Mean and standard deviation (s.d.) of five observations are 9 and 0, respectively. If on

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13.Statistics

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The mean and the standard deviation $(s.d.)$ of five observations are $9$ and $0,$ respectively. If one of the observations is changed such that the mean of the new set of five observations becomes $10,$ then their $s.d.$ is?

A

$0$

B

$4$

C

$2$

D

$1$

(JEE MAIN-2018)

Solution

Here mean $ = \bar x = 9$

$ \Rightarrow \bar x = \frac{{\sum {{x_i}} }}{n} = 9$

$ \Rightarrow \sum {{x_i}} = 9 \times 5 = 45$

Now, standard deviation $=0$

$\therefore $ all the five terms are same i.e.;$9$.

Now for changed observation

${{\bar x}_{new}} = \frac{{36 + {x_5}}}{5} = 10$

$ \Rightarrow {x_5} = 14$

$\therefore {\sigma _{new}} = \sqrt {\frac{{\sum {{{\left( {{x_i} – {{\bar x}_{new}}} \right)}^2}} }}{n}} $

$ = \sqrt {\frac{{4{{\left( {9 – 10} \right)}^2} + {{\left( {14 – 10} \right)}^2}}}{5}} $

$ = 2$

Standard 11

Mathematics

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