Question

The mean and standard deviation of $15$ observations are found to be $8$ and $3$ respectively. On rechecking it was found that, in the observations, $20$ was misread as $5$. Then, the correct variance is equal to ______.

Answer 1

We have
Variance $=\frac{\displaystyle\sum_{ r =1}^{15} x _{ r }^{2}}{15}-\left(\frac{\displaystyle\sum_{ r =1}^{15} x _{ r }}{15}\right)^{2}$
Now, as per information given in equation
$\frac{\displaystyle\sum x_{ r }^{2}}{15}-8^{2}=3^{2} \Rightarrow \displaystyle\sum x _{ r }^{2}=\log 5$
Now, the new $\displaystyle\sum x _{ r }^{2}=\log 5-5^{2}+20^{2}=1470$
And, new $\displaystyle\sum x _{ r }=(15 \times 8)-5+(20)=135$
$\therefore$ Variance $=\frac{1470}{15}-\left(\frac{135}{15}\right)^{2}=98-81=17$