In a series of 2n observations, half of them equal a and remaining half equal -a.

Question

In a series of $2n$ observations, half of them equal a and remaining half equal $-a$. If the standard deviation of the observations is 2, then $|a|$ equals.

Accepted Answer

Mean $$\left( A \right) = {{a - a} \over {2n}} = 0$$

Given standard deviation (S.D) = 2

$$\therefore\,\,\,$$ $$\sqrt {{{\sum {{{\left( {x - A} \right)}^2}} } \over {2n}}}  = 2$$

$$ \Rightarrow \,\,\,\sqrt {{{{{\left( {a - 0} \right)}^2} + {{\left( {a - 0} \right)}^2} + ..... + {{\left( {0 - a} \right)}^2}} \over {2n}}}  = 2$$

$$ \Rightarrow \,\,\,\sqrt {{{{a^2} + {a^2}........2n\,times} \over {2n}}}  = 2$$

$$ \Rightarrow \,\,\,\sqrt {{{2n\,.\,{a^2}} \over {2n}}}  = 2$$

$$ \Rightarrow \,\,\,\sqrt {{a^2}}  = 2$$

$$ \Rightarrow \,\,\,\left| a \right| = 2$$

Solutions · 1