Separate a population from its sample means

The sample mean varies across random samples. For independent observations from a population with mean μ and standard deviation σ, its mean is μ and its standard deviation is σ/ $\sqrt{n}$ .

For a normal population, the sample mean is normal for any sample size. The central limit theorem often gives an approximate normal model for sufficiently large independent samples from other finite-variance populations; strong skewness may require larger samples.

\displaystyle E(\overline X)=\mu,\quad\operatorname{SD}(\overline X)=\frac\sigma{\sqrt n}

WORKED EXAMPLE

A normal population has mean 50 and standard deviation 12. Describe the mean of 36 independent observations.

Its mean remains 50.
Its standard error is 12/ $\sqrt{36}$ = 2.
The sample mean is normally distributed with mean 50 and variance 4.

Assumed knowledge

Distinguish a population from a random sample. Calculate and interpret means, variance and standard deviation; recall independence and the normal distribution from Mathematical Methods.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Separate a population from its sample means

A normal population has mean 50 and standard deviation 12. Describe the mean of 36 independent observations.

Put the idea to work.