Recent determinations of tectonic rotation and flattening from paleomagnetic data have exaggerated the size of the confidence limits or error bars by about 25%. This has been caused by the misuse of conventional error propagation formulas to derive one degree of freedom error bars from a two dimensional cone of confidence (α95). An analytical calculation, which is valid when error bars are small, shows that the correct error bars are 75% to 80% of the values derived in previous studies. I have used numerical integration to check on the applicability of the analytical solution and to calculate error bars when the analytical approximation cannot be used. The analytical approximation may be used whenever α95 is less than 10¿ and the mean inclination is not too close to vertical. Confidence limits given for tectonic rotation and flattening in recent literature are too large and should be reduced by 20--25%. Narrowing the error bars in this way permits a more accurate interpretation of paleomagnetic data.