Sampling Distributions

What this unit covers

The topics below follow the published Statistics course framework for Unit 5. This unit is worth 7–12% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Sampling Distributions & Variability13 min · 3 objectivesDefine a sampling distribution as the distribution of a statistic over all samples · Explain what it means for a statistic to be an unbiased estimator · Describe how sample size affects the variability of a statistic
• The Sampling Distribution of a Sample Mean15 min · 3 objectivesState the mean and standard deviation of the sampling distribution of x-bar · Apply the Central Limit Theorem to determine the shape of the distribution of x-bar · Compute probabilities involving a sample mean
• The Sampling Distribution of a Sample Proportion14 min · 3 objectivesState the mean and standard deviation of the sampling distribution of p-hat · Check the conditions for approximate Normality of p-hat · Compute probabilities involving a sample proportion
• Differences & the Big Picture of Variability13 min · 3 objectivesFind the mean and standard deviation of a difference between two independent statistics · Explain how variability of a difference combines the two individual variabilities · Connect sampling distributions to the logic of inference

Formulas in Unit 5

Every term in Unit 5

All 12 terms we publish for Sampling Distributions, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Central Limit Theorem

The sampling distribution of x̄ approaches normal as n grows, whatever the population shape. Usually taken as n ≥ 30.

Sampling distribution of x̄

Center μ, standard deviation σ/√n. Normal if the population is normal, or approximately normal for large n by the Central Limit Theorem.

Sampling distribution of p̂

Center p, standard deviation √(p(1 − p)/n), approximately normal when np ≥ 10 and n(1 − p) ≥ 10.

Sampling distribution

The distribution of a statistic over all possible samples of a given size. Not the distribution of one sample, and not the population.

Unbiased estimator

A statistic whose sampling distribution is centered at the parameter. The sample mean and sample proportion are both unbiased.

Standard error

The standard deviation of a sampling distribution. It shrinks as √n, so quadrupling the sample size halves it.

Why sample size matters more than population size

Standard error depends on n, not on population size, which is why a national poll needs about the same sample as a state poll.

Difference of two proportions

Center p₁ − p₂ with standard deviation √(p₁(1−p₁)/n₁ + p₂(1−p₂)/n₂) — the variances add even though the means subtract.

Distinguishing three distributions

The population, the distribution of one sample, and the sampling distribution of a statistic are different objects. Exam questions turn on telling them apart.

Shape of the sampling distribution of p̂

Approximately normal when np and n(1 − p) are both at least 10. Otherwise it is skewed and normal-based methods do not apply.

Why standard error shrinks with n

It divides by √n, so precision improves with sample size but with diminishing returns — quadrupling n only halves the error.

Bias vs variability

Bias is being centered in the wrong place; variability is being spread out. A method can have low variability and still be badly biased.

What examiners penalize here

• Before finding a probability for x-bar, justify the Normal shape: either state the *population* is Normal, or invoke the **CLT with n ≥ 30**. Free-response answers that skip this justification lose the condition point even if the arithmetic is perfect.
• Every two-sample standard error you will meet is built by the same move: compute each group’s variance contribution, **add** them, then square-root. Memorize the pattern once and it powers two-proportion and two-mean intervals and tests alike.

Practice Statistics

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 9 units of AP Statistics

1. Unit 1 · Exploring One-Variable Data
2. Unit 2 · Exploring Two-Variable Data
3. Unit 3 · Collecting Data
4. Unit 4 · Probability & Random Variables
5. Unit 5 · Sampling Distributions
6. Unit 6 · Inference for Proportions
7. Unit 7 · Inference for Means
8. Unit 8 · Inference for Categorical Data: Chi-Square
9. Unit 9 · Inference for Quantitative Data: Slopes