Geometric Mean Calculator

About the Geometric Mean Calculator

The everyday average — add the numbers and divide by how many there are — is the arithmetic mean. It is the right choice for many things, but not for quantities that multiply rather than add: investment returns, population growth, price changes, ratios and data that spans several orders of magnitude. For those, the geometric mean gives the typical value, and using the arithmetic mean instead can overstate growth badly.

This geometric mean calculator works in two ways. For positive values, it finds their geometric mean and compares it with the arithmetic mean. For growth rates given as percentages, it finds the true average rate per period — the steady rate that would produce the same overall change — and shows how far the simple average of the rates would mislead.

How to Use the Geometric Mean Calculator

Choose Positive values or Growth rates (%).

For values, enter two or more positive numbers.

For growth rates, enter each period's percentage change, such as 10, −20, 30 for a 10% rise, a 20% fall and a 30% rise.

The geometric mean appears with the working and a comparison.

The Formula

  geometric mean = (x₁ × x₂ × … × xₙ)^(1/n)
                 = e^( (ln x₁ + ln x₂ + … + ln xₙ) ÷ n )

  average growth rate = ((1 + r₁)(1 + r₂)…(1 + rₙ))^(1/n) − 1

The calculator works through logarithms, which gives the same answer but avoids overflow when multiplying many large numbers.

Step-by-Step Example: Values

The numbers 2 and 8.

  Product:  2 × 8 = 16
  Root:     √16 = 4

The geometric mean is 4, while the arithmetic mean is 5.

The numbers 1, 3, 9 and 27.

  Product:  1 × 3 × 9 × 27 = 729
  Root:     729^(1/4) = 5.19615

The geometric mean is about 5.2, much lower than the arithmetic mean of 10, which is dragged up by the 27.

Step-by-Step Example: Growth Rates

Returns of +10%, −20% and +30% over three years.

  Factors:        1.10 × 0.80 × 1.30 = 1.144
  Mean factor:    1.144^(1/3) = 1.04586
  Average rate:   1.04586 − 1 = 4.586% a year

The investment grew by 14.4% overall, which is the same as a steady 4.586% a year. The simple average of the three rates, 6.67%, overstates the real growth: 6.67% a year for three years would give 21.4%, not 14.4%.

Why the Simple Average Misleads for Growth

Percentage changes act on different starting amounts. A 50% loss followed by a 50% gain takes 100 to 50 and then to 75 — a 25% loss overall — even though the two rates average to zero. The geometric mean handles this correctly: the factors 0.5 and 1.5 multiply to 0.75, and the square root is 0.866, an average of −13.4% a period. This is why fund managers and economists report compound annual growth rates rather than simple averages of yearly returns.

Properties of the Geometric Mean

For any set of positive numbers, the geometric mean is never larger than the arithmetic mean, and the two are equal only when every value is the same. The gap between them grows as the values become more spread out. The geometric mean is unaffected by the units you choose in one important way: multiplying every value by the same number multiplies the geometric mean by that number too.

Where It Is Used

It appears in more places than most people expect.

The geometric mean is used for average investment and economic growth rates, for index numbers and price comparisons, for averaging ratios such as speed-ups or exchange rates, and for data spread over many orders of magnitude, such as bacterial counts, sound intensities and earthquake energies. It is also used to combine scores measured on different scales, since it treats proportional changes equally.

Geometric Mean and Logarithms

Taking the geometric mean is the same as averaging on a logarithmic scale. The calculator finds the average of the natural logarithms of the values and then converts back. For 2 and 8, ln 2 is 0.693 and ln 8 is 2.079; their average is 1.386, and e to the power 1.386 is 4. This view explains why the geometric mean suits data that grows by multiplying: on a log scale, equal ratios become equal steps, so the geometric mean treats a doubling and a halving as equal and opposite changes. It is also why scientists often plot such data on logarithmic axes before summarising it.

Understanding Your Result

The headline is the geometric mean, or the average growth rate per period.

For values, the arithmetic mean, product and root and check lines compare the result with the ordinary average and show the calculation.

For growth rates, the overall change, simple average of rates and check lines show the total growth and why the simple average differs.

When Should You Use This Calculator?

Use it to find the true average return of an investment over several years.

Use it to average growth rates of sales, prices or populations.

Use it to average ratios and index numbers.

Use it for data that spans very different scales.

Common Mistakes

Averaging percentage returns directly. Use the geometric mean of the growth factors.

Including zero or negative values. Ordinary values must be positive.

Forgetting to add 1 to each rate. Work with factors such as 1.10 and 0.80.

Forgetting to subtract 1 at the end. The mean factor 1.046 means 4.6% growth.

Using it for data that adds. For sums and totals, the arithmetic mean is right.

Comparing it directly with an arithmetic mean. The geometric mean is always the smaller of the two, so a gap between them is expected, not an error.