Gujarati
13.Statistics
easy

प्रथम $n$ प्राकृत संख्याओं का मानक विचलन $(S.D.)$ है

A

$\frac{{n + 1}}{2}$

B

$\sqrt {\frac{{n(n + 1)}}{2}} $

C

$\sqrt {\frac{{{n^2} - 1}}{{12}}} $

D

इनमें से कोई नहीं

Solution

(c) प्रथम $n$ प्राकृत संख्याओं का मानक विचलन
$ = \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

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