# How to prove the mean of a sample approximates well to the mean of the population

Suppose I have a population whose distribution is definitely not normal but both the population and sample size will be large. Is there any way I can prove/ show that the mean of a large enough sample on the population will confidently approximate the mean of the entire population? Is there also a way to calculate the percentage of the confidence interval since I obviously would want the approximation to be of high confidence?

• Please research the Weak Law of Large Numbers, the Strong Law of Large Numbers, and the Central Limit Theorem. And could you explain what you mean by the "percentage of the confidence interval"?
– whuber
Jun 27, 2022 at 18:52

Supposing your samples are IID, and supposing the population variance $$\sigma^2$$ is finite, you can use the central limit theorem. Under these conditions, if the sample size $$n$$ is sufficiently large, then the sample mean $$\bar{X}_n$$ is approximately $$N(\mu, \sigma^2/n)$$ with $$\mu$$ the population mean. You can use this normal to create confidence intervals and show convergence of $$\bar{X}_n$$ to $$\mu$$ as $$n\to\infty$$.