Geometric Mean Calculator

What is the geometric mean?

The geometric mean is a kind of average built from multiplication instead of addition. Instead of adding the numbers and dividing by how many there are, you multiply them together and take the root that matches the count: the square root for two numbers, the cube root for three, and so on.

It answers the question "what single value, used in every position, would give the same product?" That makes it the natural average for quantities that combine by multiplying: growth rates, ratios, index numbers and anything measured on a scale where percentage differences matter more than absolute ones.

For two numbers the idea is easy to picture. A rectangle with sides 2 and 8 has the same area as a square with side 4, because 2 × 8 = 16 = 4 × 4. So the geometric mean of 2 and 8 is 4.

Geometric mean formula

GM = (x1 × x2 × … × xn) ^ (1 / n)Multiply n positive numbers together and raise the product to the power 1/n.

Multiplying many numbers can overflow a computer or lose precision, so the calculation is usually done with logarithms instead. The two forms are mathematically identical:

GM = exp( (ln x1 + ln x2 + … + ln xn) / n )

This calculator uses the logarithm form. In words: take the natural logarithm of every number, average those logarithms in the ordinary way, then convert the average back with the exponential function.

How to calculate it, step by step

1. Count the numbers. Call the count n.
2. Multiply all the numbers together.
3. Take the n-th root of the product (raise it to the power 1/n). A calculator for any root or the exponent calculator can do this step.

Example 1. For 4 and 9 the product is 36 and n = 2, so GM = √36 = 6.

Example 2. For 2, 4 and 8 the product is 64 and n = 3, so GM = the cube root of 64 = 4. Notice that 4 is also the middle value of this evenly scaling sequence, which is typical for a geometric progression.

Example 3. For 1, 2, 3, 4 and 5 the product is 120 and n = 5, so GM = 120^1/5 = 2.605171084697. The ordinary average of the same list is 3, a little higher.

Using the calculator

Type or paste your numbers into the box, separated by commas, spaces, tabs or new lines, and press Calculate. Use a dot for decimals. Entries that are not numbers are skipped.

The answer is rounded to 12 decimal places. Whole numbers show without decimals, results with up to two decimals show two, and longer results show 6, 10 or 12 digits, with thousands separators for large values. A zero anywhere in a list of non-negative numbers makes the geometric mean 0, because it makes the product 0.

Geometric mean versus the arithmetic and harmonic means

For any set of positive numbers the three classical means are ordered: harmonic mean ≤ geometric mean ≤ arithmetic mean. They are all equal only when every number in the list is the same.

For 2 and 8 the arithmetic mean is 5, the geometric mean is 4 and the harmonic mean is 3.2. Compare them with the mean calculator and the harmonic mean calculator.

The gap matters when you average ratios. Suppose machine A is twice as fast as machine B on one test and half as fast on another. The ratios are 2 and 0.5. Their arithmetic mean is 1.25, which wrongly suggests A is 25% faster overall. Their geometric mean is 1, which correctly says the two machines are even. For this reason the geometric mean is the standard way to summarize normalized benchmark scores.

Averaging growth rates and returns

Percentage changes compound by multiplication, so the correct average of several periods is a geometric one. Convert each change to a growth factor (a 20% rise is 1.20, a 10% fall is 0.90), take the geometric mean of the factors, then convert back.

average change = ((1 + r1) × (1 + r2) × … × (1 + rn)) ^ (1 / n) − 1Each r is a percentage written as a decimal.

Negative values. If any number in your list is negative, the calculator reads every number as a percentage change: 10 means +10% and −10 means −10%. The result is then an average percentage per period and a note is shown under it. A fall of 100% or more cannot be turned into a growth factor, so the calculator reports an error instead.

Example. An account earns +20%, then −10%, then +15%. The factors are 1.20, 0.90 and 1.15, whose product is 1.242, and the cube root of 1.242 is 1.0749. Enter 20, -10, 15 and the calculator shows 7.491436130478% per period. The plain average, (20 − 10 + 15) ÷ 3 = 8.33%, overstates the result: 10,000 grows to 12,420 over the three periods, which matches 7.49% a period, whereas 8.33% a period would give 12,714.

Growth between two values. If a value rises from 10,000 to 16,000 in 4 years, the steady yearly factor is the fourth root of 1.6, about 1.1247, an average gain of 12.47% a year. The CAGR calculator is built for exactly this question, and the compound interest calculator runs it in the other direction.

The finance figures here only illustrate the arithmetic. They are not investment advice and past returns do not predict future ones. See the disclaimer.

Zeros and negative numbers

The geometric mean is defined for positive numbers. A zero makes the whole product zero, so the geometric mean of 3, 0 and 9 is 0. That is correct but often unhelpful: one missing or zero value wipes out the information in all the others. If the zero means "no data", remove it before averaging. If it is a real measurement, consider adding a small constant to every value, or use another average.

Negative numbers are a problem in plain data, because a product with an odd number of negative factors is negative and has no real even root. This is why the calculator switches to the percentage reading described above whenever it sees a minus sign. If your negative numbers are not percentage changes, the geometric mean is probably not the right tool for the data.

Where the geometric mean is used

• Finance: average annual return, compound growth rates and inflation over several years.
• Statistics and economics: price indices, and composite indicators that combine several dimensions. The United Nations Human Development Index, for example, combines its health, education and income indices with a geometric mean.
• Biology and public health: bacteria counts and antibody titers, which vary over orders of magnitude and are summarized on a logarithmic scale.
• Computing: averaging speed-up ratios across benchmark programs.
• Everyday ratios: aspect ratios, scale factors and exchange-rate movements.

For other kinds of average, see the median calculator, the mode calculator and the weighted mean calculator.