Geometric Mean Calculator NZ

This calculator finds the geometric mean of a set of positive numbers, the correct average to use for ratios, rates of change and anything that grows by multiplication. Most people reach for the ordinary, arithmetic mean by adding the values and dividing, but that gives the wrong answer for quantities that compound. The geometric mean instead multiplies all the values together and takes the root according to how many there are, which properly captures multiplicative growth. It is the right tool for averaging investment returns over several years, population or revenue growth rates, ratios and index numbers, and any situation where each value builds on the last. A classic example: if an investment grows 10 percent one year and falls 10 percent the next, the arithmetic mean suggests no change, but the geometric mean correctly shows a small loss, because the percentages compound rather than add. This tool makes it easy. You paste or type your numbers, which must be positive, and the calculator returns the geometric mean, along with the arithmetic mean for comparison and the count of values. The geometric mean is always less than or equal to the arithmetic mean, and the gap between them grows as the numbers become more spread out. The results update as you type. Use it to find an average growth rate, to average ratios correctly, or for statistics work where the geometric mean is required. When averaging growth, enter each period as a multiplier, so 5 percent growth is 1.05, and the geometric mean of those multipliers gives the average growth factor per period. The calculation uses logarithms internally to stay accurate for long lists.

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4
geometric mean
Arithmetic mean4.667
Count3
GM / AM0.857

Geometric mean = the nth root of the product of n values. All values must be positive. It is always less than or equal to the arithmetic mean.

How it works

The geometric mean multiplies all the values together and takes the root matching how many there are. To stay accurate for long lists, the calculator works through logarithms: it averages the natural logs of the values and raises e to that average, which gives the same result without the risk of the product overflowing.

Worked example

For 2, 4 and 8, the product is 64, and the cube root of 64 is 4, so the geometric mean is 4. The arithmetic mean of the same numbers is 14 divided by 3, about 4.667. The geometric mean is lower, as it always is, and is the right average when these represent multiplicative factors.

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Frequently asked questions

When should I use the geometric mean?

Use it for quantities that compound or are multiplicative: investment returns over time, growth rates, ratios and index numbers. The arithmetic mean overstates the average in these cases, while the geometric mean correctly accounts for the way each period builds on the last.

Why must the numbers be positive?

The geometric mean involves multiplying the values and taking a root, which is not defined for negative numbers or zero in the usual real sense. For growth rates that can be negative, enter each period as a multiplier instead, so a 10 percent fall becomes 0.90.

Why is the geometric mean lower than the arithmetic mean?

For any set of positive numbers that are not all equal, the geometric mean is always less than the arithmetic mean, a result known as the AM-GM inequality. The more spread out the values, the larger the gap. They are equal only when every value is the same.