To find the midpoint \( M \) of a line segment with endpoints \( P_1 \left( \frac{11}{2}, \frac{3}{8} \right) \) and \( P_2 \left( -\frac{5}{2}, \frac{5}{8} \right) \), we use the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( (x_1, y_1) = P_1 \) and \( (x_2, y_2) = P_2 \).
Step 1: Calculate the x-coordinate of the midpoint
\[ x_1 = \frac{11}{2}, \quad x_2 = -\frac{5}{2} \]
\[ \text{Midpoint } x = \frac{\frac{11}{2} + \left( -\frac{5}{2} \right)}{2} = \frac{\frac{11 - 5}{2}}{2} = \frac{\frac{6}{2}}{2} = \frac{6}{4} = \frac{3}{2} \]
Step 2: Calculate the y-coordinate of the midpoint
\[ y_1 = \frac{3}{8}, \quad y_2 = \frac{5}{8} \]
\[ \text{Midpoint } y = \frac{\frac{3}{8} + \frac{5}{8}}{2} = \frac{\frac{3 + 5}{8}}{2} = \frac{\frac{8}{8}}{2} = \frac{1}{2} \]
Step 3: Combine the coordinates
So the midpoint \( M \) is:
\[ M = \left( \frac{3}{2}, \frac{1}{2} \right) \]
Thus, the midpoint of the line segment is
\[ \boxed{\left( \frac{3}{2}, \frac{1}{2} \right)} \]