Let in a series of 2n observations, half of them are equal to a and remaining half are equal to $-$a. Also by adding a constant b in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of a² + b² is equal to :

Given series<br><br>(a, a, a, ........ n times), ($-$a, $-$a, $-$a, ...... n times)<br><br>Now $\overline x$ = ${{\sum {{x_i}} } \over {2n}} = 0$<br><br>as, x_i $\to$ x_i + b<br><br>then $\overline x$ $\to$ $\overline x$ + b<br><br>So, $\overline x$ + b = 5 $\Rightarrow$ b = 5<br><br>No change in S.D. due to change in origin<br><br>Standard deviation ($\sigma$) = $\sqrt {{{\sum\limits_{i = 1}^{2n} {{{({x_i} - \overline x )}^2}} } \over {2n}}}$<br><br>$$= \sqrt {{{\sum\limits_{i = 1}^{2n} {x_i^2} } \over {2n}}} = \sqrt {{{2n{a^2}} \over {2n}}} = \sqrt {{a^2}} $$<br><br>$\therefore$ $20 = \sqrt {{a^2}} \Rightarrow a = 20$<br><br>$\therefore$ a² + b² = 425