Explain, by taking a suitable example, how the arithmetic mean alters by adding a constant $k$ to each term.

To do:

We have to explain how the arithmetic mean alters by adding a constant $k$ to each term.

Solution:

We know that,

Mean $\overline{X}=\frac{Sum\ of\ the\ observations}{Number\ of\ observations}$

Let $x_1, x_2, x_3, x_4$ and $x_5$ be five numbers whose mean is $\overline{X}$.

This implies,

$\overline{X}=\frac{x_1+x_2+x_3+x_4+x_5}{5}$

A constant $k$ is added to each term.

Therefore,

New mean $=\frac{\left(x_{1}+k\right)+\left(x_{2}+k\right)+\left(x_{3}+k\right)+\left(x_{4}+k\right)+\left(x_{5}+k\right)}{5}$

$=\frac{x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+5 k}{5}$

$=\frac{x_{1}+x_{2}+x_{3}+x_{4}+x_{5}}{5}+k$

$=\bar{X}+k$

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Updated on: 10-Oct-2022

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