If we add a constant (say, d) for all data values, how this will affect the geometric mean? Give an example.

If we add a constant (say, d) for all data values, how this will affect the geometric mean? Give an example.

Question
Data distributions
asked 2021-02-03
If we add a constant (say, d) for all data values, how this will affect the geometric mean? Give an example.

Answers (1)

2021-02-04

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.

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