How large a sample for the CLT to give you approximately normal sample means?

# Short answer The central limit theorem (CLT) guarantees that sample means converge to a normal distribution as sample size increases, but "how large" depends on the population. For symmetric, light-tailed populations the mean behaves normally for very small n. For heavily skewed populations, simulations here show that even n = 100 can leave the distribution of sample means visibly non-normal and that standard 95% CIs based on asymptotic normality can miss the true mean much more than 5% of the time.

# What the author did

• calculates the empirical coverage of 95% confidence intervals built by assuming the sample mean is normally distributed (using the sample SD to estimate the standard error).

The simulations use a large population (N = 1e6) and many repetitions (default 10,000) to estimate the distribution of the sample mean precisely.

• From a standard normal population with n = 5, the AD statistic was 0.45 and the 95% CI built under asymptotic normality covered the true mean about 87% of the time. Ellis notes this is because he deliberately avoids using the t-distribution (which would be more appropriate in that specific case) to test reliance on asymptotic normality alone.

• From a heavily skewed log-normal population (generated as exp(N(0,1))), a sample size of n = 5 shows clear non-normality of the sample mean. Increasing sample size to n = 100 improves things but still does not produce a "satisfactory" normal distribution of the sample means according to the author's graphical checks.

# Wider sweep of simulations

# Follow-up: bootstrap vs asymptotic CIs A short sequel tests whether bootstrap confidence intervals help. The follow-up compares the traditional asymptotic 95% CI to a bias-corrected and accelerated (BCa) bootstrap interval across the same skewed distributions and sample sizes. Results reported in that sequel: the BCa bootstrap gives substantially better coverage than naive CLT-based intervals, especially at smaller n, though it is not claimed as a universal fix for every scenario.

# Practical takeaway Don't rely on a fixed rule-of-thumb like n = 30 for the CLT to make the sample mean approximately normal in every situation. For skewed or heavy-tailed populations you may need much larger n for asymptotic normality to be a good approximation. When sample size is limited, use diagnostic plots or tests on the sampling distribution (or simulation if you can generate a plausible population), and consider bootstrap CIs (BCa) as a practical alternative to naive normal-based intervals.

# Code and reproducibility