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13.Statistics
hard
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