Geometric Mean Calculator

StatisticsLast updated: August 22, 2026

The geometric mean is the nth root of the product of n values, and it is the correct average for multiplicative (proportional) processes. Annual returns, inflation, population growth, revenue growth, and index changes accumulate by multiplication rather than addition, which is why the arithmetic mean systematically overstates real performance for such series.

This calculator accepts your values either directly or as percentage changes; when percentages are entered, each rate is converted into the factor 1 + r/100. The output brings together the geometric mean, the average growth rate, a numerical comparison with the arithmetic and harmonic means, the geometric standard deviation, a period-by-period cumulative index table, and charts. To rule out any risk of overflow, the sum of logarithms is used instead of the product.

Write the numbers separated by commas, spaces, or line breaks. If you use a decimal comma, separate the values with spaces (like 1,08 1,12).
Choose percentage change if you are entering annual returns or inflation rates; choose direct value for raw positive figures such as indices, factors, or prices.

Geometric Mean Formulas

Definition:          GM = ⁿ√(x₁ · x₂ · … · xₙ) = (Πxᵢ)^(1/n)
Logarithmic form:    GM = exp[ (1/n) · Σ ln xᵢ ]
Percentage change:   fᵢ = 1 + rᵢ/100   →   r̄ = (ⁿ√(Πfᵢ) − 1) · 100
Compound growth:     r̄ = (Last value / First value)^(1/n) − 1
Geometric std. dev.: GSD = exp( s₍ln x₎ )
Inequality:          HM ≤ GM ≤ x̄   (all xᵢ > 0; equality only when all xᵢ are equal)
Relationship:        GM² ≈ x̄ · HM   (exact for a two-value series)

The result is identical to Excel's GEOMEAN function and to R's exp(mean(log(x))) expression. To prevent overflow, the computation runs through the sum of logarithms instead of the product.

How to Calculate

  1. Paste your values into the box; commas, spaces, and line breaks are all accepted.
  2. Choose the 'Percentage change' type if you are entering annual returns, inflation, or growth rates; use 'Direct value' for indices, prices, or factors.
  3. Make sure your series contains no zero and no percentage below −100; the geometric mean is undefined for those values.
  4. Compare the geometric mean with the arithmetic mean: the gap between them is a direct measure of the volatility of the series.
  5. In percentage mode, look at the cumulative index chart; the constant geometric growth curve and the actual path must end at the same point.
  6. When reporting, use the phrase 'average compound growth per period' and always state the number of periods (n).

Worked Examples

Five-year portfolio return

For a portfolio with annual returns of 12%, −4%, 8%, 20%, and 5%, the factors give 1.12 · 0.96 · 1.08 · 1.20 · 1.05 = 1.4631, so the total five-year return is 46.31%. The geometric mean is 1.0791, which makes the average annual compound return 7.91%. The arithmetic mean gives 8.20% and exaggerates the real accumulation by 0.29 points.

Number of values (n): 5 · Geometric mean (GM): 1.0791 · Average growth rate: 7.91%

A series with an exact root

The product of the three values is 2 · 8 · 32 = 512 and its cube root is exactly 8, so the geometric mean comes out as 8.0000. For the same data the arithmetic mean is 14.0000 and the harmonic mean is 4.5714. The inequality harmonic ≤ geometric ≤ arithmetic is clearly visible, and the gap comes from the values being spread widely between 2 and 32.

Number of values (n): 3 · Geometric mean (GM): 8.0000 · Arithmetic mean (x̄): 14.0000

The compound effect of inflation

With three consecutive years of 64%, 44%, and 65% inflation, prices rise by a factor of 1.64 · 1.44 · 1.65 = 3.8966, that is, a total increase of 289.66%. The geometric mean is 1.5736, so the average annual inflation is 57.36%; the arithmetic mean stays slightly above that at 57.67%.

Number of values (n): 3 · Geometric mean (GM): 1.5736 · Average growth rate: 57.36%

Frequently Asked Questions

When is the geometric mean used instead of the arithmetic mean?
Use the geometric mean whenever values accumulate by multiplication rather than addition: annual returns, inflation, population growth, revenue growth, bacterial growth, and index changes all belong to this group. For such series the arithmetic mean always comes out larger than the real growth. For additive measurements such as height, weight, or exam scores, the arithmetic mean is the right choice.
What should I do if there are negative or zero values?
By definition, the geometric mean works only with positive values: a single zero collapses the whole product, and negative values leave the root undefined. For return data, the solution is to convert the percentages into the factors 1 + r/100; every rate above −100% produces a positive factor. For measurements that can genuinely be negative (profit and loss amounts, temperature differences), the geometric mean is not appropriate.
Why is there a gap between the arithmetic and the geometric mean?
The gap comes from the volatility of the series and is roughly half the variance (GM ≈ x̄ − s²/2x̄). The closer the values are to each other, the more the two means coincide; as volatility grows, the geometric mean falls further behind. This is why an investment that gains 50% and then loses 50% looks break-even on an arithmetic basis but actually loses 25%.
Are CAGR and the geometric mean the same thing?
Yes: the compound annual growth rate (CAGR) is obtained by subtracting 1 from the geometric mean of the period factors. An equivalent route uses only the first and last values and computes (Last/First)^(1/n) − 1; when all intermediate periods are recorded, the two methods agree. The only requirement is that the periods be of equal length.
How is it computed in Excel, SPSS, and R?
In Excel the GEOMEAN function returns the geometric mean directly. R has no built-in function for it; writing exp(mean(log(x))) is the standard approach. In SPSS you can select the Geometric Mean statistic under Analyze > Compare Means > Means. All three produce the same value as this calculator.
How is the geometric standard deviation interpreted?
The geometric standard deviation is read as a factor, not as a unit. GSD = 1 means all values are equal; a value such as 1.5 says that the data typically fluctuates between 1.5 times and 1/1.5 times the geometric mean. For log-normally distributed quantities (income, particle size, biological measurements) it is a more meaningful measure of spread than the standard deviation.
Will the computation overflow if I enter many values?
No. This calculator never takes the product directly; it sums the logarithms and then exponentiates. That way, even with hundreds of values there is neither overflow (Infinity) nor underflow to zero for very small products. The result is mathematically identical to the defining formula.