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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Statistics

NDA 2023 · Mathematics · Statistics

Let P be the median, Q be the mean and R be the mode of...

Let PP be the median, QQ be the mean and RR be the mode of observations x1,x2,x3,…xnx_1, x_2, x_3, \ldots x n. Let S=∑i=1n(2xi−a)2.SS=\sum_{i=1}^n\left(2 x_i-a\right)^2. S takes minimum value, when aa is equal to

Options

  1. APP
  2. BQ2\frac{Q}{2}
  3. C2Q2 Q
  4. DRR

Correct answer

C. 2Q2 Q

Step-by-step solution

\Sigma \left(x_i-a\right)^2 is minimum for mean=a∴mean=a\therefore \quad For Σ(2xi−a)2\Sigma\left(2 x_i-a\right)^2=\Sigma 4\left(x_i-\frac{a}{2}\right)^2=\frac{a}{2}mustbemeanst be mean=a2=Q⇒a=2Q\begin{aligned} & =\frac{a}{2}=Q \\ \Rightarrow \quad & a=2 Q\end{aligned}$
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