Textbook: Interactive Statistics, 3rd ed. (Sullivan & Woodbury), Section 8.1
Chapter Summary
Here is the idea that makes all of inferential statistics possible. Take a population with mean μ and standard deviation σ. Now imagine drawing every possible sample of size n, computing each sample’s mean x̄, and plotting all those means. The resulting distribution — the sampling distribution of the sample mean — is itself approximately normal, centered at μ, with a standard deviation of σ/√n. That last quantity has its own name: the standard error of the mean.
This is the Central Limit Theorem, and its punchline is remarkable: sample means are less spread out than individual values, and the bigger the sample, the tighter they cluster around μ. That’s why a poll of 1,000 voters predicts an election far better than asking 10 people — averaging washes out the extremes. Concretely: if individual test scores have σ = 12, a single student’s score varies a lot, but the mean of 36 students has standard error 12/√36 = 2, so sample means cluster tightly around the true average.
The practical rule: the sampling distribution of x̄ is approximately normal when n ≥ 30 (or when the population itself is normal, any n works). Everything in Chapters 9–11 — confidence intervals, hypothesis tests — is just this chapter applied: we observe one sample mean and use the sampling distribution to say how far it could plausibly be from μ.
Study Guide
Key terms and definitions
- Sampling distribution of the sample mean — the probability distribution of all possible values of x̄ from samples of size n.
- Central Limit Theorem (CLT) — as n grows, the sampling distribution of x̄ becomes approximately normal, regardless of the population’s shape (given n ≥ 30, or a normal population).
- Standard error of the mean — σ_x̄ = σ ÷ √n; the standard deviation of the sampling distribution. It shrinks as n grows.
- Unbiased estimator — the mean of the sampling distribution equals the population parameter: μ_x̄ = μ.
Essential formulas
- Mean of the sampling distribution: μ_x̄ = μ.
- Standard error: σ_x̄ = σ ÷ √n.
- z-score for a sample mean: z = (x̄ − μ) ÷ (σ/√n).
- CLT condition: use the normal model for x̄ if the population is normal or n ≥ 30.
Step-by-step procedures
- Find P(x̄ < a) or similar: (1) verify the CLT condition (normal population or n ≥ 30); (2) compute the standard error σ/√n; (3) convert to a z-score with z = (x̄ − μ)/(σ/√n); (4) find the area with the standard normal (same as Chapter 7).
- Compare individual vs. mean probabilities: note how the same cutoff gives a much smaller tail probability for a mean than for one individual — that’s the √n effect in action.
Common mistakes to avoid
- Using σ instead of σ/√n in the z-formula for a sample mean — the most common error in this chapter.
- Applying the CLT with n < 30 to a skewed population. Small samples only work if the population is already normal.
- Confusing the standard deviation (spread of individuals) with the standard error (spread of sample means).
Self-check: I can…
- [ ] Explain the Central Limit Theorem in my own words.
- [ ] State the mean and standard deviation of the sampling distribution of x̄.
- [ ] Check whether the CLT conditions are satisfied.
- [ ] Compute probabilities involving a sample mean.
- [ ] Explain why larger samples give more precise estimates.
StatCrunch How-To
- Simulate a sampling distribution: Data > Simulate > choose a distribution (e.g., Normal or Uniform) — generate many rows of n columns, then use Data > Compute to average each row into a new column. Graph > Histogram that column: watch it look normal even when the original data weren’t.
- Probabilities for x̄: Stat > Calculators > Normal — enter mean = μ and SD = σ/√n (compute the standard error first), then proceed exactly like Chapter 7.
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