Central Limit Theorem (CLT): Definition and Key Characteristics

What Is the Central Limit Theorem (CLT)?

In probability theory, the central limit theorem (CLT) states that the distribution of a sample will approximate a normal distribution (i.e., a bell curve) as the sample size becomes larger, regardless of the population's actual distribution shape.

Put another way, CLT is a statistical premise that, given a sufficiently large sample size from a population with a finite level of variance, the mean of all sampled variables from the same population will be approximately equal to the mean of the whole population. Furthermore, these samples will approximate a normal distribution, with their variances being approximately equal to the variance of the population as the sample size gets larger, according to the law of large numbers.

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. The concept can hold true regardless of whether the distribution of the population is normal or skewed.

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

The central limit theorem is often used in conjunction with the law of large numbers, which states that the average of the sample means will come closer to equaling the population mean as the sample size grows. This concept can be extremely useful in accurately predicting the characteristics of very large populations.

Although this concept was first developed by Abraham de Moivre in 1733, it was not formalized until 1920, when the Hungarian mathematician George Pólya dubbed it the central limit theorem.

Key Components of the Central Limit Theorem

The central limit theorem has several key components. They largely revolve around sampling technique.

1. Sampling is successive: This means some sample units are common with sample units selected on previous occasions.
2. Sampling is random: All samples must be selected at random so that they have the same statistical possibility of being selected.
3. Samples should be independent: The selections or results from one sample should have no bearing on future samples or other sample results.
4. Large sample size: As sample size increases, the sampling distribution should come ever closer to the normal distribution.

The Central Limit Theorem in Finance and Investing

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

Suppose, for example, that an investor wishes to analyze the overall return for a stock index that consists of 1,000 different equities. In this scenario, the investor may simply study a random sample of stocks to arrive at an estimated return for the total index. To be safe in this instance, at least 30 to 50 randomly selected stocks across various sectors should be sampled for the central limit theorem to hold.

The Bottom Line

The central limit theorem (CLT) holds that as a sample size gets larger, its mean will increasingly approximate the mean in a normal distribution. This concept can be useful in many applications, such as analyzing investment returns, because it requires only a sufficient sample size (generally interpreted as 30 or more data points) rather than the entire population.