Q253 / 270 · NDA 2016 (Phase 1) · Mathematics

The mean of the series $x_1, x_2, \ldots, x_n$ is $\overline{\mathrm{X}}$. If $\mathrm{x}_2$ is replaced by $\lambda$, then what is the new mean?

  • A
    $\overline{\mathrm{X}}-\mathrm{x}_2+\lambda$
  • B
    $\frac{\bar{X}-x_2-\lambda}{n}$
  • C
    $\frac{\bar{X}-\mathrm{x}_2+\lambda}{\mathrm{n}}$
  • Correct answer
    D
    $\frac{\mathrm{n} \overline{\mathrm{X}}-\mathrm{x}_2+\lambda}{\mathrm{n}}$

Solution

Mean of series $\left(x_1, x_2, x_3 \ldots \ldots x_{\mathrm{n}}\right)$ $\begin{aligned} & \bar{x}=\frac{x_1+x_2+x_3+\ldots . x_n}{n} \\ & \Rightarrow x_1+x_2+x_3+\ldots . x_n=n \bar{x} \end{aligned}$ Now we will replace $x_2$ by $\lambda$ so no. of elements in series will not change. New series will include $\lambda$ and exclude $x_2$ Hence new series sum : $\left(x_1+x_2+\ldots . . x_n\right)-x_2+\lambda=n \bar{x}+\lambda-x_2$ Now new mean $=\frac{n \bar{x}+\lambda-x_2}{n}=\frac{n \bar{x}-x_2+\lambda}{n}$

Hence, the correct answer is option: 'D'.

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