Solution:

For a point \( Q \) on the perpendicular bisector of an electric dipole (distance \( r \) from the center, where \( r \gg \text{dipole length} \)):

1. Electric Field on Perpendicular Bisector: The electric field \( E \) at a point on the perpendicular bisector of a dipole is given by:
\[
E \propto \frac{p}{r^3}
\]

2. Dependence:
- Directly proportional to the dipole moment \( p \).
- Inversely proportional to \( r^3 \).

Answer:
The electric field at \( Q \) is proportional to:
\[
p \quad \text{and} \quad r^{-3}
\]