This geometric mean calculator finds a multiplicative average from nonnegative values, relative weights or equal-period percentage changes. It shows the logarithmic working and an arithmetic comparison.
Average values that combine multiplicatively
The geometric mean summarizes values through multiplication. With equal weights, it is the root of their product corresponding to the observation count. For positive values, the same quantity is the exponential of their average natural logarithm. This calculator uses logarithmic working so a product can exceed finite arithmetic without destroying a useful mean. Choose raw observations or factors, known relative-weight pairs, or equal-duration percentage changes. Raw and weighted values must be nonnegative and share a common measurement unit. An included zero gives a zero product-root mean under this page’s explicit nonnegative convention. Negative raw values are outside the supported convention. A geometric average and an arithmetic average answer different questions. The tool shows a comparison on the same input basis. Percentage changes are converted to factors first; they are not treated as signed raw observations. This is generic multiplicative mathematics, without an inferred financial valuation or uncertainty model.
Use the geometric mean to summarize multiplicative factors. The arithmetic comparison summarizes additive values on the same basis.
Percentage mode converts each equal-period change to a nonnegative factor. A negative percentage can be supported even though a negative raw observation is rejected.
Weights are nonnegative relative contributions, with positive total weight. They need not sum to a percentage total.
Check the product and the geometric mean
The raw preset uses 2 8, with observation count 2. Its product is 16, and the geometric mean is 4. Repeating that mean as a factor for each observation reproduces the product. The arithmetic mean of the same observations is 5.
For positive inputs, average their natural logarithms and exponentiate the result. The raw preset’s mean logarithm is approximately 1.386294361. Working is computed before display rounding. The official SciPy geometric-mean documentation gives both the logarithmic and product-root identities.
This formula has no sample-size-minus-one variant. A sample/population label does not create an alternate geometric-mean denominator. Each equal-weight observation contributes to the full count. The original measurement unit remains the mean’s unit.
Enter weights or equal-period percentage changes
In weighted mode, enter one value and relative weight pair per line. The tool normalizes the weights before averaging logarithms. The weighted preset assigns weights 1 and 3 to values 2 and 8. Total weight is 4, weighted geometric mean is approximately 5.656854249 and the same-basis weighted arithmetic mean is 6.5. Scaling every weight by the same positive amount preserves the mean.
Every row is validated. Zero-weight rows have no mathematical influence; a zero value with positive weight instead makes the nonnegative geometric mean zero. A positive total weight is required. SciPy supplies the weighted-log formula, and R’s weighted.mean documentation explains normalized weights for the arithmetic comparison.
In percentage mode, use numeric changes, optionally with a suffix, for equal-duration periods. The preset changes 10 -10 become factors 1.1 and 0.9. Their total factor is 0.99. The geometric mean factor is approximately 0.9949874371, equivalent to -0.5012562893% per period. The arithmetic average of the supplied changes is 0%.
The factor conversion adds the fractional change to the unchanged baseline. Converting a mean factor back subtracts that baseline and applies the percentage scale. NIST SP811 defines as the scale 0.01; the factor identities follow from that definition. Period duration equality is a stated assumption, not inferred from the entered numbers. Unequal period lengths, endpoint CAGR and financial projections are outside this page.
Handle zeros, signs and numerical limits
A raw zero and a zero percentage have different meanings. Raw observations 0 4 give geometric mean 0. Percentage changes 0% 0% instead give factor 1 and equivalent change 0%. A percentage change of -100% is a zero factor; the check -100% 10% gives mean factor 0 and equivalent change -100%. Changes below that boundary would require unsupported negative factors.
The zero branch is the explicitly declared nonnegative product-root extension. Logarithms are calculated only for positive included values. Some software chooses a stricter positive-only domain; the tool does not silently replace zeros, add a pseudocount or infer a signed/complex geometric mean.
Very small percentage changes can vanish when a rounded factor is recombined around its baseline. The check 1e-100 1e-100 displays mean factor 1 but retains equivalent change 1e-100%. The engine uses log1p and expm1 for that conversion. These results use represented IEEE754 inputs; precision already lost on entry cannot be recovered.
The four-value overflow check 1e100 1e100 1e100 1e100 has geometric mean 1e+100 even though the product exceeds the finite-number range. The corresponding underflow check 1e-100 1e-100 1e-100 1e-100 has geometric mean 1e-100 while the product falls below the finite positive-number range. Details report those product limits explicitly rather than silently present infinity or an underflowed zero as a valid product.
For additive center, use the Mean Median Mode Calculator or the Weighted Average Calculator. The Standard Deviation Calculator measures absolute squared-deviation spread. The Coefficient of Variation Calculator compares SD with a nonzero mean and explains scale limitations. A multiplicative average does not supply those spread summaries.
Local limits are 5,000 observations or weighted rows, 100,000 characters and 128 characters per token, including any percentage suffix. Nonzero entered magnitudes range from 1e-100 to 1e+100. Details show the first 100 rows; all validated positive-weight rows contribute. Simple fractions, decimals and scientific notation are accepted. No spreadsheet upload, automatic deletion, offset-based temperature averaging, volatility approximation or quality verdict is inferred.
Display values use 10 significant digits after calculation. Raw and weighted positive values use centered log ratios and normalized weights; percentage changes use stable small-change functions and a direct near-total-loss factor. The arithmetic comparison uses ordinary count or the same relative-weight basis, as appropriate.
The Harmonic Mean Calculator averages positive values through their reciprocals. For speeds, it separates equal-distance averaging from equal-time arithmetic averaging and accepts relative distance weights.