Why is the central limit theorem helpfull? NO ONE HAS BEEN ABLE TO ASNSWER THIS ONE CORRECTLY SO FAR!!!?

Answer 1

The distribution of an average tends to be Normal, even when the distribution from which the average is computed is decidedly non-Normal. Furthermore, this normal distribution will have the same mean as the parent distribution, AND, variance equal to the variance of the parent divided by the sample size.

The Central Limit Theorem describes the relation of a sample mean to the population mean. If the population mean doesn't exist, then the CLT doesn't apply and the characteristics of the sample mean, Xbar, are not predictable. Attention to detail is needed here: You can always compute the numerical mean of a finite number of observations from any density (if every observation is finite). But the population mean is defined as an integral, which diverges for the Cauchy, so even though a sample mean is finite, the population mean is not. The Cauchy has another interesting property - the distribution of the sample average is that same as the distribution of an individual observation, so the scatter never diminishes, regardless of sample size.

Answer 2

Most of the methods one learns about in a standard statistics course work because of the central limit theorem. For example, whenever you read in the paper that 54% of voters support such and such with a margin of error of plus of minus 3% (usually with the implied understanding that the confidence level is 95%) then while the 54% comes from the raw data, the margin of error and the confidence level come from treating the sample proportion as if it had a normal distribution. The central limit theorem assures that this is valid for large samples.