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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Statistics

NDA 2024 · Mathematics · Statistics

The mean of the series x_1, x_2, , x_n is x . If x_n is replaced by...

The mean of the series x1,x2,…,xnx_1, x_2, \ldots, x_n is xˉ\bar{x}. If xnx_n is replaced by kk, then what is the new mean?

Options

  1. Axˉ−xn+k\bar{x}-x_n+k
  2. Bnxˉ−xˉ+kn\frac{n \bar{x}-\bar{x}+k}{n}
  3. Cxˉ−xn−kn\frac{\bar{x}-x_n-k}{n}
  4. Dnxˉ−xn+kn\frac{n \bar{x}-x_n+k}{n}

Correct answer

D. nxˉ−xn+kn\frac{n \bar{x}-x_n+k}{n}

Step-by-step solution

Given that, xˉ=x1+x2+x3+……+xnn⇒x1+x2+x3+……+xn=nx‾\bar{x}=\frac{x_1+x_2+x_3+\ldots \ldots+x_n}{n}\Rightarrow \mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\ldots \ldots+\mathrm{x}_{\mathrm{n}}=\mathrm{n} \overline{\mathrm{x}} Mean when xn\mathrm{x}_{\mathrm{n}} is replaced by k .  New mean =x1+x2+x3……+xn−1+kn=x1+x2+x3+…xn−1+xn−xn+kn=nxˉ−xn+kn\begin{aligned} & \text{ New mean }=\frac{x_1+x_2+x_3 \ldots \ldots+x_{n-1}+k}{n} \\ & =\frac{x_1+x_2+x_3+\ldots x_{n-1}+x_n-x_n+k}{n} \\ & =\frac{n \bar{x}-x_n+k}{n} \end{aligned}
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