### What Is the Central Limit Theorem (CLT)?

In the study of probability theory, the central limit theorem (CLT) states that the distribution of sample means approximates a normal distribution (also known as a “bell curve”), as the sample size becomes larger, assuming that all samples are identical in size, and regardless of the population distribution shape.

Said another way, CLT is a statistical theory stating 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.

Although this concept was first developed by Abraham de Moivre in 1733, it wasn’t formally named until 1930, when noted Hungarian mathematician George Polya officially dubbed it the Central Limit Theorem.

#### Central Limit Theorem

### Understanding the Central Limit Theorem (CLT)

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. In other words, the data is accurate whether the distribution is normal or aberrant.

As a general rule, sample sizes equal to or greater than 30 are deemed sufficient for the CLT to hold, meaning that the distribution of the sample means is fairly normally distributed. Therefore, the more samples one takes, the more the graphed results take the shape of a normal distribution.

Central Limit Theorem exhibits a phenomenon where the average of the sample means and standard deviations equal the population mean and standard deviation, which is extremely useful in accurately predicting the characteristics of populations.

### Key Takeaways

- The central limit theorem (CLT) states that the distribution of sample means approximates a normal distribution as the sample size gets larger.
- Sample sizes equal to or greater than 30 are considered sufficient for the CLT to hold.
- A key aspect of CLT is that the average of the sample means and standard deviations will equal the population mean and standard deviation.
- A sufficiently large sample size can predict the characteristics of a population accurately.

### The Central Limit Theorem in Finance

The CLT is useful when examining the returns of an individual stock or broader indices, because analysis is simple, due to the relative ease of generating the necessary financial data. Consequently, investors of all types rely on the CLT to analyze stock returns, construct portfolios, and manage risk.

Say, for example, an investor wishes to analyze the overall return for a stock index that comprises 1,000 equities. In this scenario, that investor may simply study a random sample of stocks, to cultivate estimated returns of the total index. At least 30 randomly selected stocks, across various sectors must be sampled, for the central limit theorem to hold. Furthermore, previously-selected stocks must be swapped out with different names, to help eliminate bias.