Mean $=5,$ Variance $=10$
$\frac{3+7+x+y}{4}=5,10=\frac{\sum x_i^2}{4}-{\left(\sum x\right)}^2$

$10+x+y=20, 10=\frac{\sum x_i^2}{4}-25$

$x+y=10$

$y=10-x, 40=\sum x_i^2-100$

$140=\sum x_i^2$

$140=9+49+x^2+y^2$
$140=58+x^2+y^2$
$x^2+y^2=82$
${\left(x+y\right)}^2=x^2+y^2+2xy$
$100-82=2xy$
$18=2xy$
$xy=9$
$x\left(10-x\right)=9$
$\Rightarrow x^2-10x+9=0$
$=x=1,9$
$y=9,1$

As $x>y$

So, $x=9, y=1$

$\overline{x}=\frac{3+2x+7+2y+x+y+x-y}{4}$

$\Rightarrow \frac{10+4x+2y}{4}$

$\Rightarrow \frac{5+2x+y}{2}$

$\Rightarrow \frac{5+18+1}{2}=\frac{24}{2}=12$