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If V is the variance and M is the mean of first 15 natural numbers then what is $V+M^{2}$ equal to ?
Given :
V is the variance and M is the mean of first 15 natural numbers.
To find :
We have to find $V+M^{2}$.
Solution :
The first 15 natural numbers are 1,2,3,....14,15.
$ Mean = \frac{1}{N}\sum x_{i}$
$ =\frac{1}{15}( 1+2+3+....+15)$
$ =\frac{1}{15} \times \frac{( 15\times 16)}{2}$
$ = 8$
$Variance = \frac{1}{N}\sum ( x_{i} -\mu )^{2} $
$ =\frac{1}{15} \times \left[( 1-8)^{2} +( 2-8)^{2} +....+( 15-8)^{2}\right]$
$=\frac{1}{15} \times \left[ 7^{2} +6^{2} +5^{2} +4^{2} +3^{2} +2^{2} +1^{2} +0^{2} +1^{2} +...+7^{2}\right]$
$ =\frac{1}{15} \times 2[ 49+36+25+16+9+4+1]$
$=\frac{1}{15} \times 2\times 140$
$ = \frac{56}{3}$
Therefore,
$V+M^{2} =\frac{56}{3} +8^{2}$
$ = \frac{56}{3} +64$
$=\frac{56+64\times 3}{3}$
$=\frac{56+192}{3}$
$ =\frac{248}{3}$ .
The value of $V+M^{2}$ is $\frac{248}{3}$