Geometric Mean Calculator

What is the geometric mean?

The geometric mean is the average for numbers that multiply. Where the ordinary (arithmetic) mean asks which single number, added n times, gives the same total, the geometric mean asks which single number, multiplied n times, gives the same product. For 2, 8 and 32 the product is 512, and 8 × 8 × 8 is also 512, so the geometric mean is 8.

It is also the typical ratio. In 2, 8, 32 each number is 4 times the one before, and the geometric mean sits at the same ratio from both ends: 8 ÷ 2 = 32 ÷ 8. That is why it is the natural center for growth rates, price changes, ratios and data that spreads over several orders of magnitude. The arithmetic mean of the same list, 14, is pulled toward the largest value.

The geometric mean formula

The first line is the definition: multiply, then take the nth root, where n is how many numbers there are. The second line says the same thing with logarithms: the log of the geometric mean is the ordinary average of the logs. It works with any base, as long as you undo the log with the same base at the end.

How to calculate the geometric mean

Two numbers

Multiply them and take the square root. For 14 and 20, 14 × 20 = 280. Since 280 = 4 × 70, the square root simplifies to 2√70, which is about 16.7332. Geometry classes usually want the simplified root; the calculator gives both.

Show the geometric mean of 14 and 20

Three or more numbers

Multiply all of them and take the root that matches the count: the cube root for three numbers, the fourth root for four, and so on. On a calculator, raise the product to the power 1/n. For 2, 3 and 36, the product is 216 and ∛216 = 6.

Show 2, 3 and 36 in the calculator

Long lists: average the logarithms

The product of many numbers quickly gets too large or too small for a calculator to hold. Take the log of each number instead, average the logs, and raise 10 to that average. For 2, 8 and 32: log 2 ≈ 0.301, log 8 ≈ 0.903 and log 32 ≈ 1.505; their average is 0.903, and 10^0.903 ≈ 8. The calculator shows this route under every answer.

Geometric mean of two numbers: examples

The geometric mean of a and b is √(ab). It is a whole number when ab is a perfect square; otherwise it is a root, shown here in simplest form with its decimal. Each pair opens in the calculator.

The geometric mean theorem (right triangles)

In a right triangle, draw the altitude from the right angle to the hypotenuse. It cuts the hypotenuse into two pieces, p and q, and the altitude is their geometric mean:

If the pieces are 4 and 9, the altitude is √36 = 6, the hypotenuse is 13, and the legs are √52 = 2√13 ≈ 7.21 and √117 = 3√13 ≈ 10.82. A check with Pythagoras: 52 + 117 = 169 = 13². The calculator draws this for any two numbers: a semicircle over p + q, where the height at the join is the geometric mean and the radius is the arithmetic mean. Because the radius is the tallest line that fits, the geometric mean can never beat the arithmetic mean.

Draw the semicircle for 4 and 9

Average growth rates and investment returns

Percentage changes compound, so they have to be averaged with the geometric mean. Say an investment gains 50% one year and loses 50% the next. The simple average is 0%, yet 100 became 150 and then 75. Turn each change into a growth factor (1.5 and 0.5), take their geometric mean, √0.75 ≈ 0.866, and subtract 1: the true average is about -13.4% a year. Two years at that rate also turn 100 into 75.

Average +50% and −50%

The gap between the two averages grows with the swings. For +10%, −20% and +30%, the simple average is 6.67% but the geometric average is only about 4.59%. This geometric average is what funds report as an annualized return. When you only know the start and end values, the same idea gives the compound annual growth rate (CAGR): (end ÷ start)^(1/years) − 1. Growing from 400 to 750 in 5 years is about 13.4% a year.

Find the CAGR from 400 to 750

Weighted geometric mean

When some values count more than others, or come from a frequency table, raise each value to its weight before taking the root of the total weight:

A growth factor of 1.05 that happened 3 times, 1.10 twice and 0.95 once has a weighted geometric mean of about 1.0488. Counts work as weights, so this is also how to find the geometric mean of grouped data.

Show the weighted example

Geometric vs arithmetic vs harmonic mean

For positive numbers the three classic means always come in the same order, AM ≥ GM ≥ HM, and they are equal only when every number is the same. Which one to use depends on how the numbers combine.

For two numbers the three are linked: GM² = AM × HM, so 6² = 36 = 6.5 × 5.5385. Our average calculator and harmonic mean calculator cover the other two.

Zeros and negative numbers

A single 0 makes the product 0, so the geometric mean is 0 however large the other numbers are. When a 0 really means “below what we could measure”, replace it with a small value first; the log method cannot handle a 0 at all. Negative numbers have no geometric mean, because a root of a negative product has no single real value. Losses are still fine as growth rates: −20% is the factor 0.8, which is positive.

Geometric mean in Excel and Google Sheets

GEOMEAN takes positive numbers and returns an error for a 0 or a negative value. The second line is the weighted version, with values in column A and weights in column B. RRI(periods, start, end) returns the CAGR. For growth rates in percent, put the growth factors (1 + rate) in the cells and subtract 1 from GEOMEAN. The calculator writes the matching formula under each answer.