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If both the mean and the standard deviation of 50 observations x1, x2, …,x50 are equal to 16, then the mean of(x1 - 4)2, (x2 - 4)2,….(x50 - 4)2 is:
1. 400
2. 380
3. 525
4. 480

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Correct Answer - Option 1 : 400

Given, mean and standard deviation of the 50 observations are equal to 16.

Mean, \(\left( \mu \right) = \frac{{\sum {x_i}}}{{50}} = \frac{{{x_1} + {x_2} + \ldots .{x_{50}}}}{{50}} = 16\) 

∴ ∑xi = 16 × 50

Standard deviation, \(\left( \sigma \right) = \sqrt {\frac{{\sum x_i^2}}{{50}} - {{(\mu )}^2}} = 16\) 

\( \Rightarrow \frac{{\sum x_i^2}}{{50}} = \)256 × 2

Required mean \( = \frac{{{{\left( {{x_1} - 4} \right)}^2} + {{\left( {{x_2} - 4} \right)}^2} + \ldots {{\left( {{x_{50}} - 4} \right)}^2}}}{{50}}\)

\( = \frac{{\sum {{\left( {{x_i} - 4} \right)}^2}}}{{50}}\) 

\( = \frac{{\sum x_i^2 + 16 \times 50 - 8\sum {x_i}}}{{50}}\) 

= 256 × 2 + 16 - 8 × 16

= 528 - 128 = 400

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