Central Limit Theorem (CLT) is a statistical theory that states that given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population. Furthermore, all of the samples will follow an approximately normal distribution pattern, with all variances being approximately equal to the variance of the population divided by each sample’s size. According to the central limit theorem, the mean of a sample of data will be closer to the mean of the overall population in question as the sample size increases, notwithstanding the actual distribution of the data, and whether it is normal or non-normal. As a general rule, sample sizes equal to or greater than 30 are considered sufficient for the central limit theorem to hold, meaning the distribution of the sample means is fairly normally distributed. The central limit theorem is the basis for sampling in statistics, so it holds the foundation for sampling and statistical analysis in finance as well. Investors of all types rely on the central limit theorem to analyze stock returns, construct portfolios and manage risk.

CLT along with Cochran’s theorem, are among the main theorems of statistics. It tells us that sample means and many other important quantities such as maximum likelihood estimators in regular problems have sampling distributions that converge to the Gaussian. Given that the sampling distributions of most quantities we want to use are essentially intractable, this turns out to be highly fortunate.

It is:
– Used in hypothesis testing
– Used for confidence intervals

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