# 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.
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SoosteethicU

Geometric mean of a data distribution ${a}_{1},{a}_{2},{a}_{3},..,{a}_{n}$ is ${n}^{th}$ square root of product of all data points and is calculated using the below mentioned formula $GM=\sqrt[n]{{a}_{1}\cdot {a}_{2}\cdot \dots \cdot {a}_{n}}={\left({x}_{1}\cdot {x}_{2}\cdot \dots \cdot {x}_{n}\right)}^{\frac{1}{n}}$
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
$GM=\sqrt[n]{{a}_{1}\cdot {a}_{2}\cdot \dots \cdot {a}_{n}}$
$=\sqrt[5]{2\cdot 4\cdot 8\cdot 16\cdot 32}$
$=\sqrt[5]{32768}$
$=8$
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:
$GM=\sqrt[n]{{a}_{1}\cdot {a}_{2}\cdot \dots \cdot {a}_{n}}$
$=\sqrt[5]{4\cdot 6\cdot 10\cdot 18\cdot 34}$
$=\sqrt[5]{146880}$
$=10.7992$
Here it is clearly shown that adding a constant to all data values increases the geometric mean of the data distribution.