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In probability theory, the central limit theorem (CLT) states that as the sample size increases, the distribution of the sample variables approaches a normal distribution (i.e., the "bell curve"), regardless of the actual distribution shape of the population.
In probability theory, the central limit theorem (CLT) states that as the sample size increases, the distribution of the sample variables approaches a normal distribution (i.e., the "bell curve"), regardless of the actual distribution shape of the population.
In other words, CLT is a statistical premise that when a sufficiently large sample size is obtained from a population with limited variance, the mean of the selected sample variable will be approximately equal to the mean of the entire population. In addition, as the sample size increases, these samples will also approximate a normal distribution, and their variance will also approach the variance of the population, all of which is consistent with the law of large numbers.
Although the concept was first proposed by Abraham de Moffre in 1733, it was not officially called the Central Limit Theorem until 1920 by the Hungarian mathematician George Polya. [1]
According to the central limit theorem, as the sample size increases, the mean of the data sample will become closer to the mean of the population in question, regardless of the actual distribution of the data. In other words, the data is accurate regardless of whether the data distribution is normal or abnormal.
Generally, a sample size of around 30 is generally considered a sufficient condition for CLT to hold, which means that the distribution of the sample mean is relatively close to a normal distribution. Therefore, the more samples there are, the more the plot will resemble a normal distribution. [3]
The central limit theorem is often used in conjunction with the law of large numbers, which states that as the sample size increases, the average of the sample means will get closer and closer to the population mean, which is extremely useful for accurately predicting population characteristics.
The central limit theorem consists of several key features. These characteristics mainly revolve around sample, sample size and data population.
CLT is useful when examining the returns of individual stocks or broader indices because the analysis is relatively simple and it is relatively easy to generate the required financial data. As a result, investors of all types rely on CLT to analyze stock returns, construct portfolios, and manage risk.
For example, if an investor wishes to analyze the overall return of a stock index consisting of 1,000 stocks, the investor can randomly select a subset of stocks to estimate the expected return of the entire index. To ensure reliability, at least 30 to 50 random stocks from different sectors should be sampled for the central limit theorem to hold. Additionally, previously selected stocks should be replaced with different names to help eliminate bias.
The central limit theorem is very useful when analyzing large data sets because it allows one to assume that the sampling distribution of the sample mean will be normally distributed in most cases. This makes statistical analysis and inference simpler. For example, investors can use the central limit theorem to aggregate performance data for individual securities and generate a sample mean distribution that is representative of the overall security return distribution.
In statistics, a sample size of 30 is considered a common minimum for applying the central limit theorem. [6] The higher the sample size, the more likely the sample is representative of the overall data set.
The central limit theorem has no specific formula for practical applications. The principles are simple and applicable. As long as the sample size is large enough, the sample distribution will approximate a normal distribution and the sample mean will be close to the population mean. Therefore, if our sample size is at least 30, we can start to treat the data analysis as conforming to a normal distribution.
[1] Hans Fischer. "A History of the Central Limit Theorem." Page 1. Springer, 2011.
[2] Stark, Benjamin A. "Studying Moments of the Central Limit Theorem." The Mathematics Enthusiast, Vol 14, No. 1, 2017, pp. 53-76.
[3] Boston University School of Public Health. "Central Limit Theorem."
[4] University of Massachusetts Amherst. "What Is Central Limit Theorem? Properties, Best Practices, Examples & Everything To Know."
[5] Emory University. "Final Summary The Central Limit Theorem."
[6] 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.
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