Lesson 2: The Central Limit Theorem and standard error

Explain the Central Limit Theorem and calculate the standard error of the mean SE=SD/√n.

The standard error shrinks as the sample grows

SE = SD / √n with SD = 10

The Central Limit Theorem guarantees: the larger the sample, the more precise the estimate of the mean. SE shrinks with the square root of n.

Line graph of the standard error decreasing as the sample size n grows. At n=10 SE is 3.16; at n=25 SE=2; at n=50 SE=1.41; at n=100 SE=1; at n=400 SE=0.5. The curve is convex and decreases toward zero — With SD = 10: doubling the precision requires quadrupling the sample size

Lesson notes

The CLT and the standard error of the mean

The Central Limit Theorem (CLT) is one of the most important results in statistics. In short: if you take repeated random samples of size n from any distribution (with a finite mean and variance), then for large enough n the sampling distribution of the sample mean approaches a normal distribution. This holds even when the original data are far from normal — skewed or with several peaks. In practice, n ≥ 30 is usually considered “large enough.” The standard error of the mean (SE) shows how far the sample mean deviates from the true population mean. Formula: SE = SD / √n, where SD is the population standard deviation (or its estimate from the sample) and n is the sample size. Example: SD = 20, n = 100 → SE = 20 / √100 = 20 / 10 = 2. If you increase the sample to n = 400: SE = 20 / √400 = 20 / 20 = 1. Conclusion: the larger the sample, the smaller the standard error and the more precise the estimate of the mean. Making the sample 4 times larger cuts SE in half — precision grows slowly, with the square root of n.