Geometric Mean Calculator
This page opens the average calculator with the geometric mean as its headline figure instead of the arithmetic mean: every value multiplied together, then the nth root of that product, which is the average that fits growth rates, ratios and anything else that compounds. 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 the reason the geometric mean is undefined there rather than a figure that would mean nothing. The median, mode, range, both standard deviations, both variances, the harmonic mean and the sum are all worked out on the same paste.
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.
(18 + 22 + 31 + … + 83) ÷ 12 = 600 ÷ 12 = 50 The arithmetic mean: the one most people mean by “average”.
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
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.
Largest − smallest = 83 − 18 = 65
√(sample variance) = √428.72727272727275 = 20.705730432111608 Divides by n − 1. Use this when your numbers are a sample drawn from something larger.
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.
√(population variance) = √393 = 19.82422760159901 Divides by n. Use this when your numbers ARE the whole group.
Σ(x − mean)² ÷ n = 4716 ÷ 12 = 393 Divides by n. Use this when your numbers ARE the whole group.
(18 × 22 × 31 × … × 83)^(1/12) = 45.487558964974305 The geometric mean: the right average for growth rates, ratios and anything that multiplies.
12 ÷ (1/18 + 1/22 + … + 1/83) = 12 ÷ 0.29584565082861597 = 40.56169143061571 The harmonic mean: the right average for speeds, rates and other per-unit quantities.
18 + 22 + 31 + … + 83 = 600
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?
- Type them as multipliers, not as percentages: a 10% rise is 1.10, a 3% fall is 0.97. A percent sign is read off and ignored, so 10% goes in as the number 10, and a list with a fall written as -10 has a negative value in it, which leaves the geometric mean with no real answer. The tool gives you the average multiplier, and turning that back into a rate is one subtraction you do yourself: 1.05, 1.10, 0.97 and 1.08 give a geometric mean of 1.0488040808247006, so the average growth is about 4.88% a period. The arithmetic mean of those same four multipliers is 1.05, and compounding 1.05 four times overshoots the actual total, which is the mistake the geometric mean exists to avoid.
- Why does it say the geometric mean is not defined?
- Because a zero or a negative value is in the list. A single zero drags the product to zero no matter what the other numbers are, so the tool says so and suggests dropping the zero or quoting the arithmetic mean. A negative value means the nth root of the product has no real answer, and the message names the root it would have taken, counts the negatives and adds that for data which goes negative, such as profits and losses or temperature changes, the arithmetic mean is the honest one to quote. Either way the row prints the reason rather than a number, and the mean, median and the rest of the statistics on the page are still computed.
- 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.
Exact for median, mode, minimum, maximum and range. The mean, variances and standard deviations use compensated summation and are accurate to roughly 15 significant digits, not claimed as exact, because double-precision arithmetic is not.