Let in a series of 2n observations, half of them are equal to a and remaining half are equ…

Question

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 :

Accepted Answer

Given series

(a, a, a, ........ n times), (-a, -a, -a, ...... n times)

Now $$\overline x $$ = $${{\sum {{x_i}} } \over {2n}} = 0$$

as, xi $$ \to $$ xi + b

then $$\overline x $$ $$ \to $$ $$\overline x $$ + b

So, $$\overline x $$ + b = 5 $$ \Rightarrow $$ b = 5

No change in S.D. due to change in origin

Standard deviation ($$\sigma$$) = $$\sqrt {{{\sum\limits_{i = 1}^{2n} {{{({x_i} - \overline x )}^2}} } \over {2n}}}  $$

$$= \sqrt {{{\sum\limits_{i = 1}^{2n} {x_i^2} } \over {2n}}}  = \sqrt {{{2n{a^2}} \over {2n}}}  = \sqrt {{a^2}} $$

$$ \therefore $$ $$20 = \sqrt {{a^2}}  \Rightarrow a = 20$$

$$ \therefore $$ a2 + b2 = 425

Solutions · 1