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Introductory Statistics Unit 5: Sampling distributions

A statistic changes from sample to sample. Its sampling distribution shows that variation, and it's the foundation for every confidence interval and test.

Covers: Sampling distributions of proportions and means, Central Limit Theorem

Study guide

A parameter describes a population (p, μ, σ); a statistic describes a sample (p̂, x̄, s). Statistics vary from sample to sample, and the sampling distribution is the distribution of a statistic's values over all possible samples of the same size. It's the foundation of inference.

For a sample proportion p̂, the sampling distribution has mean p (p̂ is unbiased) and standard deviation √(p(1 − p)/n). It's approximately normal when the Large Counts condition holds: np ≥ 10 and n(1 − p) ≥ 10. The standard deviation formula requires the 10% condition, that the sample is no more than 10% of the population when sampling without replacement.

For a sample mean x̄, the sampling distribution has mean μ and standard deviation σ/√n. Larger samples give less variability: quadrupling n cuts the standard deviation in half. If the population is normal, x̄ is exactly normal for any n.

The Central Limit Theorem says that for large samples (a common rule of thumb is n ≥ 30), the sampling distribution of x̄ is approximately normal no matter what shape the population has. That's why normal calculations work for means of many kinds of data.

Bias and variability are separate ideas: an unbiased statistic is centered on the parameter, and low variability means its values cluster tightly. Random sampling controls bias; larger samples reduce variability.

A dotplot of a statistic from many simulated samples approximates its sampling distribution. To judge an observed value, find the fraction of simulated values at least as extreme. A small fraction means the observed value would be unusual if the simulation's assumptions were true. For independent samples, p̂₁ − p̂₂ has mean p₁ − p₂ and standard deviation √(p₁(1 − p₁)/n₁ + p₂(1 − p₂)/n₂). Likewise x̄₁ − x̄₂ has mean μ₁ − μ₂ and standard deviation √(σ₁²/n₁ + σ₂²/n₂).

Individual values vary more than sample means. A single observation has standard deviation σ, while the mean of n observations has standard deviation σ/√n.

Key ideas

• A parameter describes a population; a statistic describes a sample.
• Larger samples give less variable statistics.
• Central Limit Theorem: for large n, the sampling distribution of x̄ is approximately normal whatever the population's shape.
• Large Counts condition for proportions: np ≥ 10 and n(1 − p) ≥ 10.
• 10% condition: the sample is at most 10% of the population when sampling without replacement.

Formulas and rules

Common mistakes

• Thinking the CLT makes the population normal. It's about the sampling distribution.
• Forgetting the √n: quadrupling n only halves the SD.

Words to know

Parameter vs statistic

A parameter describes a population; a statistic describes a sample.

SD of the sampling distribution of p̂

√(p(1 − p)/n)

SD of the sampling distribution of x̄

σ/√n

Central Limit Theorem

For large n, x̄ is approximately normal whatever the population's shape.

Large Counts condition

np ≥ 10 and n(1 − p) ≥ 10

Sample size and standard error

σ/√n: quadrupling n halves the standard error.

Large sample (CLT rule of thumb)

n ≥ 30

Parameter

A number describing a population (p, μ, σ).

Statistic

A number describing a sample (p̂, x̄, s).

Sampling distribution

The distribution of a statistic over all possible samples of the same size.

Unbiased estimator

A statistic whose sampling distribution is centered on the parameter.

10% condition

The sample is no more than 10% of the population.

Variability of a statistic

Decreases as sample size increases.

Sample practice questions

19 more questions like these, from level 1 to 5, are in the practice set.