Q75 / 160 · AP EAMCET 2019 (21 Apr Shift 1) · Mathematics

For a data consisting of 15 observations $x_i$, $i=1,2,3, \ldots, 15$ the following results are obtained : $\sum_{i=1}^{15} x_i=170 ; \sum_{i=1}^{15} x_i^2=2830$. If one of the observation namely 20 was found wrong and was replaced by its correct value 30 , then the corrected variance is

  • A
    80
  • Correct answer
    B
    78
  • C
    76
  • D
    75

Solution

Given, $$ \Sigma x^2=2830 \text { and } \Sigma x=170 $$ Increase in $\Sigma x=10$ and increase in $\Sigma x^2=(30)^2-(20)^2$ $$ \begin{aligned} & =900-400=500 \\ & \Sigma x^{\prime}=170+10=180 \\ & \text { and } \Sigma x^{\prime 2}=2830+500=3330 \\ & \text { We know that, variance }=\frac{\Sigma x^2}{n}-\left(\frac{\Sigma x}{n}\right)^2 \\ & =\frac{3330}{15}-\left(\frac{180}{15}\right)^2 \quad \text{[here, $n=15$ ]}\\ & =222-144=78 \\ \end{aligned} $$

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

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