Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in statistics and probability theory that describes the behavior of the sample mean of a sufficiently large number of independent, identically distributed random variables. It is one of the most important theorems in statistics and has far-reaching applications in various fields, including hypothesis testing, confidence intervals, and inferential statistics.

The Central Limit Theorem states that regardless of the underlying distribution of the population from which the random variables are drawn, the distribution of the sample means tends to follow a normal (Gaussian) distribution as the sample size increases, even if the original population distribution is not normal.

Key characteristics of the Central Limit Theorem include:

1. Large Sample Size: 

The CLT holds true when the sample size is sufficiently large (typically n ≥ 30). The larger the sample size, the closer the sample mean distribution approaches a normal distribution.

2. Independent and Identically Distributed (IID) Samples: 

The random variables in the sample must be independent of each other, and each variable must be drawn from the same population distribution.

3. Convergence to a Normal Distribution: 

As the sample size increases, the distribution of the sample means approaches a normal distribution with the same mean as the original population and a standard deviation equal to the population standard deviation divided by the square root of the sample size (σ / √n).

Mathematically, if X₁, X₂, ..., Xₙ are independent and identically distributed random variables with mean μ and standard deviation σ, then the sample mean X̄ is defined as:

X̄ = (X₁ + X₂ + ... + Xₙ) / n

The Central Limit Theorem allows statisticians to make inferences about the population mean even when the underlying distribution is unknown or non-normal. It forms the basis for many statistical methods and is widely used in hypothesis testing, confidence intervals, and other inferential techniques.

However, it's important to note that the CLT works best when the sample size is reasonably large, and there are some conditions and limitations to its applicability, especially when dealing with heavily skewed or fat-tailed distributions.