Sampling means and standard error
State the sampling model before using its distribution
- iid finite variance The sample mean has expected value μ, variance equal to the population variance divided by n, and standard error equal to the population standard deviation divided by the square root of n
- Normal population The iid mean is exactly normal for every positive n
- Non-normal population A normal approximation is a large-sample limit; adequacy depends on shape and tails
- Estimated SE When σ is unknown, s/√n estimates the standard error
- Simulation n observations form one mean; R repetitions produce R means
Worked example
Independent observations follow a normal distribution with mean 18 and variance 36. For a sample of 16, find the probability that the sample mean exceeds 21
\[\overline X\sim N(18,9/4)\]
- \(\displaystyle \operatorname{SE}=6/\sqrt{16}=3/2\)Use the SD of the mean
- \(\displaystyle z=(21-18)/(3/2)=2\)Standardise the sample-mean event
- \(\displaystyle \Pr(\overline X>21)=1-\Phi(2)\approx0.0228\)Take the upper tail and round at the end
Common mistakes
Wrong spread
✕Divide the population standard deviation by the sample size✓Divide the population standard deviation by the square root of the sample size
It is variance that is divided by nWrong count
✕Use the number of repeated samples to calculate the standard error of each mean✓The SE of each mean depends on n
R changes the empirical approximation, not the fixed-n sampling distributionAutomatic normality
✕A sample size of at least 30 guarantees exact normality✓Exact normality follows from iid normal inputs; other finite-variance inputs may permit an approximation
Finite-sample accuracy depends on the underlying populationDifficulty levels
introKnown normal mean standard error
challengeEstimated standard error from s and n
Questions about this topic
Can an individual sample mean differ from μ?+
Yes. E(X̄)=μ describes repeated sampling, not each realised mean
Does more data fix selection bias?+
No. The target population and selection mechanism still matter
Does σ²/n apply without replacement?+
Not exactly in general for a finite population; the draws are dependent, and a census has zero sampling spread
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Specialist Mathematics, chapter 8: Probability and statistical inference
In this chapter
Standardise tails of three independent normal variablesCombine independent random variables and normal distributionsConstruct finite independent sums and compare repeated copies with scalingCalculate probabilities for normal distributions8B Distribution of sample means: Use normal sampling distributions and standard errorsConstruct sample means and interpret finite sampling distributionsConstruct and interpret confidence intervals for a population mean8C Confidence intervals for population means: Use normal sampling distributions and standard errorsPredict interval coverage and plan a width reductionForm and interpret hypothesis tests for a population mean8D Hypothesis tests for population means: Use normal sampling distributions and standard errorsCalculate theoretical test errors and supplied-tail boundaries