric mean of 1 and 4
The product is 4 and its positive square root is 2.
√(1×4) = 2
Supported inputs
Precision and limits
List size
Add 1–100 labelled value rows.
Decimal size
Each value may contain up to 200 digits and 100 decimal places.
Exactness
Exact rational results are never replaced by a rounded decimal; approximations are labelled.
Real domain
Negative values are rejected. Zero is supported and produces exact zero.
Approximation
General positive results are numerical approximations formatted to 15 significant digits; an identical-input shortcut remains exact.
