If the standard deviation of the numbers -1,0,1,k is √5 where k0, then k is equal to:

Question

If the standard deviation of the numbers $-1,0,1,k$ is $\sqrt5$ where $k>0$, then k is equal to:

Accepted Answer

standard deviation = $$\sqrt 5$$

$$ \therefore $$ Variance = $${\left( {\sqrt 5 } \right)^2}$$ = 5

Also variance = $${{\sum {x_i^2} } \over N} - {\mu ^2}$$

Where $$\mu $$ = Mean = $${{ - 1 + 0 + 1 + k} \over 4}$$ = $${k \over 4}$$

$$ \therefore $$ Variance = $${{{{\left( { - 1} \right)}^2} + 0 + {1^2} + {k^2}} \over 4}$$ - $${{{k^2}} \over {16}}$$

$$ \Rightarrow $$ 5 = $${{2 + {k^2}} \over 4}$$ - $${{{k^2}} \over {16}}$$

$$ \Rightarrow $$ 8 + 3k2 = 80

$$ \Rightarrow $$ k2 = 24 = 2$$\sqrt 6 $$

Solutions · 1