Let $\rm a = \frac{15}{4}$ and $\rm b = \frac{19}{4}$.

The arithmetic mean between two numbers a and b is given by $\rm m = \frac{a + b}{2}$.

$\rm m = \frac{ \frac{15}{4} + \frac{19}{4} }{2}$

$\rm = \frac{ \frac{15 + 19}{4} }{2}$

$\rm = \frac{ 34}{8}$

$\rm = \frac{17}{4}$

Hence, the required arithmetic mean between the given two numbers is $\rm \frac{17}{4}.