Change of variance due to addition, subtraction and multiplication of the old set

Question Sample Titled 'Change of variance due to addition, subtraction and multiplication of the old set'

If the variance of the five numbers ${x}-{15}$, ${x}-{14}$, ${x}-{4}$, ${x}+{16}$ and ${x}+{17}$ is ${v}$ , which of the following must be true?

 I. The variance of ${3}{x}-{49}$, ${3}{x}-{46}$, ${3}{x}-{16}$, ${3}{x}+{44}$ and ${3}{x}+{47}$ is ${3}{v}$ . II. The variance of ${x}-{19}$, ${x}-{18}$, ${x}-{8}$, ${x}+{12}$ and ${x}+{13}$ is ${v}$ . III. The variance of ${4}{x}-{15}$, ${4}{x}-{14}$, ${4}{x}-{4}$, ${4}{x}+{16}$ and ${4}{x}+{17}$ is ${4}{v}$ .

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

 Observe the ralations of each numbers in both sets. Recall that: 1. If each datum is added or subtracted by a constant ${k}$, then the variance remains unchanged. 2. If each datum is multipied by a constant ${k}$, then the variance is multipied by ${k}^{{2}}$ . I. ${x}-{19}$ $={3}{\left({x}-{15}\right)}-{4}$ ${x}-{18}$ $={3}{\left({x}-{14}\right)}-{4}$ ${x}-{8}$ $={3}{\left({x}-{4}\right)}-{4}$ ${x}+{12}$ $={3}{\left({x}+{16}\right)}-{4}$ ${x}+{13}$ $={3}{\left({x}+{17}\right)}-{4}$ Each number is first multiplied by ${3}$ and then subtracted by ${4}$ . ∴  The new variance $={\left({3}^{{2}}\right)}{v}={9}{v}$ I is false. II. ${3}{x}-{49}$ $={\left({x}-{15}\right)}-{4}$ ${3}{x}-{46}$ $={\left({x}-{14}\right)}-{4}$ ${3}{x}-{16}$ $={\left({x}-{4}\right)}-{4}$ ${3}{x}+{44}$ $={\left({x}+{16}\right)}-{4}$ ${3}{x}+{47}$ $={\left({x}+{17}\right)}-{4}$ Each number is subtracted by ${4}$ . ∴  The new variance remains unchanged. II is true. III. ${4}{x}-{15}$ $={\left({x}-{15}\right)}+{3}{x}$ ${4}{x}-{14}$ $={\left({x}-{14}\right)}+{3}{x}$ ${4}{x}-{4}$ $={\left({x}-{4}\right)}+{3}{x}$ ${4}{x}+{16}$ $={\left({x}+{16}\right)}+{3}{x}$ ${4}{x}+{17}$ $={\left({x}+{17}\right)}+{3}{x}$ Each number is added by ${3}{x}$ . ∴  The new variance remains unchanged. III is false.

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