Geometric vs Arithmetic Mean: Which to Use

How each mean is computed

The arithmetic mean of n numbers is their sum divided by n. For 4, 9, and 17 it is (4 + 9 + 17) / 3 = 10. The geometric mean multiplies all n numbers and takes the nth root: for 4, 9, and 17 it is the cube root of 4 x 9 x 17 = the cube root of 612, about 8.49. The two agree only when every value is identical.

The geometric mean is always less than or equal to the arithmetic mean for positive numbers. The bigger the spread between values, the wider the gap. This is a known inequality and a quick sanity check: if your geometric mean comes out larger than the arithmetic one, you made an arithmetic slip.

Why growth rates need the geometric mean

Suppose an investment grows 50% one year and falls 50% the next. The arithmetic mean of +50% and -50% is 0%, suggesting you broke even. But 100 becomes 150, then 75 — a real loss. The geometric mean of the growth factors 1.5 and 0.5 is the square root of 0.75, about 0.866, meaning roughly -13.4% per year, which matches the actual outcome.

A quick rule of thumb

Use the arithmetic mean when the values add up in a meaningful way — test scores, heights, temperatures. Use the geometric mean when they multiply or compound — annual returns, population growth, index ratios, or scale factors. If your data are percentages of change or the answer must respect ratios, reach for the geometric mean.

Frequently asked questions

Can the geometric mean handle negative numbers?

Not directly — a product with negatives can turn negative or make the root undefined. That is why growth is expressed as factors (1.05 for +5%) rather than raw percentages, keeping every value positive.

Is the geometric mean the same as the median?

No. The median is the middle value when data are sorted; the geometric mean is a computed product-and-root. They can be close for smooth data but measure different things and often differ.

Why is the geometric mean lower than the arithmetic mean?

Multiplying is more sensitive to small values than adding, so a low value drags the geometric mean down harder. The two are equal only when all values match; any spread pushes the geometric mean below.

Which mean do stock indexes use?

For summarizing multi-year returns, the geometric mean gives the true average compound rate. Some price indexes use it too, precisely because it respects ratios and avoids the overstatement the arithmetic mean produces.