Question

For a data consisting of $15$ observations $x_i,i=1,2,3,....,15$ the following results are obtained: ${\sum }_{i=1}^{15}{x}_i=170;{\sum }_{i=1}^{15}{x}_1^2=2830$ .If one of the observation namely $20$ was found wrong and was replaced by its collect value $30$. then the corrected variance is

• A. 80
• B. 78
• C. 76
• D. 75

Answer 1

Answer: (B)

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