Summary: Iterative methods for finding accurate estimates of percentage points are usually based on numerical root finding techniques applied to the distribution function. In this paper we take a different approach by creating an auxiliary function, whose fixed point is shown to be the desired percentage point, and then applying Steffenson's acceleration technique to find the fixed point. For the auxiliary function used, the conditions for convergence of Steffensen's method are mild and are satisfied by many percentage point problems arising in practice. The method is tested by calculating four-place \(\chi^ 2\) percentage points and two-place \(F\) percentage points agree with those in published tables.