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13.Statistics
easy
The $S.D$. of the first $n$ natural numbers is
A
$\frac{{n + 1}}{2}$
B
$\sqrt {\frac{{n(n + 1)}}{2}} $
C
$\sqrt {\frac{{{n^2} - 1}}{{12}}} $
D
None of these
Solution
(c) $S. D.$ of first $n$ natural numbers $ = \sqrt {\frac{1}{n}\Sigma {x^2} – {{\left( {\frac{{\Sigma x}}{n}} \right)}^2}} $,
$ = \sqrt {\frac{{n(n + 1)(2n + 1)}}{{6n}} – {{\left[ {\frac{{n(n + 1)}}{{2n}}} \right]}^2}} $
$ = \sqrt {\frac{{(n + 1)(2n + 1)}}{6} – {{\left( {\frac{{n + 1}}{2}} \right)}^2}} = \sqrt {\frac{{n + 1}}{2}\left( {\frac{{2n + 1}}{3} – \frac{{n + 1}}{2}} \right)} $
$ = \sqrt {\frac{{n + 1}}{2}\left( {\frac{{4n + 2 – 3n – 3}}{6}} \right)} $
$ = \sqrt {\frac{{{n^2} – 1}}{{12}}} $.
Standard 11
Mathematics
Similar Questions
The data is obtained in tabular form as follows.
| ${x_i}$ | $60$ | $61$ | $62$ | $63$ | $64$ | $65$ | $66$ | $67$ | $68$ |
| ${f_i}$ | $2$ | $1$ | $12$ | $29$ | $25$ | $12$ | $10$ | $4$ | $5$ |
hard
