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JEE Main 20206 SeptemberMorning Shift
Exam
JEE Main
Year
2020
Date
6 September
Shift
Morning Shift
Subject
Mathematics
Chapter
Statistics

JEE Main 2020 · 6 September · Morning Shift · Mathematics · Statistics

If Σ _ i = 1 ^ n ( x _ i - a) = n and Σ _ i = 1 ^ n ( x _ i - a) ^...

If Σi=1n(xi−a)=n\mathop{Σ}_{i = 1}^{n}\left({x}_{i} - a\right) = n and Σi=1n(xi−a)2=na,(n,a>1)\mathop{Σ}_{i = 1}^{n}{\left({x}_{i} - a\right)}^{2} = na, \left(n, a > 1\right), then the standard deviation of nn observations x1,x2,…,xn{x}_{1}, {x}_{2}, \ldots , {x}_{n} is

Options

  1. Aa−1a - 1
  2. Bn(a−1)n\sqrt{( a - 1 )}
  3. Cn(a−1)\sqrt{n ( a - 1 )}
  4. D(a−1)\sqrt{( a - 1 )}

Correct answer

D. (a−1)\sqrt{( a - 1 )}

Step-by-step solution

SD, σ=∑xi2n−(∑xin)2\sigma = \sqrt{\dfrac{∑ {x}_{i}^{2}}{n} - {\left(\dfrac{∑ {x}_{i}}{n}\right)}^{2}}

SD is independent of change of origin, so, for given data
σ=Σ(xi−a)2n−(Σ(xi−a)n)2\sigma = \sqrt{\dfrac{Σ{\left({x}_{i} - a\right)}^{2}}{n} - {\left(\dfrac{Σ\left({x}_{i} - a\right)}{n}\right)}^{2}}
=nan−(nn)2=a−1= \sqrt{\dfrac{na}{n} - {\left(\dfrac{n}{n}\right)}^{2}} = \sqrt{a - 1}

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