Geometric mean of a data distribution \(\displaystyle{a}_{{1}},{a}_{{2}},{a}_{{3}},..,{a}_{{n}}\) is \(n^{th}\) square root of product of all data points and is calculated using the below mentioned formula \(\displaystyle{G}{M}={\sqrt[{{n}}]{{{a}_{{1}}\cdot{a}_{{2}}\cdot\ldots\cdot{a}_{{n}}}}}={\left({x}_{{1}}\cdot{x}_{{2}}\cdot\ldots\cdot{x}_{{n}}\right)}^{{\frac{{1}}{{n}}}}\)

When a constant d is added to all data values, the product of all numbers will increase, so the geometric mean of the data distribution also increases by adding a constant to all values.

Let us consider an example data distribution 2,4,8,16,32

The geometric mean is calculated as shown below

\(\displaystyle{G}{M}={\sqrt[{{n}}]{{{a}_{{1}}\cdot{a}_{{2}}\cdot\ldots\cdot{a}_{{n}}}}}\)

\(\displaystyle={\sqrt[{{5}}]{{{2}\cdot{4}\cdot{8}\cdot{16}\cdot{32}}}}\)

\(\displaystyle={\sqrt[{{5}}]{{{32768}}}}\)

\(=8\)

If we add a constant 2 to all data points then the data distribution will be 4 ,6 ,10, 18, 34. The geometric mean is calculated as shown below:

\(\displaystyle{G}{M}={\sqrt[{{n}}]{{{a}_{{1}}\cdot{a}_{{2}}\cdot\ldots\cdot{a}_{{n}}}}}\)

\(\displaystyle={\sqrt[{{5}}]{{{4}\cdot{6}\cdot{10}\cdot{18}\cdot{34}}}}\)

\(\displaystyle={\sqrt[{{5}}]{{{146880}}}}\)

\(=10.7992\)

Here it is clearly shown that adding a constant to all data values increases the geometric mean of the data distribution.