### Effects of data change on measures of dispersion

Question Sample Titled 'Effects of data change on measures of dispersion'

Let $\overline{{x}}_{{1}}$ and $\sigma_{{1}}$ be the mean and the standard deviation of a group of numbers ${\left\lbrace{a}_{{1}},{a}_{{2}},{a}_{{3}},\ldots,{a}_{{23}}\right\rbrace}$ respectively while $\overline{{x}}_{{2}}$ and $\sigma_{{2}}$ be the mean and the standard deviation of a group of numbers ${\left\lbrace{a}_{{1}},{a}_{{2}},{a}_{{3}},\ldots,{a}_{{22}}\right\rbrace}$ respectively. If $\sigma_{{1}}\ne{0}$ and ${a}_{{23}}=\overline{{x}}_{{1}}$ , which of the following must be true?

 I. $\overline{{x}}_{{1}}=\overline{{x}}_{{2}}$ II. $\sigma_{{1}}<\sigma_{{2}}$ III. The standard score of ${a}_{{1}}$ in the first set is the same as that in the second set.

A
I and II only
B
II and III only
C
I, II and III
D
I and III only

 $\overline{{x}}_{{1}}$ $=\dfrac{{{a}_{{1}}+{a}_{{2}}+{a}_{{3}}+\ldots+{a}_{{23}}}}{{23}}$ ${a}_{{1}}+{a}_{{2}}+{a}_{{3}}+\ldots+{a}_{{23}}$ $={23}\overline{{x}}_{{1}}$ ${a}_{{1}}+{a}_{{2}}+{a}_{{3}}+\ldots+{a}_{{22}}$ $={23}\overline{{x}}_{{1}}-{a}_{{23}}$  $={22}\overline{{x}}_{{1}}$  $\overline{{x}}_{{2}}$ $=\dfrac{{{a}_{{1}}+{a}_{{2}}+{a}_{{3}}+\ldots+{a}_{{22}}}}{{22}}$  $=\dfrac{{{22}\overline{{x}}_{{1}}}}{{22}}$  $=\overline{{x}}_{{1}}$ ∴  $\overline{{x}}_{{1}}$ $=\overline{{x}}_{{2}}$

 $\sigma_{{1}}$ $=\sqrt{{\dfrac{{{\left({a}_{{1}}-\overline{{x}}_{{1}}\right)}^{{2}}+{\left({a}_{{2}}-\overline{{x}}_{{1}}\right)}^{{2}}+\ldots+{\left({a}_{{23}}-\overline{{x}}_{{1}}\right)}^{{2}}}}{{{23}}}}}$  $=\sqrt{{\dfrac{{{\left({a}_{{1}}-\overline{{x}}_{{1}}\right)}^{{2}}+{\left({a}_{{2}}-\overline{{x}}_{{1}}\right)}^{{2}}+\ldots+{\left(\overline{{x}}_{{1}}-\overline{{x}}_{{1}}\right)}^{{2}}}}{{{23}}}}}$  $=\sqrt{{\dfrac{{{\left({a}_{{1}}-\overline{{x}}_{{1}}\right)}^{{2}}+{\left({a}_{{2}}-\overline{{x}}_{{1}}\right)}^{{2}}+\ldots+{\left({a}_{{22}}-\overline{{x}}_{{1}}\right)}^{{2}}}}{{{23}}}}}$ $\sigma_{{2}}$ $=\sqrt{{\dfrac{{{\left({a}_{{1}}-\overline{{x}}_{{1}}\right)}^{{2}}+{\left({a}_{{2}}-\overline{{x}}_{{1}}\right)}^{{2}}+\ldots+{\left({a}_{{22}}-\overline{{x}}_{{1}}\right)}^{{2}}}}{{{22}}}}}$ $\sqrt{{22}}\sigma_{{2}}$ $=\sqrt{{23}}\sigma_{{1}}$ $\sigma_{{1}}:\sigma_{{2}}$ $=\sqrt{{22}}:\sqrt{{23}}$ ∴  $\sigma_{{1}}$ $<\sigma_{{2}}$

 Let ${z}_{{1}}$ and ${z}_{{2}}$ be the standard score of ${a}_{{1}}$ in the first set and the second set respectively. ${z}_{{1}}$ $=\dfrac{{{a}_{{1}}-\overline{{x}}_{{1}}}}{\sigma_{{1}}}$ ${z}_{{2}}$ $=\dfrac{{{a}_{{1}}-\overline{{x}}_{{2}}}}{\sigma_{{2}}}$  $=\dfrac{{{a}_{{1}}-\overline{{x}}_{{1}}}}{\sigma_{{2}}}$ $\sigma_{{2}}{z}_{{2}}$ $=\sigma_{{1}}{z}_{{1}}$ ${z}_{{1}}:{z}_{{2}}$ $=\sigma_{{2}}:\sigma_{{1}}$ ∵  $\sigma_{{2}}$ $>\sigma_{{1}}$ ∴  ${z}_{{1}}$ $>{z}_{{2}}$

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