The mean and variance of 7 observations are 8 and 16, respectively. If five observations are 2, 4, 10, 12, 14, then the absolute difference of the remaining two observations is :

$\overline x = {{2 + 4 + + 10 + 12 + 14 + x + y} \over 7} = 8$<br><br>x + y = 14 ....(i)<br><br>$${(\sigma )^2} = {{\sum {{{({x_i})}^2}} } \over n} - {\left( {{{\sum {{x_i}} } \over n}} \right)^2}$$<br><br>$\Rightarrow$ $16 = {{4 + 16 + 100 + 144 + 196 + {x^2} + {y^2}} \over 2} - {8^2}$<br><br>$\Rightarrow$ $16 + 64 = {{460 + {x^2} + {y^2}} \over 7}$<br><br>$\Rightarrow$ 560 = 460 + x² + y²<br><br>$\Rightarrow$ x² + y² = 100 ......(ii)<br><br>Clearly by (i) and (ii), <br><br>(x + y)² - 2xy = 100 <br><br>$\Rightarrow$ (14)² - 2xy = 100 <br><br>$\Rightarrow$ 2xy = 96 <br><br>$\Rightarrow$ xy = 48 <br><br>Now, |x - y| = $\sqrt {{{\left( {x + y} \right)}^2} - 4xy}$ <br><br>= $\sqrt {196 - 192}$ <br><br>= 2