### Effects on measures of dispersion from some changes of each datum

Question Sample Titled 'Effects on measures of dispersion from some changes of each datum'

Let $\overline{{x}}_{{1}}$ , ${I}{Q}{R}_{{1}}$ and $\sigma_{{1}}$ be the mean, the inter-quartile range and the standard deviation of a data set ${A}$ respectively. Now, ${8}$ is added from each number of the data set ${A}$ and then all of them are multiplied by $\dfrac{{6}}{{7}}$ to form a new data set ${B}$ . If $\overline{{x}}_{{2}}$, ${I}{Q}{R}_{{2}}$ and $\sigma_{{2}}$ are the mean, the inter-quartile range and the standard deviation of the data set ${B}$ respectively, then

A
$\overline{{x}}_{{2}}=\dfrac{{6}}{{7}}\overline{{x}}_{{1}}+\dfrac{{48}}{{7}}$ , ${I}{Q}{R}_{{2}}=\dfrac{{6}}{{7}}{I}{Q}{R}_{{1}}$ , $\sigma_{{2}}=\dfrac{{6}}{{7}}\sigma_{{1}}$
B
$\overline{{x}}_{{2}}=\dfrac{{6}}{{7}}\overline{{x}}_{{1}}+\dfrac{{48}}{{7}}$ , ${I}{Q}{R}_{{2}}=\dfrac{{6}}{{7}}{I}{Q}{R}_{{1}}+\dfrac{{48}}{{7}}$ , $\sigma_{{2}}=\dfrac{{6}}{{7}}\sigma_{{1}}$
C
$\overline{{x}}_{{2}}=\overline{{x}}_{{1}}$ , ${I}{Q}{R}_{{2}}={I}{Q}{R}_{{1}}$ , $\sigma_{{2}}=\sigma_{{1}}$
D
$\overline{{x}}_{{2}}=\dfrac{{6}}{{7}}\overline{{x}}_{{1}}$ , ${I}{Q}{R}_{{2}}=\dfrac{{6}}{{7}}{I}{Q}{R}_{{1}}$ , $\sigma_{{2}}=\dfrac{{6}}{{7}}\sigma_{{1}}$

When a constant is added or deducted from each data, the mean will change, but the ${I}{Q}{R}$ and the standard deviation will not change.

When each data is multiplied by a factor, the mean, the ${I}{Q}{R}$ and the standard deviation will be multiplied by the same factor.

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