JEE MAIN - Mathematics (2004 - No. 8)
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
2
$$\sqrt 2 $$
$${1 \over n}$$
$${{\sqrt 2 } \over n}$$
Explanation
Mean $$\left( A \right) = {{a - a} \over {2n}} = 0$$
Given standard deviation (S.D) = 2
$$\therefore\,\,\,$$ $$\sqrt {{{\sum {{{\left( {x - A} \right)}^2}} } \over {2n}}} = 2$$
$$ \Rightarrow \,\,\,\sqrt {{{{{\left( {a - 0} \right)}^2} + {{\left( {a - 0} \right)}^2} + ..... + {{\left( {0 - a} \right)}^2}} \over {2n}}} = 2$$
$$ \Rightarrow \,\,\,\sqrt {{{{a^2} + {a^2}........2n\,times} \over {2n}}} = 2$$
$$ \Rightarrow \,\,\,\sqrt {{{2n\,.\,{a^2}} \over {2n}}} = 2$$
$$ \Rightarrow \,\,\,\sqrt {{a^2}} = 2$$
$$ \Rightarrow \,\,\,\left| a \right| = 2$$
Given standard deviation (S.D) = 2
$$\therefore\,\,\,$$ $$\sqrt {{{\sum {{{\left( {x - A} \right)}^2}} } \over {2n}}} = 2$$
$$ \Rightarrow \,\,\,\sqrt {{{{{\left( {a - 0} \right)}^2} + {{\left( {a - 0} \right)}^2} + ..... + {{\left( {0 - a} \right)}^2}} \over {2n}}} = 2$$
$$ \Rightarrow \,\,\,\sqrt {{{{a^2} + {a^2}........2n\,times} \over {2n}}} = 2$$
$$ \Rightarrow \,\,\,\sqrt {{{2n\,.\,{a^2}} \over {2n}}} = 2$$
$$ \Rightarrow \,\,\,\sqrt {{a^2}} = 2$$
$$ \Rightarrow \,\,\,\left| a \right| = 2$$
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