MHT CET · Maths · Statistics
For 20 observations of variable \(x\), if \(\sum\left(x_{\mathrm{i}}-2\right)=20\) and \(\sum\left(x_{\mathrm{i}}-2\right)^2=100\), then the standard deviation of variable \(x\) is
- A \(2\)
- B \(3\)
- C \(4\)
- D \(9\)
Answer & Solution
Correct Answer
(A) \(2\)
Step-by-step Solution
Detailed explanation
Note that standard derivation is independent of change of origin.
\(\therefore\) S.D. of \(x_{\mathrm{i}}=\) S.D. of \(\left(x_{\mathrm{i}}-2\right)\)
\(\therefore\) S.D. of \(\left(x_{\mathrm{i}}-2\right) =\sqrt{\frac{1}{\mathrm{n}} \sum_{\mathrm{i}}^{20}\left(x_{\mathrm{i}}-2\right)^2-\left[\frac{\sum\left(x_{\mathrm{i}}-2\right)}{\mathrm{n}}\right]^2} \)
\( =\sqrt{\frac{100}{20}-(1)^2} \)
\( =2 \)
\(\Rightarrow\) Required S.D \(=2\)
\(\therefore\) S.D. of \(x_{\mathrm{i}}=\) S.D. of \(\left(x_{\mathrm{i}}-2\right)\)
\(\therefore\) S.D. of \(\left(x_{\mathrm{i}}-2\right) =\sqrt{\frac{1}{\mathrm{n}} \sum_{\mathrm{i}}^{20}\left(x_{\mathrm{i}}-2\right)^2-\left[\frac{\sum\left(x_{\mathrm{i}}-2\right)}{\mathrm{n}}\right]^2} \)
\( =\sqrt{\frac{100}{20}-(1)^2} \)
\( =2 \)
\(\Rightarrow\) Required S.D \(=2\)
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