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Explain, by taking a suitable example, how the arithmetic mean alters by dividing each term by a non-zero constant $k$.
To do:
We have to explain how the arithmetic mean alters by dividing each term by a non-zero constant $k$.
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 multiplied with each term.
Therefore,
New mean $=\frac{\left(x_{1}\div k\right)+\left(x_{2}\div k\right)+\left(x_{3}\div k\right)+\left(x_{4}\div k\right)+\left(x_{5}\div k\right)}{5}$
$=\frac{(x_{1}+x_{2}+x_{3}+x_{4}+x_{5})\div k}{5}$
$=\frac{1}{k}\times \frac{x_{1}+x_{2}+x_{3}+x_{4}+x_{5}}{5}$
$=\frac{1}{k}\bar{X}$
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