Calculate the geometric mean of a set of positive numbers using the log-sum method, with arithmetic mean for comparison.
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8
Geometric mean
12.4
Arithmetic mean
5
Count
| Geometric mean | 8 |
| Arithmetic mean | 12.4 |
| Minimum | 2 |
| Maximum | 32 |
| Count | 5 |
Also in Statistics
Means & Averages
The geometric mean multiplies n values together and takes the nth root of the product. Unlike the arithmetic mean, it is the appropriate average when your data represents multiplicative quantities — growth rates, investment returns, ratios, or any measurement that compounds over successive periods.
Given n positive values x₁, x₂, …, xₙ, the geometric mean is GM = (x₁ × x₂ × … × xₙ)^(1/n). For two values it is simply the square root of their product: GM(2, 8) = √(2 × 8) = √16 = 4.
The geometric mean is always less than or equal to the arithmetic mean (AM–GM inequality), with equality only when all values are identical. This property ensures the geometric mean never overstates the central value of a skewed multiplicative dataset.
Suppose an investment grows by 50% one year (+1.5×) and shrinks by 33% the next (×0.67). The arithmetic mean return is (1.5 + 0.67) / 2 ≈ 1.085, suggesting ~8.5% average growth per year. But $100 → $150 → $100.50 is barely breakeven. The geometric mean gives GM(1.5, 0.67) ≈ 1.001, or about 0.1% per year — the correct compound annual growth rate (CAGR).
Whenever the question is "what single rate, applied repeatedly, produces the same end result?", use the geometric mean. CAGR, average index returns, average population growth rates, and average inflation rates all use the geometric mean.
The geometric mean is also the correct average for ratios. If one lab reports a concentration 10× baseline and another reports 0.1× baseline, the arithmetic mean is (10 + 0.1) / 2 = 5.05×, which is clearly biased toward the high outlier. The geometric mean is √(10 × 0.1) = √1 = 1×, correctly identifying a neutral centre. This is why many index-number calculations, including the Consumer Price Index and geometric mean price indices, use the geometric mean.
Direct multiplication of many numbers can overflow floating-point precision. This calculator uses the log-sum method: GM = exp(mean(ln(xᵢ))). The logarithm converts multiplication to addition, making it numerically stable even for very large or very small values.
All values must be strictly positive — zero and negative values are undefined for the geometric mean because the logarithm is only defined for positive numbers.
Frequently asked questions
The logarithm of zero is undefined (negative infinity), and the logarithm of a negative number is not a real number. Even without the log method, multiplying an even number of negative values yields a positive product that has no meaningful relationship to the original data. The geometric mean is defined only for positive values.
Use the geometric mean when your values are growth factors (returns, rates, ratios) or when the values span several orders of magnitude and you want a "typical" value that is not dominated by large outliers. Use the arithmetic mean when values are additive — sums of items, average counts, temperatures.
CAGR (Compound Annual Growth Rate) is the geometric mean of annual growth factors. For example, if an investment grows by factors r₁, r₂, …, rₙ over n years, the CAGR = GM(r₁, r₂, …, rₙ) − 1. It is the single constant rate that would produce the same total return.
No. The median is the middle value when data is sorted. The geometric mean is a calculated value based on the product of all values. They can be similar for lognormal distributions but are conceptually and numerically distinct.
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