Q75 / 160 · AP EAMCET 2018 (22 Apr Shift 2) · Mathematics

The arithmetic mean and standard deviation of a data of nine numbers are 13 and 5 respectively. If 3 is included as the 10th item of the data, then the variance of the data of ten number is

  • A
    23.5
  • B
    21.5
  • Correct answer
    C
    31.5
  • D
    27

Solution

$\because \frac{\sum_{i=1}^9 x_i}{9}=13 \Rightarrow \sum_{i=1}^9 x_i=117$. $\begin{array}{rlrl} & & & \\ \text { and } & & \sigma^2=25=\frac{\sum_{i=1}^9 x_i^2}{9}-(13)^2 \\ \Rightarrow & \quad \sum_{i=1}^9 x_i^2=9[25+169]\end{array}$ Now, after including $10^{\text {th }}$ item as ' 3 ' $$ \begin{gathered} \text { New mean } \bar{y}=\frac{\sum_{i=1}^9 x_i+3}{10}=\frac{117+3}{10}=12 \\ \text { and new variance }=\frac{\left(\sum_{i=1}^9 x_i^2+9\right)}{10}-(12)^2 \\ =\frac{(169+25+1) 9}{10}-144=\frac{1755-1440}{10}=31.5 \end{gathered} $$

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

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