In probability theory, the Central Limit Theorem (CLT) posits that the distribution of a sample variable approximates a normal distribution (i.e., a ‘bell curve’) as the sample size increases, regardless of the population’s actual distribution shape. This means that, with 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 close to the mean of the whole population. These samples approximate a normal distribution, with variances being approximately equal to the population variance as the sample size grows, adhering to the law of large numbers.
The concept of the Central Limit Theorem was first developed by Abraham de Moivre in 1733 and was formalized in 1920 by Hungarian mathematician George Pólya.
Key Takeaways
- The CLT states that the distribution of sample means approximates a normal distribution as the sample size increases, regardless of the population’s distribution.
- Sample sizes equal to or greater than 30 are typically considered sufficient for the CLT to hold.
- The average of sample means and standard deviations will equal the population mean and standard deviation as sample sizes become large enough.
- A sufficiently large sample size can predict population characteristics more accurately.
- CLT is vital in finance for estimating portfolio distributions and traits, such as returns, risk, and correlation.
Understanding the Central Limit Theorem (CLT)
According to the central limit theorem, the mean of a sample of data will approach the mean of the overall population as the sample size increases, regardless of the data’s distribution. In simple terms, the data becomes more accurate whether the distribution is normal or abnormal.
Generally, sample sizes of 30 are considered sufficient for the CLT to hold, meaning that the distribution of the sample means is fairly normally distributed. Therefore, the more samples you take, the more the graphed results resemble a normal distribution.
The central limit theorem often works in conjunction with the law of large numbers, stating that the average of the sample means will approach the population mean as the sample size grows. This is crucial for predicting population characteristics accurately.
Key Components of the Central Limit Theorem
The CLT comprises several key characteristics focused on samples, sample sizes, and population data:
- Sampling is successive: Some sample units are common with units selected in previous sampling occasions.
- Sampling is random: All samples must be selected at random, ensuring they have the same statistical probability of being selected.
- Samples should be independent: The results of one sample should not affect subsequent samples or their results.
- Large sample size: As sample size increases, the sampling distribution begins to approach a normal distribution.
The Central Limit Theorem in Finance
The CLT proves useful when analyzing individual stock returns or broader indices due to the relative ease of generating the necessary financial data. Investors rely on the CLT for analyzing stock returns, constructing portfolios, and managing risk.
For instance, suppose an investor wants to analyze the overall return for a stock index comprising 1,000 equities. They may study a random sample of stocks to estimate the returns of the total index. To be thorough, at least 30-50 randomly selected stocks across various sectors should be sampled for the CLT to hold. Additionally, previously selected stocks must be replaced with different names to eliminate bias.
Why Is the Central Limit Theorem Useful?
The CLT is valuable for analyzing large datasets because it allows for assuming that the sampling distribution of the mean will be normally-distributed in most cases. This facilitates easier statistical analysis and inference. For example, investors can use the CLT to aggregate individual security performance data, generating a distribution of sample means that represents the larger population distribution for security returns over a period.
Why Is the Central Limit Theorem’s Minimize Sample Size 30?
A sample size of 30 is commonly applied across statistics as the minimum for the CLT’s application. The larger the sample size, the more representative it is of the population set.
What Is the Formula for Central Limit Theorem?
The CLT doesn’t have a specific formula for its practical application. Its principle is implicit: with a sufficiently large sample size, the sample distribution will approximate a normal distribution, and the sample mean will approach the population mean. Thus, with a sample size of at least 30, you can begin to analyze the data as if it fitted a normal distribution.
Related Terms: Normal Distribution, Law of Large Numbers, Variance, Mean.
References
- Hans Fischer. “A History of the Central Limit Theorem”. Page 1. Springer, 2011.
- Stark, Benjamin A. “Studying Moments of the Central Limit Theorem”. *The Mathematics Enthusiast,*Vol 14, No. 1, 2017, pp. 53-76.
- Boston University School of Public Health. “Central Limit Theorem”.
- University of Massachusetts Amherst. “What Is Central Limit Theorem? Properties, Best Practices, Examples & Everything To Know”.
- Emory University. “Final Summary The Central Limit Theorem”.
- Chang, H. J., K. Huang, and C. Wu. “Determination of Sample Size in Using Central Limit Theorem for Weibull Distribution”. *International Journal of Information and Management Sciences,*Vol. 17, No. 3. 2006, pp. 153-174.
