The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation by mistake was taken as 8 instead of 12. The correct standard deviation is

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Solution

Q.
The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation by mistake was taken as 8 instead of 12. The correct standard deviation is [2024]

1 1.8

2 1.94

3 $\sqrt{3.86}$

4 $\sqrt{3.96}$

Ans.
(4)

Incorrect mean, $\bar{x} = 10$

$\Rightarrow Incorrect \frac{\sum x_{i}}{20} = 10$

Incorrect sum, $\sum x_{i} = 200$

$Correct sum = 200 - 8 + 12 = 204$

$Correct mean = \frac{204}{20} = 10.2$

Incorrect S.D. = 2

Incorrect variance = 4

$\Rightarrow \frac{\sum x_{i}^{2}}{20} - {(\frac{\sum x_{i}}{20})}^{2} = 4 \Rightarrow \frac{\sum x_{i}^{2}}{20} - 100 = 4$

$\Rightarrow \sum x_{i}^{2} = 2080$

Incorrect $\sum x_{i}^{2} = 2080$

Correct $\sum x_{i}^{2} = 2080 - 8^{2} + 12^{2} = 2160$

Correct Variance $= \frac{2160}{20} - {(10.2)}^{2} = 108 - 104.04 = 3.96$

Correct S.D. = $\sqrt{3.96}$

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