Central Limit Theorem › The Standard Error

A critical component of the Central Limit Theorem is the standard error of the mean, which quantifies the variability of sample means around the population mean. The standard error is defined as:

This formula reveals an important relationship between sample size and precision. Doubling the sample size does not double the precision; rather, it improves precision by a factor of $\sqrt{2} \approx 1.41$. To cut the standard error in half and achieve twice the precision, we need to quadruple the sample size. This square-root relationship has practical implications for study design and resource allocation in research projects.

In practice, the population standard deviation $\sigma$ is typically unknown and must be estimated from the sample using the sample standard deviation $s$. When we substitute $s$ for $\sigma$, we obtain the estimated standard error:

The use of the estimated standard error leads to the Student's t-distribution rather than the normal distribution for small samples, but as sample size increases, the t-distribution converges to the normal distribution, consistent with the Central Limit Theorem.