A) \[a\sigma +b\]
B) \[|a|\sigma \]
C) \[|a|\sigma +b\]
D) \[{{a}^{2}}\sigma \]
Correct Answer: B
Solution :
Let \[{{x}_{1}},{{x}_{2}},.........,{{x}_{n}}\] be n values of x. Then \[{{\sigma }^{2}}=\frac{1}{n}\sum\limits_{i=1}^{n}{({{x}_{i}}-\bar{x}}{{)}^{2}}\] ??(i) The variable \[ax+b\] takes values \[a{{x}_{1}}+b,\] \[a{{x}_{2}}+b,.....,a{{x}_{n}}+b\]with mean \[a\overline{x}+b.\] \[\therefore \] \[Var(ax+b)=\frac{1}{n}\sum\limits_{i=1}^{n}{{{\{(a{{x}_{i}}-b)-(a\bar{x}+b)\}}^{2}}}\] \[=\frac{{{a}^{2}}}{n}\sum\limits_{i=1}^{n}{({{x}_{i}}-\overline{x}}{{)}^{2}}\] \[\Rightarrow \] SD of \[(ax+b)=\sqrt{{{a}^{2}}.\frac{1}{n}\sum\limits_{i=1}^{n}{{{({{x}_{i}}-\overline{x})}^{2}}}}\] \[=|a|\sigma \]
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