For a data consisting of 15 observations $x_i$, $i=1,2,3, \ldots, 15$ the following results are obtained :…
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
80
78
76
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}
$