We propose an analytic equation for approximating expected root mean square (RMS) target registration error (TRE) for rigid shape-based registration where measured noisy data points are matched to a rigid shape. The noise distribution of the data points is assumed to be zero-mean, independent, and non-identical; i.e., the noise covariance may be different for each data point. The equation was derived by extending a previously published spatial stiffness model of registration. The equation was validated by performing registration experiments with both synthetic registration data and data collected using an optically tracked pointing stylus. The synthetic registration data were generated from the surface of an ellipsoid. The optically tracked data were collected from three plastic replicas of human radii and registered to isosurface models of the radii computed from CT scans. Noise covariances for the data points were computed by considering the pose of the tracked stylus, the positions of the individual fiducial markers on the stylus coordinate reference frame, and the calibrated position of the stylus tip; these quantities and an estimate of the fiducial localization covariance of the tracking system were used as inputs to a previously published algorithm for estimating the covariance of TRE for point-based (fiducial) registration. Registration simulations were performed using a modified version of the iterated closest point algorithm and the resulting RMS TREs were compared to the values predicted by our analytic equation.