When measuring the form errors of precision optics with an interferometer, calibration of the reference wavefront is of central importance. In recent years, ball averaging, or random ball testing, has emerged as a robust method for calibrating spherical reference wavefronts in converging beams. We describe a simple instrument, consisting of an air bearing and two electric motors, that can rotate the test ball around three axes as required for a ball averaging test. The performance of the instrument is demonstrated by using it to calibrate a concave transmission sphere. Further we discuss the effects of image sampling at random locations or on uniform grids, and the effect of correlated measurements. Finally, we describe a method to determine the number of measurements which are sufficient for a ball averaging calibration.© (2005) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.