6.4.4—Normal distribution

For normal data, X̄ is exactly normal; for large independent samples it may be approximately normal by the central limit effect. Its mean is μ and variance σ²/n.

Standardise with the standard error, state whether σ is known or estimated, and check independence and sample-size assumptions before using normal probabilities.

With μ=100, σ=15 and n=25, X̄~N(100,9), so a one-standard-error interval is 100±3.

The sample mean does not inherit the population variance unchanged; averaging reduces variance by n.