Definition of the Central Limit Theorem
The central limit theorem (CLT) is a fundamental theorem in probability and statistics that states the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution, provided the samples are independent and identically distributed.
Key Principles and Conditions
The CLT relies on several key principles: the sample size n should be sufficiently large (typically n ≥ 30); the population variance must be finite; and observations must be independent. It explains why many natural phenomena and statistical estimates behave normally, enabling the use of normal distribution approximations for inference.
Practical Example
Consider measuring the heights of adults in a population with a skewed distribution. If random samples of size 50 are taken repeatedly, the means of these samples will form a bell-shaped normal distribution, allowing statisticians to predict confidence intervals for the true population mean height using standard normal tables.
Importance and Applications
The CLT is crucial for statistical inference, hypothesis testing, and confidence intervals, as it justifies using parametric methods like t-tests on non-normal data with large samples. It applies in fields such as quality control, polling, and finance, where it underpins reliable predictions from sample data.