Central Limit Theorem (CLT)

The central limit theorem states that the distribution of sample means approximates a normal distribution as the sample size gets larger (assuming that all samples are identical in size), regardless of population distribution shape.

Said another way, 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 the samples will follow an approximate 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. This concept was first discovered by Abraham de Moivre in 1733 though it wasn't named until George Polya, in 1920, dubbed it the Central Limit Theorem.

As a general rule, sample sizes equal to or greater than 30 are considered sufficient for the CLT to hold, meaning the distribution of the sample means is fairly normally distributed. So, the more samples that one takes results in the graph looking more and more like that of a normal distribution.

Another key aspect of CLT is that the average of the sample means and standard deviations will equal the population mean and standard deviation. This is extremely useful in predicting the characteristics of a population with a high degree of accuracy.

The CLT is useful when examining returns for a stock or index because it simplifies many analysis procedures. An appropriate sample size depends on the data available, but generally, having a sample size of at least 50 observations is sufficient. Due to the relative ease of generating financial data, it is often easy to produce much larger sample sizes. The CLT is the basis for sampling in statistics, so it holds the foundation for sampling and statistical analysis in finance too. Investors of all types rely on the CLT to analyze stock returns, construct portfolios and manage risk.

If an investor is looking to analyze the overall return for a stock index made up of 1,000 stocks, he or she can take random samples of stocks from the index to get an estimate for the return of the total index. The samples must be random, and he or she must evaluate at least 30 stocks in each sample for the central limit theorem to hold. Random samples ensure a broad range of stock across industries and sectors is represented in the sample. Stocks previously selected must also be replaced for selection in other samples to avoid bias. The average returns from these samples approximates the return for the whole index and are approximately normally distributed. The approximation holds even if the actual returns for the whole index are not normally distributed.