Central Limit Theorem

• 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 bell curve) as the sample size becomes larger, regardless of the population's actual distribution share.
  • 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 number.

KEY TAKEAWAYS:

• The central limit theorem (CLT) states that the distribution of sample means approximates a normal distribution as the sample size gets larger, regardless of the population's distribution

• A sufficiently large sample size can predict the characteristics of a population more accurately.

• Sample sizes equal to or greater than 30 are often 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 to the population mean and standard deviation.

• CLT is useful in finance and investing when analyzing a large collection of securities to estimate portfolio distributions and traits for return, risk, and correlation.

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 graphed results should take the shape of aa 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 a very large populations.

Key Components of the Central Limit Theorem

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 1000 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

Why Is the Central Limit Theorem Useful

• The central limit theorem is useful when analyzing large data sets because it allows one to assume that the mean will be normally distributed in most cases. This allows for easier statistical analysis and inference. For example, investor can use central limit theorem to aggregate individual security performance data and generate distribution of sample means that represent a larger population distribution for security returns over a period of time.

What is the Formula for Central Limit Theorem?

• The central limit theorem doesn't have a formula used in its practical application. Its principle is simply applied. With a sufficiently large sample size, the sample distribution will approximate a normal distribution, and the sample mean will approach the population mean. It suggests that if we have a sample size of at least 30, we can begin to analyze the data as if it fit a normal distribution

Why is the Central Limit Theorem's Minimum Sample Size 30?

• A sample size of 30 or more is fairly common across statistics as the minimum for applying the central limit theorem. The greater your sample size, the more likely the sample will be representative of your population set.

What is the Law of Large Numbers?

• In the probability and statistics, the law of large numbers states that the larger sample size, the more likely its mean is to reflect the mean of the entire population
• In business, the law of large numbers can have a different meaning, specifically that as a company grows in size, maintaining its rate of growth in percentage terms becomes more difficult.

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 (30 data points) rather than the entire population.