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Sampling and the sample mean

Choose a sampling model, estimate population parameters and distinguish exact normal results from CLT approximations.

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By the end, you can

  • Separate a population parameter from a sample statistic
  • Find the mean and standard error of a sample mean
  • Use unbiased variance estimation and the CLT with care

about 35 minutes for the lesson · practise at your own pace.

The central idea

Population parameters are fixed; sample statistics vary across repeated samples. For iid observations with mean μ\mu and finite variance σ2\sigma^2, E(Xˉ)=μE(\bar X)=\mu and Var⁡(Xˉ)=σ2/n\operatorname{Var}(\bar X)=\sigma^2/n. An exact normal population gives an exact normal mean; otherwise a suitable large-sample CLT approximation may apply. Use n−1n-1 in the unbiased variance estimate, and keep sampling bias separate from sampling variability.

Compare all five sampling distributions

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Five rows compare n=1, 5, 10, 50 and 100. Left: exact means of iid uniform integers 1–10, centre 5.5 and variance 8.25/n, with an orange normal approximation. Right: exact normal means with centre 5 and variance 9/n. All panels share horizontal −5–16 and density 0–1.5 scales; discrete mass is multiplied by n for comparison. The flat n=1 uniform distribution becomes bell shaped; normal means are normal at every n. Both spreads shrink. Independent mathematical reconstruction; original source simulation images are retained separately.
Five rows compare n=1, 5, 10, 50 and 100. Left: exact means of iid uniform integers 1–10, centre 5.5 and variance 8.25/n, with an orange normal approximation. Right: exact normal means with centre 5 and variance 9/n. All panels share horizontal −5–16 and density 0–1.5 scales; discrete mass is multiplied by n for comparison. The flat n=1 uniform distribution becomes bell shaped; normal means are normal at every n. Both spreads shrink. Independent mathematical reconstruction; original source simulation images are retained separately.
Five rows compare n=1, 5, 10, 50 and 100. Left: exact means of iid uniform integers 1–10, centre 5.5 and variance 8.25/n, with an orange normal approximation. Right: exact normal means with centre 5 and variance 9/n. All panels share horizontal −5–16 and density 0–1.5 scales; discrete mass is multiplied by n for comparison. The flat n=1 uniform distribution becomes bell shaped; normal means are normal at every n. Both spreads shrink. Independent mathematical reconstruction; original source simulation images are retained separately.
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Explore

The source compares independent samples from XX uniform on {1,2,…,10}\{1,2,\ldots,10\} and Y∼N(5,9)Y\sim N(5,9) for n=1,5,10,50,100n=1,5,10,50,100.

x12345678910
P(X=x)P(X=x)0.10.10.10.10.10.10.10.10.10.1

Here E(X)=5.5E(X)=5.5, Var⁡(X)=8.25\operatorname{Var}(X)=8.25, so the mean has variance 8.25/n8.25/n. The n=1n=1 distribution is flat rather than normal; as nn grows, the exact discrete mean distribution becomes bell shaped and concentrates near 5.5. For YY, the mean is exactly N(5,9/n)N(5,9/n) for every listed nn. Both spreads shrink like 1/n1/\sqrt n.

nVar⁡(Xˉ)\operatorname{Var}(\bar X)Var⁡(Yˉ)\operatorname{Var}(\bar Y)
18.259
51.651.8
100.8250.9
500.1650.18
1000.08250.09

The figure is an independent mathematical reconstruction: exact discrete convolutions are shown for Xˉ\bar X and exact normal densities for Yˉ\bar Y, with a normal approximation overlaid for comparison.

It does not recreate the source’s random simulation run. Its ten original simulation chart images remain in the review evidence.

For a spreadsheet experiment, generate many independent size-nn samples, calculate each mean, and plot the means while keeping the population distribution fixed. Compare the observed centre and variance with the table.

The lecture links two external simulations: uniform discrete population and normal population. These are preserved source references; their external contents have not been reviewed. The Example 10 enrichment QR decodes to estimation theory; unbiased estimation is one form of estimation, and the destination is also outside this source review.

Before viewing each comparison, predict the centre, variance and shape of the distribution of means. In a spreadsheet, keep a separate plot of the individual observations. Explain why increasing the number of repeated samples makes the plot more stable, while increasing the size of each sample changes the spread of the means.

Worked example

Independent component measurements are normally distributed with population mean 100 and known standard deviation 6. Let XX be one measurement and let Xˉ\bar X be the mean of nine independent measurements. Then X∼N(100,36)X\sim N(100,36), while Xˉ∼N(100,36/9)=N(100,4)\bar X\sim N(100,36/9)=N(100,4) exactly.

For a mean above 104, the standardised boundary is (104−100)/2=2(104-100)/2=2, giving P(Xˉ>104)≈0.0228P(\bar X>104)\approx0.0228.

For one measurement above 104, it is (104−100)/6=2/3(104-100)/6=2/3, giving P(X>104)≈0.252P(X>104)\approx0.252.

Sketch both curves around the same centre: the mean has the smaller spread. No CLT approximation is needed because the population is normal.

The standard deviation of a mean is $\sigma/\sqrt n$, not $\sigma/n$. More observations reduce the variability of the mean; they do not change the distribution of a single measurement.

Check your understanding

Five independent randomly selected observations from a population of unspecified shape give ∑x=50\sum x=50 and ∑x2=516\sum x^2=516. Estimate the population mean and variance. Can these summaries justify calculating an exact normal probability for the sample mean? If the same sampling process is used with 20 observations, what happens to the theoretical standard error, and would that repair a biased sampling frame?

Mastery learning objectives

At the end of this chapter, I should be able to:

Sampling: after lecture and after tutorial

Understand population, random and non-random samples, distinguishing the finite without-replacement design from an iid observation model.

Calculate unbiased estimates of population mean and variance from raw, frequency and summarized data: ∑x,∑x2\sum x,\sum x^2 or ∑(x−a),∑(x−a)2\sum(x-a),\sum(x-a)^2.

Understand Xˉ\bar X as a random variable with E(Xˉ)=μE(\bar X)=\mu and Var⁡(Xˉ)=σ2/n\operatorname{Var}(\bar X)=\sigma^2/n under the independent common-variance model.

Solve problems involving sample means from a normal population, using exact normality.

Use CLT to approximate sample-mean distributions for sufficiently large samples; interpret the source’s n≥30n\ge30 as a guideline rather than a universal guarantee.

Choose a practice question

Common question typeLecture examplesDiscussionReview follow-up
Unbiased estimates using calculator2, 311
Unbiased estimates using summarized data2, 3, 4, 10 first part22
Means from normal populations5, 63(b), 4(iv), 5(b)(i), 67, 10
CLT for means7, 8, 9, 105(a),(b)(ii)(iii), 73, 4(ii), 6(vi), 5(b), 7(a), 9(iii), 10(iv)
Prior topic: discrete random variables——9(i)(ii)
Prior topic: binomial distribution—4(ii)4(i), 5(a)
Prior topic: normal distribution—3(a), 4(i)–(iii)5(b), 6(i)–(v), 8(ii), 10

The printed CLT row labels Example 10 “1st part”; its first part estimates parameters and the later probability part uses CLT. The map preserves the reference and explains its intended skill. The printed map also places Review 5(b) in normal prerequisites although its binomial daily count needs a CLT approximation; follow the model explained in that question.

Both lecture copies repeat all seven Discussion questions on PDF25–26 and all ten Review questions on PDF27–30. These links point to the already source-compared questions, preserving separate source occurrences through aliases.

Discussion short answers agree with the retained solutions.

The Review short key mislabels 5(b) as (c), 6(vi) as (iv), and 8(ii)(a)/(b) as (b)(i)/(ii); it labels 10(ii) .0882 as(i) and 10(iii) .375 as(ii), and omits 10(i)’s roots.

Canonical working retains the actual asks and the complete answer. Native keys stay unchanged. No marks are printed for the lecture examples or Discussion; the repeated Review retains 75 marks in its canonical records.

Practice questions

Practice section 1 · Questions 1–6

Select thirty distinct students fairly · Estimate mean and variance from seven raw values · +4 more

Practice section 2 · Questions 7–12

Separate a binomial mean approximation from its exact total · Approximate mean daily wine production · +4 more

Practice section 3 · Questions 13–18

Separate individual observations from a normal sample mean · Combine normal observations and choose a minimum sample size · +4 more

Practice section 4 · Questions 19–24

Define a random sample and use shifted totals · Approximate a binomial sample mean and track its endpoints · +4 more

Practice section 5 · Questions 25–27

Recover normal parameters and combine independent masses · Find a repeat-draw distribution and approximate its sample mean · +1 more

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