Concept Explainer
Concept Explainer - Central Limit Theorem
Introduction
The central limit theorem is a core concept within statistics which speaks to the idea that any given sample distribution will converge with a normal distribution as the number of samples increase. We shall look into this theorem a bit more in depth below.
In-Depth Description
The Laws of Large Numbers state that as a sample size increases, the sample mean will become equal to/converge with the population mean.
The Central Limit Theorem states that as the sample size increases, the distribution of the mean of samples will become equal to/converge with a normal distribution, regardless of the population’s distribution. This means that even if the population is skewed, the resulting sample distribution of means will not be.
A sample size of 30 or greater is generally considered to be sufficient for the Central Limit Theorem to hold.However, large sample sizes greater than 30 will lead to the distribution of the sample means fitting a normal distribution increasingly well.
Mathematical Definition:
This formula represents the Central Limit Theorem, depicting that as the limit of the sample size (n) approaches infinity, the probability that the sample mean will be less than or equal to any given value will be equal to the CDF (cumulative distribution function) of the standard normal distribution. This means that the sample distribution converges with the normal distribution.
Standard Deviation Vs Sample Size
The standard deviation of the sampling distribution is:
\(\sigma/n\) Where \(\sigma\) is the population standard deviation, and \(n\) is the number of samples.
This formula provides some insight about what occurs with the standard deviation as the sample size changes. The sample size acts as the denominator here, so an increase in sample size leads to a decrease in the standard deviation of the sampling distribution. Therefore, the result is that an increasing sample size leads to the sample means being more tightly distributed about the mean.
This plot demonstrates the Central Limit Theorem in action. In each of the three cases, 1000 samples were taken at random among a population, and the mean of each individual sample was calculated. The distribution of these sample means is what this plot displays. The only changing parameter in this example is the sample size itself. As the sample size increases, it becomes visually clear that the distribution of the sample means begins to fit a normal distribution.This is despite the fact that the original population data was right-skewed, as can be seen from the cases where n=3 or n=5.
Conclusion
The Central Limit Theorem is a key theorem within the world of statistics. It is useful in many situations, as it allows statisticians to evaluate sample results even when the distribution of the population is unknown or not normal.
References
Ganti, A. (2023). Central Limit Theorem (CLT): Definition and Key Characteristics. Investopedia. https://www.investopedia.com/terms/c/central_limit_theorem.asp
Frost, J. (2018). Central Limit Theorem Explained. Statistics by Jim. https://statisticsbyjim.com/basics/central-limit-theorem/
Anonymous. (n.d.). Central Limit Theorem: Definitions and Examples. Statistics How To. https://www.statisticshowto.com/probability-and-statistics/normal-distributions/central-limit-theorem-definition-examples/
OpenAI. (2023). ChatGPT (Mar 14 version) [Large language model]. https://chat.openai.com/chat (This was used specifically only for coding assistance when creating the CLT demonstration graph).