The mean and variance of eight observations are $9$ and $9.25,$ respectively. If six of the observations are $6,7,10,12,12$ and $13,$ find the remaining two observations.
Let the remaining two observations be $x$ and $y$.
Therefore, the observations are $6,7,10,12,12,13, x, y$
Mean, $\bar{x}=\frac{6+7+10+12+12+13+x+y}{8}=9$
$\Rightarrow 60+x+y=72$
$\Rightarrow x+y=12$ ...........$(1)$
Variance $ = 9.25 = \frac{1}{n}\sum\limits_{i = 1}^8 {{{\left( {{x_i} - \bar x} \right)}^2}} $
$9.25=\frac{1}{8}[(-3)^{2}+(-2)^{2}+(1)^{2}+(3)^{2}+(4)^{2}$
$+x^{2}+y^{2}-2 \times 9(x+y)+2 \times(9)^{2}]$
$9.25=\frac{1}{8}\left[9+4+1+9+9+16+x^{2}+y^{2}-18(12)+162\right]$ ........[ using $(1)$ ]
$9.25=\frac{1}{8}\left[48+x^{2}+y^{2}-216+162\right]$
$9.25=\frac{1}{8}\left[x^{2}+y^{2}-6\right]$
$\Rightarrow x^{2}+y^{2}=80$ .........$(2)$
From $(1),$ we obtain
$x^{2}+y^{2}+2 x y=144$ ........$(3)$
From $(2)$ and $(3),$ we obtain
$2 x y=64$ ..........$(4)$
Subtracting $(4)$ from $(2),$ we obtain
$x^{2}+y^{2}-2 x y=80-64=16$
$\Rightarrow x-y=\pm 4 $ ...........$(5)$
Therefore, from $(1)$ and $(5),$ we obtain
$x=8$ and $y=4,$ when $x-y=4$
$x=4$ and $y=8,$ when $x-y=-4$
Thus, the remaining observations are $4$ and $8$
Let the observations $\mathrm{x}_{\mathrm{i}}(1 \leq \mathrm{i} \leq 10)$ satisfy the equations, $\sum\limits_{i=1}^{10}\left(x_{i}-5\right)=10$ and $\sum\limits_{i=1}^{10}\left(x_{i}-5\right)^{2}=40$ If $\mu$ and $\lambda$ are the mean and the variance of the observations, $\mathrm{x}_{1}-3, \mathrm{x}_{2}-3, \ldots ., \mathrm{x}_{10}-3,$ then the ordered pair $(\mu, \lambda)$ is equal to :
Find the mean and variance for the first $n$ natural numbers
The mean and variance of $7$ observations are $8$ and $16$ respectively. If two observations are $6$ and $8 ,$ then the variance of the remaining $5$ observations is:
Mean and standard deviation of 100 observations were found to be 40 and 10 , respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively, find the correct standard deviation.