The central limit theorem says that for sufficiently large sample size n, the sampling distribution of the sample mean approaches normal with mean mu and standard error sigma/sqrt(n), even when the population is not normal, under common regularity conditions.
- CLT applies to the distribution of sample means, not raw individual data.
- Standard error shrinks as n increases, specifically with 1/sqrt(n).
- Independence and sampling design assumptions still matter.
What converges to normal
The theorem concerns repeated samples and the distribution of xbar, the sample mean. Individual observations can remain skewed while xbar becomes approximately normal for large n.
A common practical rule is that n >= 30 may be adequate for moderate skew, but stronger skew or heavy tails may require larger n.
Why CLT matters in practice
Many confidence intervals and tests for means rely on normal approximations for xbar. CLT explains why these methods often work beyond strictly normal populations.
If observations are dependent or sampling is highly biased, CLT-based inference can still fail, so design quality checks remain necessary.
- Sampling distribution mean: E(xbar)=mu.
- Sampling distribution standard deviation: SD(xbar)=sigma/sqrt(n).
- Larger n narrows uncertainty around xbar.
Approximate mean shipping time variability
Population standard deviation of shipping time is known as 12 hours. Sample size is n=36.
- Use CLT for sample mean xbar with mean mu and standard error sigma/sqrt(n).
- Compute standard error: 12/sqrt(36)=12/6=2 hours.
- Use this SE in normal-based probability or interval calculations for xbar.
- Interpret as variability of sample means across repeated samples of size 36.
Common mistakes
- Applying CLT directly to one raw observation.
- Confusing standard deviation with standard error.
- Assuming CLT fixes bias from poor sampling.
- Using very small n with extreme skew and no caution.
Try one
If sigma is fixed, how does doubling n affect standard error of the mean?
It multiplies SE by 1/sqrt(2), so SE decreases but not by half.
Sources
- OpenStax: Introductory StatisticsOpen textbook covering descriptive statistics, probability, inference, and regression fundamentals.
- NIST/SEMATECH e-Handbook of Statistical MethodsAuthoritative reference on probability models, inference procedures, and practical statistical diagnostics.