An approximation for the error of the normal approximation to a linear combination of independently distributed random variables

A new approximation introduced elsewhere is employed to approximate the error associated with the central limit approximation. In particular, the respective error obtained on approximating a linear combination of n independently distributed random variables (S_n) is examined, and it is shown that the unstandardized error is approximately independent of n. Two examples, for a discrete and for a continuous S_n, demonstrate that if a correction term, based on the above error approximation, is added to the traditional central limit approximation a remarkable improvement of accuracy ensues. Some implications of the new approximation are probed.