In probability theory and statistics, the geometric standard deviation (GSD) describes how spread out are a set of numbers whose preferred average is the geometric mean. For such data, it may be preferred to the more usual standard deviation. Note that unlike the usual arithmetic standard deviation, the geometric standard deviation is a multiplicative factor, and thus is dimensionless, rather than having the same dimension as the input values. Thus, the geometric standard deviation may be more appropriately called geometric SD factor. When using geometric SD factor in conjunction with geometric mean, it should be described as "the range from (the geometric mean divided by the geometric SD factor) to (the geometric mean multiplied by the geometric SD factor), and one cannot add/subtract "geometric SD factor" to/from geometric mean.

If the geometric mean of a set of numbers ${\textstyle {A_{1},A_{2},...,A_{n}}}$ is denoted as ${\textstyle \mu _{\mathrm {g} }}$, then the geometric standard deviation is

$${\displaystyle \sigma _{\mathrm {g} }=\exp {\sqrt {{1 \over n}\sum _{i=1}^{n}\left(\ln {A_{i} \over \mu _{\mathrm {g} }}\right)^{2}}}\,.}$$

If the geometric mean is

$${\displaystyle \mu _{\mathrm {g} }={\sqrt[{n}]{A_{1}A_{2}\cdots A_{n}}}}$$

then taking the natural logarithm of both sides results in

$${\displaystyle \ln \mu _{\mathrm {g} }={1 \over n}\ln(A_{1}A_{2}\cdots A_{n}).}$$

The logarithm of a product is a sum of logarithms (assuming ${\textstyle A_{i}}$ is positive for all ${\textstyle i}$), so