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In a data with 15 number of observations $x_1$, $x_2, x_3, \ldots, x_{15}, \sum_{i=1}^{15} x_i^2=3600$ and $\sum_{i=1}^{15} x_i=175$. If the value of one observation 20 was found wrong and was replaced by its correct value 40 , then the corrected variance of that data is
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Verified Answer
The correct answer is:
151
Given,
$$
\sum_{i=1}^{15} x_i^2=3600, \sum_{i=1}^{15} x_i=175
$$
When 20 is replace by 40 , then
$$
\begin{gathered}
\sum_{i=1}^{15} x_i=175-20+40=195 \\
\sum_{i=1}^{15} x_i^2=3600-(20)^2+(40)^2=4800 \\
\therefore \text { Corrected variance }=\frac{\sum x_i^2}{n}-\left(\frac{\sum x_i}{n}\right)^2 \\
=\frac{4800}{15}-\left(\frac{195}{15}\right)^2=320-169=151
\end{gathered}
$$
$$
\sum_{i=1}^{15} x_i^2=3600, \sum_{i=1}^{15} x_i=175
$$
When 20 is replace by 40 , then
$$
\begin{gathered}
\sum_{i=1}^{15} x_i=175-20+40=195 \\
\sum_{i=1}^{15} x_i^2=3600-(20)^2+(40)^2=4800 \\
\therefore \text { Corrected variance }=\frac{\sum x_i^2}{n}-\left(\frac{\sum x_i}{n}\right)^2 \\
=\frac{4800}{15}-\left(\frac{195}{15}\right)^2=320-169=151
\end{gathered}
$$
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