If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the followin…

Question

If the standard deviation of the numbers $2, 3, a$ and $11$ is 3.5, then which of the following is true?

Accepted Answer

The formula for standard deviation (S.D)

$$ = \sqrt {{{\sum {x_i^2} } \over n} - {{\left( {{{\sum {{x_i}} } \over n}} \right)}^2}} $$

Where $$\sum {x_i^2 = } $$ Sum of square of the numbers

$= {2^2} + {3^2} + {a^2} + {11^2}$

$= 4 + 9 + {a^2} + 121$

$= 134 + {a^2}$

$$\sum {{x_i}}  = $$ Sum of numbers

$= 2 + 3 + a + 11$

$= 16 + a$

$$\therefore\,\,\,$$ $$SD = \sqrt {{{134 + a{}^2} \over 4} - {{\left( {{{16 + a} \over 4}} \right)}^2}} $$

$$ \Rightarrow \sqrt {{{134 + {a^2}} \over 4} - {{\left( {{{16 + a} \over 4}} \right)}^2}}  = 3.5 = {7 \over 2}$$ (given)

$$ \Rightarrow {{134 + {a^2}} \over 4} - {\left( {{{16 + a} \over 4}} \right)^2} = {{49} \over 4}$$

$$ \Rightarrow 4\left( {134 + {a^2}} \right) - \left( {256 + 32a + {a^2}} \right) = 4 \times 49$$

$$ \Rightarrow 3{a^2} - 32a + 84 = 0$$

Solutions · 1