MCQ
The variance of first $50$ even natural numbers is
- A$437$
- B$\frac{{437}}{4}$
- C$\frac{{833}}{4}$
- ✓$833$
✓
Answer
Correct option: D.
$833$
d
$2,4,6,8,......,98,100$
$2,4,6,8,......,98,100$
${\sigma ^2} = \frac{{\sum x_1^2}}{n} - {\left( {\overline {.x} } \right)^2}$
$\frac{{{2^2} + {4^2} + {6^2} + .... + {{100}^2}}}{{50}}$$ - {\left( {\frac{{2 + 4 + 6 + .... + 100}}{{50}}} \right)^2}$
${i_1} = \frac{{{2^2} + {4^2} + {6^2} + .... + {{100}^2}}}{{50}}$
$ = {2^2}\frac{{{1^2} + {2^2} + {3^2} + ... + {{50}^2}}}{{50}}$
$ = \frac{{{2^2}}}{{50}} \times 50\left( {50 + 1} \right)\left( {100 + 1} \right)$
$ = 3434$
${i_2} = {\left( {\frac{{2 + 4 + 6 + ..... + 100}}{{50}}} \right)^2}$
$ = {\left( {\frac{{50 \times \frac{{2 + 100}}{2}}}{{50}}} \right)^2}$
$ = {\left( {51} \right)^2}$
${\sigma ^2} = 3434 - 2661 = 833$
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