If standard deviation of numbers -1, 0, 1, k is √ 5 where k

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13.Statistics

hard

If the standard deviation of the numbers $-1, 0, 1, k$ is $\sqrt 5$ where $k > 0,$ then $k$ is equal to

$4\sqrt {\frac {5}{3}}$

$\sqrt 6$

$2\sqrt 6$

$2\sqrt {\frac {10}{3}}$

(JEE MAIN-2019)

Solution

$S.D. = \sqrt {\frac{{\sum {{{\left( {x – \bar x} \right)}^2}} }}{n}} $

$\bar x = \frac{{\sum x }}{4} = \frac{{ – 1 + 0 + 1 + k}}{4} = \frac{k}{4}$

Now $\sqrt 5 = \sqrt {\frac{{{{\left( { – 1 – \frac{k}{4}} \right)}^2} + {{\left( {0 – \frac{k}{4}} \right)}^2} + {{\left( {1 – \frac{k}{4}} \right)}^2} + {{\left( {k – \frac{k}{4}} \right)}^2}}}{4}} $

$ \Rightarrow 5 \times 4 = 2{\left( {1 + \frac{k}{{16}}} \right)^2} + \frac{{5{k^2}}}{8}$

$ \Rightarrow 18 = \frac{{3{k^2}}}{4}$

$ \Rightarrow {k^2} = 24$

$ \Rightarrow k = 2\sqrt 6 $

Standard 11

Mathematics

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If the standard deviation of the numbers $-1, 0, 1, k$ is $\sqrt 5$ where $k > 0,$ then $k$ is equal to

Solution

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Calculate the mean, variance and standard deviation for the following distribution:

Class $30-40$ $40-50$ $50-60$ $60-70$ $70-80$ $80-90$ $90-100$

$f_i$ $3$ $7$ $12$ $15$ $8$ $3$ $2$