JEE MAIN - Mathematics (2022 - 26th July Evening Shift - No. 17)
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 $$\sigma$$ is the standard deviation of the data after omitting the two wrong observations from the data, then $$38 \sigma^{2}$$ is equal to ___________.
Answer
238
Explanation
$$\mu = {{\sum {{x_i}} } \over {40}} = 30 \Rightarrow \sum {{x_i} = 1200} $$
$${\sigma ^2} = {{\sum {x_i^2} } \over {40}} - {(30)^2} = 25 \Rightarrow \sum {x_i^2 = 37000} $$
After omitting two wrong observations
$$\sum {{y_i} = 1200 - 12 - 10 = 1178} $$
$$\sum {y_i^2 = 37000 - 144 - 100 = 36756} $$
Now $${\sigma ^2} = {{\sum {y_i^2} } \over {38}} - {\left( {{{\sum {{y_i}} } \over {38}}} \right)^2}$$
$$ = {{36756} \over {38}} - {\left( {{{1178} \over {38}}} \right)^2} = - {31^2}$$
$$ = 38{\sigma ^2} = 36756 - 36518 = 238$$
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