Geometric Mean Calculator

About Geometric Mean Calculator

Geometric mean is useful for rates of change and ratios. It's always ≤ the arithmetic mean (AM-GM inequality).

How to use this calculator

1. Enter the positive numbers you want to compare.
2. Count how many numbers are in the set, this is the value of \(n\).
3. Use the calculator to multiply all the numbers, then take the \(n\)th root of that product.
4. Read the geometric mean, which tells you the multiplicative average of the set.

The formula explained

The formula \(G = \sqrt[n]{x_1 \times x_2 \times \cdots \times x_n}\) computes the geometric mean, where \(G\) is the result and \(x_1, x_2, \ldots, x_n\) are the positive numbers in the set. It multiplies all the values together and then takes the \(n\)th root of that product.

• G = the geometric mean
• n = the number of values in the set
• \(x_1, x_2, \ldots, x_n\) = the positive numbers being averaged

Step by step method

1. Multiply all the numbers together to get the product.
2. Find how many numbers are in the set, then take the \(n\)th root of the product.
3. Check that all values are positive, because the geometric mean is defined for positive numbers in this calculator.

Worked example

Problem. Find the geometric mean of \(4\), \(9\), and \(16\).

1. First multiply the numbers, \(4 \times 9 \times 16 = 576\).
2. There are \(3\) numbers, so take the cube root of \(576\), which gives \(\sqrt[3]{576} = 8.32\) approximately.
3. So the geometric mean is about \(8.32\).

Tips and common mistakes

• Make sure every number is positive, because zero or negative values are not valid for this calculator.
• Do not add the numbers first, because the geometric mean uses multiplication, not addition.

Frequently asked questions