Sampling Distributions: Precision, Normality, and the Law of Large Numbers

A sampling distribution describes how a statistic varies across repeated samples. It is different from a histogram of the individual measurements in one sample.

Separate center, spread, and shape

Consider independent observations from a population with mean 80 and standard deviation 12. The sample mean has mean 80 and standard deviation $12/\sqrt{n}$:

These moment calculations do not require a normal population. Approximate normality is an additional shape claim that needs justification.

The law of large numbers and the CLT do different jobs

The law of large numbers explains why the sample mean concentrates near the population mean as the sample grows. The central limit theorem describes the limiting shape of the standardized fluctuations around that mean.

A smaller standard error does not imply that the raw observations have become less variable. Nor does a histogram of 100 sample means change the sample size used inside each mean: the number of repetitions and the number of observations per sample are different quantities.

Use the appropriate random variable

A question about one future measurement uses the population distribution. A question about an average uses the sampling distribution of the mean. A question about a total uses the distribution of the sum.

Before applying a normal approximation, inspect how observations were sampled and whether extreme skewness, heavy tails, or dependence threaten the model. Increasing sample size cannot repair systematic selection bias.