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Construct sample means and interpret finite sampling distributions

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One mean per repetition

Repeat the complete sampling procedure R times, keeping the within-sample size n fixed. Calculate one mean from each sample, then plot those R means. A dot represents a sample mean. R controls how many dots approximate the sampling distribution; n controls how much information goes into each mean

Year 12, VCE Units 3 & 4, Chapter 8, 3 difficulty levels, Calculator allowedBuild a worksheet

Sampling distributions of the mean

Plot one mean per repeated sample and retain the sampling assumptions

  1. Empirical distribution Observed sample means from a finite number of repetitions
  2. Exact distribution All possible means with their probabilities under the sampling model
  3. Independent equal-distribution draws The mean equals the population mean; the variance equals the population variance divided by sample size; the SD equals the population SD divided by the square root of sample size
  4. n versus R n is observations per sample; R is the number of sample means plotted
  5. Interpretation Greater n reduces theoretical spread under the model; no realised sample is guaranteed closer

Worked example

Each independent observation is 0 or 4 with equal probability. Construct the distribution of the mean of two observations

\[\mu=2,\qquad\sigma^2=4\]
  1. \(\displaystyle \Pr(\overline X=0)=\frac14,\quad\Pr(\overline X=2)=\frac12,\quad\Pr(\overline X=4)=\frac14\)Group ordered pairs by their mean; the mixed pairs both give 2
  2. \(\displaystyle E(\overline X)=2,\quad E(\overline X^2)=6,\quad\operatorname{Var}(\overline X)=6-4=2,\quad\frac{\sigma^2}{n}=\frac{4}{2}=2\)This agrees with the population variance divided by sample size. The exact sampling distribution is discrete, not normal

Common mistakes

Wrong plotted object
✕plot all nR observations and call them sample means✓calculate and plot exactly R means
A sampling distribution concerns a statistic across repeated samples
Wrong denominator
✕relative frequency of a plotted mean = count/n✓relative frequency = count/R
The dot plot has one entry per sample, not per observation within a sample
Guaranteed closeness
✕a larger sample mean is always nearer μ✓the distribution is less variable under the stated model
A statement about repeated-sample spread does not determine any one outcome

Difficulty levels

standardEmpirical frequencies of supplied sample means
challengeExact finite mean probabilities
examTheoretical spread under an iid model

Questions about this topic

Is a sample-mean distribution always normal?+

No. Small finite models can have discrete, asymmetric distributions. No normal approximation is used here

Does a more detailed empirical plot change the theoretical distribution?+

Increasing repetitions R gives more information about the same model at fixed n. Changing n changes the sampling distribution itself

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