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પ્રથમ $20$ પ્રાકૃતિક સંખ્યાઓનું વિચરણ શોધો.
$\frac{{133}}{4}$
$\frac{{379}}{{12}}$
$\frac{{133}}{2}$
$\frac{{399}}{4}$
Solution
$\because \,\,\,{\sigma ^2}\, = \,\,\,\frac{{\Sigma x_i^2}}{n}\,\, – \,\,{\left( {\frac{{\Sigma {x_i}}}{n}} \right)^2}$
$ = \,\,\frac{1}{{20}}\,[{1^2}\, + \,\,{2^2}\, + \,\,……\,\, + \,\,{20^2}]\,\, – \,\,{\left[ {\frac{1}{{20}}(1\,\, + \,\,2\,\, + \,\,….\,\, + \,\,20)} \right]^2}$
$ = \,\,\frac{1}{{20}}\,\,\,\frac{{20\,\, \times \,\,21(2\,\, \times \,\,20\,\, + \,\,1)}}{6}\,\,\, – \,\,{\left[ {\frac{1}{{20}}\,\,\frac{{20\,\, \times \,\,21}}{2}} \right]^2}\,\,\,$
$\, = \,\,\frac{{7\,\, \times \,\,41}}{2}\,\, – \,\,\frac{{441}}{2}\,\,\,\,\, = \,\,\frac{{133}}{4}\,.$
હકીકત, માં પ્રથમ $n\, – $ પ્રાકૃતિક સંખ્યાઓનું વિચરણ $ \frac{{{n^2}\, – \,\,1}}{{12}}$ થાય છે.
Similar Questions
નીચે આપેલ માહિતી માટે વિચરણ અને પ્રમાણિત વિચલન શોધો :
| ${x_i}$ | $4$ | $8$ | $11$ | $17$ | $20$ | $24$ | $32$ |
| ${f_i}$ | $3$ | $5$ | $9$ | $5$ | $4$ | $3$ | $1$ |
