Geometric Mean Calculator

Geometric mean = ⁿ√(x₁ · x₂ · … · xₙ) = exp(mean(ln xᵢ))

xᵢ

One value in the list, and every one of them has to be greater than zero. A zero makes the product zero and the answer zero whatever the rest of the list says; a negative value makes an even root of a negative product, which is not a number at all

n

How many values there are, printed on the results panel. It is the exponent of the root: two values take a square root, three a cube root, five a fifth root. Counting it wrongly is the usual way a hand calculation of this goes wrong, so it is shown rather than left to be counted off the list

x₁ · x₂ · … · xₙ

The product of every value. Only the product matters, which is why the answer does not depend on the order the values were written in — and why a list of two values is the familiar square root of their product

mean(ln xᵢ)

The other way to write the same thing, and the one a calculator actually uses: take the natural logarithm of each value, average those, then undo the logarithm. It gives the same answer as multiplying first, without the overflow that multiplying a long list of large values runs into

Use the geometric mean when the numbers multiply rather than add up — percentages that compound, ratios, index levels, growth factors. A fund that gains 10%, gains 20% and loses 10% ends the three years up 18.8%, which is 1.1 × 1.2 × 0.9; the average yearly growth factor is the cube root of that, 1.0591, so the honest annual figure is 5.91% and not the 6.67% a plain average of the three percentages gives. The same reasoning covers anything measured on a multiplicative scale: bacterial counts, earthquake magnitudes, the sizes of a set of items that spans several orders of magnitude, and a set of ratios such as price-to-earnings figures where one extreme value would otherwise dominate a plain average. Do not use it for values that can add up — a list of temperatures, marks or prices has no meaningful product, and its geometric mean would be a number with no interpretation. The one comparison worth remembering is that the geometric mean is always the lower of the two, and by more the more the values differ, so the gap between them is a rough measure of how uneven the list is.

1. Five test scores

   1. Multiply the values: 85 × 90 × 78 × 92 × 88 = 4,830,883,200
   2. There are 5 values, so take the fifth root: ⁵√4,830,883,200 = 86.458
   3. The arithmetic mean of the same list is 86.6

   The two answers differ by 0.142, which is small because these five scores are close together — the geometric mean falls below the arithmetic mean by an amount that grows with how uneven the list is. That inequality is not a rounding artefact: for positive values the geometric mean can never exceed the arithmetic mean, and it matches it only when every value is identical. Scores are not really a multiplicative quantity, so this example is here to show the mechanism rather than a natural use of it; the fund returns in the next example are what the tool is for.

2. A fund up 10%, up 20%, down 10%

   1. Multiply the growth factors: 1.1 × 1.2 × 0.9 = 1.188
   2. There are 3 factors, so take the cube root: ³√1.188 = 1.0591
   3. The arithmetic mean is 1.0667 — higher, and wrong for this question

   The three years together are up 18.8%, and 1.188 is the only number that says so. The average yearly factor is what multiplies to 1.188 over three years, and that is the cube root, 1.0591 — a compound annual growth rate of 5.91%. A plain average of the three percentages gives 6.67%, and the reason it is too high is that percentages do not add: a 10% fall undoes more than a 10% rise, because it is taken off a larger base. The gap here is 0.76 of a percentage point a year, which over twenty years is the difference between a portfolio tripling and one multiplying by 3.6.

3. Two numbers: the mean proportional

   1. Multiply the two values: 2 × 8 = 16
   2. There are 2 values, so take the square root: √16 = 4
   3. The arithmetic mean of the same pair is 5

   With two values the geometric mean is the square root of their product, which is the number the ancient geometers called the mean proportional: 2 is to 4 as 4 is to 8. It is also the side of the square that has the same area as the 2-by-8 rectangle — 16 either way — where the arithmetic mean of 5 is the side of the square with the same perimeter, a 5-by-5 square whose area is 25. This is the clearest case of the two averages disagreeing, and it shows why the geometric one is the right answer to a question about areas, scales or growth, and the arithmetic one to a question about totals.

Every value has to be greater than zero, and the page refuses a list containing a zero or a negative rather than returning something meaningless — a single zero makes the product zero and the answer zero no matter what the rest of the list holds, and an even root of a negative product is not a real number. For a list with an odd count of negative values there is a real answer in principle, but the sign of a geometric mean built from negatives carries no interpretation, so those lists are refused as well, and the fix is to work with the growth factors rather than the percentage changes. This mean is only meaningful for quantities that multiply; for marks, temperatures or prices it is a legitimate calculation with no useful reading, which is why the arithmetic mean is printed beside it rather than replaced. It cannot be used when some values are zero, which rules out a set of returns that includes a total loss. The list is capped at 200 values, and a token like 1,500 is refused rather than guessed at, because a comma between digits is a decimal comma in much of the world; write 1500 or 1.5.

How do I calculate the geometric mean?

Multiply all the values together, then take the root whose index is how many values there are. For 2 and 8 the product is 16 and there are two values, so the geometric mean is √16 = 4; for five values you would take the fifth root. A calculator does the same thing through logarithms, averaging the logs and undoing the log at the end, which avoids the enormous intermediate product.

Why is the geometric mean lower than the arithmetic mean?

Because multiplication treats a fall and a rise asymmetrically. A value that drops 10% and then rises 10% ends below where it started, of 1.1 × 0.9 = 0.99, so any list that moves both ways will have a geometric mean below its arithmetic mean. The two agree only when every value is the same, and the further apart the values are the wider the gap — which is why the difference between the two numbers is itself a rough reading of how uneven the list is.

When should I use the geometric mean rather than the arithmetic one?

When the values multiply instead of adding up. Growth rates, percentages that compound year on year, ratios and index levels are all multiplicative, and averaging them with a plain mean overstates the answer. Amounts that add — marks, temperatures, prices, counts of things — are additive, and the arithmetic mean is the right one for those. A useful test is whether doubling every value should double the answer: if it should, the quantity is multiplicative.

Can I use it on a list that contains a zero or a negative number?

No, and the page refuses it. A single zero drives the product to zero, so the answer would be zero whatever the rest of the list says, which is not an average of anything. A negative value gives a negative product, and an even root of that is not a real number. The practical fix for a set of percentage changes that includes a total loss is to work with the growth factors — 0.5 for a 50% loss — and to accept that a factor of zero, a total loss, has no geometric mean at all.

How is this different from a compound annual growth rate?

The compound annual growth rate is the geometric mean of the yearly growth factors, expressed as a percentage. If an investment multiplies by 1.1, 1.2 and 0.9 over three years, the product is 1.188, the cube root is 1.0591, and the compound annual growth is 5.91%. Everything in the calculator is already in that form: enter the factors rather than the percentages, and the number the panel shows is the growth factor whose percentage is the rate.

Does the order of the values matter?

No. The geometric mean is built from the product and the count, and multiplication does not care about order, so 2, 8 and 4 gives the same answer as 8, 4 and 2. That is worth knowing because the geometric mean often describes a sequence in time — years of returns — and the answer it gives is deliberately blind to which good year came first. It answers what the average year looked like, not what happened in any particular year.

• Geometric Mean — MathWorld (the definition, the mean proportional, and the inequality with the arithmetic mean) — Wolfram MathWorld
• Geometric Mean — Dataplot Reference Manual, auxiliary chapters (why rates and ratios call for the geometric mean rather than the arithmetic one) — National Institute of Standards and Technology (NIST)