How do you calculate the geometric mean?

Multiply all n values together and take the nth root of the product. For 1.10, 1.20 and 0.90 that is 1.1 × 1.2 × 0.9 = 1.188, and the cube root of 1.188 is 1.059104500597819, which is exactly what the line under the big figure prints: (1.1 × 1.2 × 0.9)^(1/3) = 1.188^(1/3) = 1.059104500597819. The product is shown as its own step for up to eight values, so the arithmetic can be checked on a pocket calculator. Past eight the working shows the first three values, an ellipsis and the last one, because a product of fifty numbers is not a step anyone can verify by eye.

When should I use the geometric mean instead of the arithmetic mean?

Use it when the numbers multiply rather than add: year-on-year growth multipliers, investment returns, index numbers, price ratios, and any average of quantities measured on different scales. Averaging 10 and 1000 arithmetically gives 505, which sits far closer to the larger number than to the smaller; the geometric mean is 100, the number that is ten times one and one tenth of the other. For positive values the three means always order the same way, and this page shows all three at once: on the twelve numbers it opens with, the harmonic mean is 40.56169143061571, the geometric mean is 45.487558964974305 and the arithmetic mean is 50.

How do I average annual growth rates or percentages?

This geometric-mean page expects positive multipliers: a 10% rise is 1.10 and a 3% fall is 0.97. A percent sign is removed by the numeric list parser, so entering 10% here means the number 10, not the multiplier 1.10. For 1.05, 1.10, 0.97 and 1.08, the geometric mean is about 1.0488040808247006; subtract 1 and multiply by 100 to get about 4.880408% per period. For supplied percentage changes directly, use Growth rates: it converts the observations, shows the arithmetic mean and compound comparison, and distinguishes years from months. For known starting and ending amounts instead of period observations, use the CAGR page in ROI Calculator.

Why does it say the geometric mean is not defined?

This tool uses the strictly positive-input convention for the geometric mean. A negative value is refused even when an even number of negatives makes a positive product with a real root. Under the nonnegative convention a list containing zero has geometric mean zero, but that is not the convention used by this page. Do not drop observations just to get a number. For compound growth, convert a percentage change r to a growth factor 1 + r/100, or choose the observed Growth rates workflow, which performs that conversion and handles a -100% total-loss observation explicitly.

Can I check the answer by hand for two numbers?

Yes, and two numbers is the case worth checking: the geometric mean of a pair is the square root of their product. Paste 4, 9 and the answer is exactly 6, with 36^(1/2) = 6 written under it, while the arithmetic mean of the pair is 6.5. Paste 1.5, 6 and the answer is exactly 3. Very large and very small lists work too, because the values are split into a mantissa and a power of two before the logarithms are averaged, so 1e200, 1e250 and 1e300 return 1e+250 even though their product is far past what a browser can hold. In that case the product step is left out of the working rather than printed as something misleading, and the same happens when the product is so small that it rounds to zero.

What else does this page show, and will it stay on the geometric mean?

Every statistic the average calculator computes is listed underneath on the same paste: mean, median, mode, range, sample and population standard deviation, sample and population variance, the harmonic mean and the sum, each with its own arithmetic written out, and Copy all puts the lot on the clipboard. The Featured result menu holds those nine and moves any one of them to the headline. This page always opens on the geometric mean, even if you last chose a different statistic or last used the weighted mean, because the page you asked for wins. Your last list of numbers does come back, so you can carry on where you left off.

How do I calculate the geometric mean in Excel?

For strictly positive numeric multipliers, use =GEOMEAN(A2:A20). To turn annual percentage observations stored as Excel decimals into multipliers, enter =1+D2 in a helper column and copy down; then =GEOMEAN(E2:E20)-1 gives the compound mean annual rate, formatted as a percentage. This assumes one observation per equal-length year. The browser Growth rates workflow accepts numeric percentages directly and can use equal-length months. Weighted geometric means, symbolic fractions or radicals and triangle theorem inputs are different jobs from this positive numeric list.

Geometric Mean Calculator

Find the geometric mean of your numbers: every value multiplied together, then the nth root of that product. It is the average that fits growth rates, ratios and anything else that compounds, and this page opens the average calculator with it as the headline figure instead of the arithmetic mean. On a first visit the box arrives filled with twelve numbers, 18, 22, 31, 39, 44, 46, 51, 58, 62, 70, 76 and 83, the label reads Geometric mean · 12 values, and the big figure reads 45.487558964974305 with the working under it: (18 × 22 × 31 × … × 83)^(1/12). The arithmetic mean of those same numbers is 50 and the harmonic mean is 40.56169143061571, and both stay listed below, because for positive numbers the geometric mean always sits between the two. Paste your own numbers over the sample. Up to eight values the working prints the product as its own step too, whenever that product is a number a browser can hold, so 1.10, 1.20 and 0.90 read 1.188^(1/3) = 1.059104500597819, and a list holding a zero or a negative value gets an explanation of this tool’s strictly positive-input convention. The median, mode, range, both standard deviations, both variances, the harmonic mean and the sum are all worked out on the same paste.

Geometric mean · 12 values45.487558964974305(18 × 22 × 31 × … × 83)^(1/12) = 45.487558964974305 The geometric mean: the right average for growth rates, ratios and anything that multiplies.

What to work out

Values: 18, 22, 31, 39, 44, 46, 51, 58, 62, 70, 76, 83

Featured result

Sample or population

Count: 12

Commas, spaces, tabs, semicolons and new lines all separate values, so a column pasted straight out of a spreadsheet works. Currency symbols, percent signs and accounting parentheses are understood; anything left over is listed rather than silently dropped.

Meansum ÷ count
50
(18 + 22 + 31 + … + 83) ÷ 12 = 600 ÷ 12 = 50 The arithmetic mean: the one most people mean by “average”.
Medianthe middle of the sorted list
48.5
Sorted: 18, 22, 31, …, 83, an even count (12), so the median is the average of the 6th and 7th values: (46 + 51) ÷ 2 = 48.5
Modeno value repeats
None
All 12 values appear exactly once, so nothing is more common than anything else: there is no mode. No mode is a real answer, not a failure. Adding a repeat would create one.
Rangelargest − smallest
65
Largest − smallest = 83 − 18 = 65
Standard deviation (sample)divides by n − 1 · your choice
20.705730432111608
√(sample variance) = √428.72727272727275 = 20.705730432111608 Divides by n − 1. Use this when your numbers are a sample drawn from something larger.
Variance (sample)divides by n − 1 · your choice
428.72727272727275
Σ(x − mean)² ÷ (n − 1) = 4716 ÷ 11 = 428.72727272727275 Divides by n − 1. Use this when your numbers are a sample drawn from something larger.
Standard deviation (population)divides by n
19.82422760159901
√(population variance) = √393 = 19.82422760159901 Divides by n. Use this when your numbers ARE the whole group.
Variance (population)divides by n
393
Σ(x − mean)² ÷ n = 4716 ÷ 12 = 393 Divides by n. Use this when your numbers ARE the whole group.
Geometric meanfor growth rates and ratios
45.487558964974305
(18 × 22 × 31 × … × 83)^(1/12) = 45.487558964974305 The geometric mean: the right average for growth rates, ratios and anything that multiplies.
Harmonic meanfor speeds and other per-unit rates
40.56169143061571
12 ÷ (1/18 + 1/22 + … + 1/83) = 12 ÷ 0.29584565082861597 = 40.56169143061571 The harmonic mean fits positive rates across equal quantities, such as speeds over equal distances; equal-time speeds instead use the arithmetic mean.
Sumevery value added up
600
18 + 22 + 31 + … + 83 = 600

Sample or population is not a formatting choice. Sample divides by n − 1, population by n, and on eight numbers the two answers differ by about 7%. Use sample when your numbers are a subset of something larger, which is almost always. A tool that shows one of them and calls it “standard deviation” has picked for you without saying so. Both are above, each labelled with the divisor it used.

Common questions

How do you calculate the geometric mean?: Multiply all n values together and take the nth root of the product. For 1.10, 1.20 and 0.90 that is 1.1 × 1.2 × 0.9 = 1.188, and the cube root of 1.188 is 1.059104500597819, which is exactly what the line under the big figure prints: (1.1 × 1.2 × 0.9)^(1/3) = 1.188^(1/3) = 1.059104500597819. The product is shown as its own step for up to eight values, so the arithmetic can be checked on a pocket calculator. Past eight the working shows the first three values, an ellipsis and the last one, because a product of fifty numbers is not a step anyone can verify by eye.
When should I use the geometric mean instead of the arithmetic mean?: Use it when the numbers multiply rather than add: year-on-year growth multipliers, investment returns, index numbers, price ratios, and any average of quantities measured on different scales. Averaging 10 and 1000 arithmetically gives 505, which sits far closer to the larger number than to the smaller; the geometric mean is 100, the number that is ten times one and one tenth of the other. For positive values the three means always order the same way, and this page shows all three at once: on the twelve numbers it opens with, the harmonic mean is 40.56169143061571, the geometric mean is 45.487558964974305 and the arithmetic mean is 50.
How do I average annual growth rates or percentages?: This geometric-mean page expects positive multipliers: a 10% rise is 1.10 and a 3% fall is 0.97. A percent sign is removed by the numeric list parser, so entering 10% here means the number 10, not the multiplier 1.10. For 1.05, 1.10, 0.97 and 1.08, the geometric mean is about 1.0488040808247006; subtract 1 and multiply by 100 to get about 4.880408% per period. For supplied percentage changes directly, use Growth rates: it converts the observations, shows the arithmetic mean and compound comparison, and distinguishes years from months. For known starting and ending amounts instead of period observations, use the CAGR page in ROI Calculator.
Why does it say the geometric mean is not defined?: This tool uses the strictly positive-input convention for the geometric mean. A negative value is refused even when an even number of negatives makes a positive product with a real root. Under the nonnegative convention a list containing zero has geometric mean zero, but that is not the convention used by this page. Do not drop observations just to get a number. For compound growth, convert a percentage change r to a growth factor 1 + r/100, or choose the observed Growth rates workflow, which performs that conversion and handles a -100% total-loss observation explicitly.
Can I check the answer by hand for two numbers?: Yes, and two numbers is the case worth checking: the geometric mean of a pair is the square root of their product. Paste 4, 9 and the answer is exactly 6, with 36^(1/2) = 6 written under it, while the arithmetic mean of the pair is 6.5. Paste 1.5, 6 and the answer is exactly 3. Very large and very small lists work too, because the values are split into a mantissa and a power of two before the logarithms are averaged, so 1e200, 1e250 and 1e300 return 1e+250 even though their product is far past what a browser can hold. In that case the product step is left out of the working rather than printed as something misleading, and the same happens when the product is so small that it rounds to zero.
What else does this page show, and will it stay on the geometric mean?: Every statistic the average calculator computes is listed underneath on the same paste: mean, median, mode, range, sample and population standard deviation, sample and population variance, the harmonic mean and the sum, each with its own arithmetic written out, and Copy all puts the lot on the clipboard. The Featured result menu holds those nine and moves any one of them to the headline. This page always opens on the geometric mean, even if you last chose a different statistic or last used the weighted mean, because the page you asked for wins. Your last list of numbers does come back, so you can carry on where you left off.
How do I calculate the geometric mean in Excel?: For strictly positive numeric multipliers, use =GEOMEAN(A2:A20). To turn annual percentage observations stored as Excel decimals into multipliers, enter =1+D2 in a helper column and copy down; then =GEOMEAN(E2:E20)-1 gives the compound mean annual rate, formatted as a percentage. This assumes one observation per equal-length year. The browser Growth rates workflow accepts numeric percentages directly and can use equal-length months. Weighted geometric means, symbolic fractions or radicals and triangle theorem inputs are different jobs from this positive numeric list.

Uses browser double precision with compensated sums and stable variance calculations. Median positions and mode frequencies are exact for the parsed numbers; arithmetic results can round. Unrepresentable inputs and intermediate results are explained.