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# If an experiment with 15 observations on $x$ the following results were available: $\Sigma x^2=2830 \: \Sigma x = 170$. One observation that was 20 was found to be wrong and was replaced by the correct value 30. The corrected variance is

$\begin {array} {1 1} (A)\;8.33 & \quad (B)\;78.00 \\ (C)\;188.66 & \quad (D)\;177.33 \end {array}$

Toolbox:
• $\large\frac{1}{n}$$\Sigma x^{2} - \bigg( \large\frac{1}{n}$$ \Sigma x \bigg)^2$
Given: $\Sigma x = 170 \: \Sigma x^2 = 2830$ and increase in $\Sigma x=10$
$\Rightarrow\: \Sigma x' = 170+10=180$
Increase in $\Sigma x^2=900-400=500$
$\Rightarrow\: \Sigma x'^2=2830+500=3330$
Correct Variance = $\large\frac{1}{n}$$\Sigma x^{'2} - \bigg( \large\frac{1}{n}$$ \Sigma x' \bigg)^2$
$= \large\frac{1}{15}$$\times 3330 - \bigg( \large\frac{1}{15}$ $\times 180 \bigg)^2$
$= 222-144$
$= 78$
Ans : (B)
edited Mar 26, 2014