Sampling Distributions: every key term you need

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.

More AP Statistics guides

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