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:

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

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

∴ ∑x_i = 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