What is the central limit theorem relevant for?

The central limit theorem is significant because 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 becomes larger, typically if the sample size is 30 or more. This theorem allows statisticians to make inferences about population parameters based on sample statistics, providing a foundation for hypothesis testing and confidence intervals.

For example, if a researcher collects sample data to estimate the average height of individuals in a large population, even if the height distribution in that population is not normal, the distribution of the average heights from multiple samples will tend to form a normal distribution as the sample sizes increase. This property is crucial for applying statistical methods that assume normality, especially in cases where the population distribution is unknown.

Understanding this concept is essential for interpreting results in statistics, as it justifies the use of normal probability models in various applications, even when the underlying data do not exhibit normal characteristics.