centralelimietetheorema

The Central Limit Theorem is a fundamental concept in probability theory and statistics. It states that if you take a sufficiently large random sample from any population with a known finite mean and variance, regardless of the population's original distribution, the distribution of the sample means will be approximately normally distributed. This approximation becomes better as the sample size increases.

The theorem is crucial because it allows us to make inferences about a population using sample data, even when the population's distribution is unknown or not normal. For instance, if we want to estimate the average height of all adults in a country, we can take a sample of adult heights. According to the Central Limit Theorem, the distribution of the means of many such samples will tend to be bell-shaped (normal), centered around the true population mean.

This property is what underlies many statistical methods, such as hypothesis testing and confidence interval construction. It provides a theoretical justification for using statistical techniques that assume normality, even when the underlying data might not be perfectly normal, as long as the sample size is adequate. The "sufficiently large" sample size is often considered to be around 30 or more, though this can vary depending on the skewness of the original distribution.