We know the Sample mean estimates the true population mean . We also know by the Law of large numbers that as approaches infinity, approaches .

However, our sample size is usually never infinite. Without using LLN, can we still get an accurate measurement of the error of without ?


does not need to approach infinity. However, we can say that as the sample size increases, will approach the Normal, even if are not Normal.

In other words, for a large enough size of , we have approximately


Let be the standardized version (see Standardization) of . That is, given and , we have

Then for every ,

We say that converges in distribution (or in law) to . That is, for large enough values of , is approximately equal to .