If the mean and the standard deviation of the data 3,5,7,a,b are 5 and 2 respectively, then a and b are the roots of the equation:

$\text{ Given that the mean of the data is }5,\Rightarrow \frac{3+5+7+a+b}{5}=5\Rightarrow a+b=10$

The standard deviation is 2, the variance is 4

          Hence,

$\frac{9+25+49+a^2+b^2}{5}-{\left(5\right)}^2=4\Rightarrow a^2+b^2=62$

$\text{ By solving the above two equations }ab=19$

$\text{ Therefore, the equation whose roots are }a,b\text{ is }{x}^2-10x+19=0$