If each observation of a raw data whose variance is σ2 ismultiplied by h, then the variance of the new set is

# If each observation of a raw data whose variance is ${\sigma }^{2}$ ismultiplied by $h,$ then the variance of the new set is

1. A

${\sigma }^{2}$

2. B

${h}^{2}{\sigma }^{2}$

3. C

4. D

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### Solution:

Let ${x}_{1},{x}_{2},\dots ,{x}_{n}$ be a raw data. Then,

${\sigma }^{2}=\frac{1}{n}\sum _{i=1}^{n} {\left({x}_{i}-\overline{X}\right)}^{2}$

If each value is multiplied by $h$, then the values are

$h{x}_{1},h{x}_{2},\dots ,h{x}_{n}$.The AM of the new values is

$\frac{h{x}_{1}+h{x}_{2}+\dots +h{x}_{n}}{n}=h\overline{X}.$

The variance ${\sigma }^{2}$ of the new set of values is given by

${\sigma }_{1}^{2}=\frac{1}{n}\sum _{i=1}^{n} {\left(h{x}_{i}-h\overline{X}\right)}^{2}={h}^{2}\left\{\frac{1}{n}\sum _{i=1}^{n} {\left({x}_{i}-\overline{X}\right)}^{2}\right\}={h}^{2}{\sigma }^{2}$.

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