Sampling Distributions: every key term you need (+ practice quiz)

More AP Statistics guides

• Designing Studies
• Exploring Bivariate Data
• Probability
• Random Variables & Distributions
• Inference for Means

Sampling Distributions

Probability distribution of sample statistic (like sample mean) across all possible samples. Key to inference: relates sample to population.

Sample Mean Distribution

Distribution of x̄ from all possible samples of size n. Center: μ_x̄ = μ. Spread: σ_x̄ = σ/√n (standard error).

Standard Error

Standard deviation of sample statistic; measures sampling variability. SE = σ/√n for means. Smaller n = larger SE.

Central Limit Theorem (CLT)

For large n, sample mean distribution approaches normal regardless of population shape. Critical for inference; justifies normal approximation.

CLT Conditions

Random sample, independence (large population or sampling with replacement), large n (n≥30 usually sufficient). Enables use of z-procedures.

Distribution of Sample Proportion

p̂ ≈ normal with mean p and standard error √(p(1-p)/n). Holds if np≥10 and n(1-p)≥10.

Unbiased Estimator

Sample statistic whose expected value equals population parameter. x̄ unbiased for μ, p̂ unbiased for p. Small mean error.

Biased Estimator

Expected value ≠ population parameter. Example: s (sample SD) slightly underestimates σ (though rarely significantly).

Variability of Estimators

Smaller SE = more precise estimate. Increased n reduces SE by factor 1/√n (diminishing returns; doubling n reduces SE by 1/√2).

t-Distribution

Used when σ unknown; similar to normal but with heavier tails. Approaches normal as df increases. t-values larger than z for same confidence.

Degrees of Freedom (df)

Number of independent pieces of information; df = n-1 for sample mean. More df = closer to normal distribution.

Sampling from Finite Population

Use finite population correction factor: SE = (σ/√n)√((N-n)/(N-1)). Only necessary if n/N > 0.05.

Unit 5 Summary

Sampling distributions describe variability of sample statistics. CLT ensures sample means approximately normal. SE quantifies precision; smaller SE = better estimate.

Sampling Distribution

The distribution of a statistic (x̄, p̂, etc.) over ALL possible samples of size n from a population. Different from the population distribution and from one sample's data distribution.

Three Distributions

Population distribution (all individuals), sample data distribution (one sample's values), sampling distribution (values of a statistic across many samples). Keep them separate.

Sampling Distribution of p̂

Mean = p (unbiased). SD = sqrt(p(1-p)/n) if the 10% condition holds. Approximately Normal if np ≥ 10 and n(1-p) ≥ 10.

Sampling Distribution of x̄

Mean = μ (unbiased). SD = σ/sqrt(n) if the 10% condition holds. Normal if the population is Normal; approximately Normal for large n by the CLT.

Central Limit Theorem

For any population shape with finite SD, the sampling distribution of x̄ becomes approximately Normal as n grows. Rule of thumb n ≥ 30 for moderate skew.

CLT Does Not Change the Mean or SD

The CLT is only about SHAPE. Mean μ and SD σ/sqrt(n) of x̄ hold for any n; the CLT adds approximate Normality for large n.

Square Root Law

To reduce the SD of x̄ or p̂ by a factor of k, multiply n by k^2. Quadruple the sample to halve the standard deviation.

Bias of an Estimator

An estimator is unbiased if the mean of its sampling distribution equals the parameter. Sample mean, sample proportion, and sample variance s^2 are unbiased.

Bias vs Variability Targets

Picture a target: bias = shots centered off the bullseye; high variability = shots widely scattered. Good estimators are both unbiased and low-variability.

Population Size Rarely Matters

Variability of a statistic depends on sample size n, not population size, as long as the population is at least 10 times n.

Sample Median and Range

The sample median is roughly unbiased for a symmetric population; the sample range is a biased (too small) estimator of the population range.

Difference of Two Proportions

p̂1 - p̂2 has mean p1 - p2 and SD sqrt(p1(1-p1)/n1 + p2(1-p2)/n2) for independent samples; Normal if all four counts ≥ 10.

Difference of Two Means

x̄1 - x̄2 has mean μ1 - μ2 and SD sqrt(σ1^2/n1 + σ2^2/n2) for independent samples; Normal if both populations Normal or both n large.

Standardizing a Sample Mean

z = (x̄ - μ)/(σ/sqrt(n)). Common error: dividing by σ instead of σ/sqrt(n) when the question is about an average.

Standardizing a Sample Proportion

z = (p̂ - p)/sqrt(p(1-p)/n). Use the claimed population p, not p̂, in the denominator.

Simulation of a Sampling Distribution

Repeatedly draw samples of size n from a model, compute the statistic each time, and graph the results; a dot plot of many statistics approximates the sampling distribution.

Sample Size and Shape

For a skewed population, small-n sampling distributions of x̄ retain some skew; larger n both narrows and normalizes the distribution.

Sample Proportion as Sample Mean

p̂ is the mean of 0/1 values, which is why its sampling distribution also obeys the CLT.

Sample Variance Uses n-1

Dividing by n-1 makes s^2 an unbiased estimator of σ^2. Dividing by n would systematically underestimate.

Interpreting SD of x̄

'In repeated random samples of size n, the sample mean typically varies from μ by about σ/sqrt(n).'