Module 5: Sampling Distributions
Master the Central Limit Theorem and understand how samples behave
Prerequisites: Modules 2, 3, & 4
This module builds on concepts from Module 2: Descriptive Statistics (mean, standard deviation), Module 3: Probability Basics (probability distributions, expected value), and Module 4: The Normal Distribution (z-scores, normal probabilities). Make sure you're comfortable with these topics before starting.
Learning Objectives
By the end of this module, you will be able to:
- Distinguish between parameters and statistics
- Explain what a sampling distribution is and why it matters
- Describe the sampling distribution of sample means (x̄)
- State and apply the Central Limit Theorem
- Calculate the mean (μₓ̄ = μ) and standard error (σₓ̄ = σ/√n) for sample means
- Use the normal approximation for sample means and proportions
- Determine when the Central Limit Theorem applies
- Calculate probabilities using sampling distributions of means
- Apply sampling distribution concepts to real-world problems
Lessons
Introduction to Sampling Distributions
Learn the difference between parameters and statistics, and understand what sampling distributions are.
- Parameters vs. statistics (μ vs. x̄, σ vs. s, p vs. p̂)
- What is a sampling distribution?
- Sampling variability
- Sampling distribution of sample means (x̄)
The Central Limit Theorem
Master one of the most important theorems in statistics and apply it to sample means.
- Statement of the Central Limit Theorem
- Conditions for CLT (n ≥ 30 rule)
- Sampling distribution of x̄: μₓ̄ = μ and σₓ̄ = σ/√n
- Calculating probabilities for sample means
Sampling Distribution of Proportions
Learn how sample proportions behave and when to use normal approximation.
- Distribution of sample proportions (p̂)
- Conditions: np ≥ 10 and n(1-p) ≥ 10
- Mean and standard error for proportions
- Normal approximation for proportions
Standard Error and Applications
Understand standard error and apply sampling distributions to real problems.
- What is standard error?
- Standard error vs. standard deviation
- Calculating and interpreting standard error
- Real-world applications of sampling distributions
Assessments & Practice
Test your understanding and practice what you've learned.
Study Tips for Module 5
- Understand the "why": The Central Limit Theorem is powerful—understand why it works, not just how to use it
- Parameters vs. statistics: Keep the notation straight (μ for population, x̄ for sample)
- Check conditions: Always verify CLT conditions before applying it (n ≥ 30 or population is normal)
- Standard error matters: SE = σ/√n shows larger samples give more precise estimates
- Practice, practice: Sampling distributions can be abstract—work through many examples to build intuition