The mean and standard deviation of 40 observations are 30 and 5 respectively. It was noticed that two of these observations 12 and 10 were wrongly recorded. If $σ$ is the standard deviation of the data after omitting the two wrong observations from the data, then $38 σ^{2}$ is equal to___________.

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answer is 238.

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Detailed Solution

Wrong mean $= μ_{1} = 30$

Wrong $S.D = σ_{1} = 5$

$\begin{array}{l} \frac{\sum x_{i}}{40} = 30 \\ \Rightarrow \sum x_{i} = 1200 \\ σ_{1}^{2} = 25 \\ \Rightarrow \frac{\sum x_{i}^{2}}{40} - 30^{2} = 25 \\ \Rightarrow \sum x_{i}^{2} = 925 \times 40 = 37000 \end{array}$

New sum $= \sum x_{i}^{'} = 1200 - 10 - 12 = 1178$

New mean $= μ_{1}^{'} = \frac{1178}{38} = 31$

New $\sum x_{i}^{2} = 37000 - (10)^{2} - (12)^{2} = 36756$

New S.D, $σ_{1}^{'} = \sqrt{\frac{36756}{38} - (31)^{2}} = σ$

$\begin{array}{l} 36756 - (31)^{2} \times 38 = 38 σ^{2} \\ \Rightarrow 38 σ^{2} = 238 \end{array}$

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