20.3 The Central Limit Theorem (CLT)

The Central Limit Theorem (CLT) is a fundamental concept in statistics that describes the behavior of sample means. It states that, regardless of the original distribution of the population, the distribution of sample means will approach a normal distribution as the sample size increases.

Key Principles of the Central Limit Theorem

• Independent and Random Samples: The CLT applies to samples that are drawn independently and randomly from the population. This means that the selection of one sample member does not influence the selection of another.
• Robustness to Population Distribution: A remarkable aspect of the CLT is its applicability even when the underlying population distribution is not normal. This is crucial for many statistical inference methods.
• Sample Size Requirement: For the CLT to hold well and for the sampling distribution of the sample mean to closely approximate a normal distribution, the sample size ($n$) typically needs to be 30 or greater. Smaller sample sizes might require the population distribution to be closer to normal.
• Mean of the Sampling Distribution: The mean of the sampling distribution of the sample mean is equal to the population mean ($\mu$).
• Standard Deviation of the Sampling Distribution (Standard Error): The standard deviation of the sampling distribution of the sample mean, often referred to as the "standard error," is calculated by dividing the population standard deviation ($\sigma$) by the square root of the sample size ($n$).

Importance of the Central Limit Theorem

The CLT is vital for several reasons:

• Justification for Normal Probability Models: It provides the theoretical basis for using normal probability models for statistical inference concerning sample means. This is because even with non-normal populations, the distribution of sample means will be approximately normal, allowing us to apply the well-understood properties of the normal distribution.
• Enabling Statistical Inference: The CLT empowers statisticians to perform hypothesis testing and construct confidence intervals for population means, even when the population distribution is unknown or non-normal, provided the sample size is sufficiently large.

Formula for Standard Error

The standard error (SE) of the sample mean is calculated as:

SE = σ / √n

Where:

• $\sigma$ (sigma) = the population standard deviation.
• $n$ = the sample size.

Illustrative Example

Imagine a population of test scores that is heavily skewed to the right (e.g., most students score low, but a few score very high).

• Population Distribution: Not Normal (skewed).
• Take many random samples: If you were to take many random samples of size $n=5$ from this population and calculate the mean for each sample, the distribution of these sample means would still likely be somewhat skewed.
• Increase sample size: However, if you increase your sample size to $n=30$ (or more) and repeat the process of taking many random samples and calculating their means, the distribution of those sample means would begin to look very much like a normal distribution, centered around the true population mean.

Interview Questions Related to CLT

Here are common interview questions that test understanding of the Central Limit Theorem:

• What is the Central Limit Theorem (CLT)?
• Why is the Central Limit Theorem important in statistics?
• How does the CLT apply to non-normal population distributions?
• What is the typical sample size required for the CLT to hold?
• How is the standard error calculated according to the CLT?
• What does the sampling distribution of the sample mean represent?
• How does the CLT justify using the normal distribution for inference?
• Can you explain how CLT is used in hypothesis testing?
• What are the assumptions required for the Central Limit Theorem?
• How does increasing sample size affect the sampling distribution?