1. Parameters vs. Statistics

Parameter: A numerical summary of a population. Fixed but usually unknown.

Statistic: A numerical summary of a sample. Calculated from data, varies from sample to sample.

Notation

Measure	Population Parameter	Sample Statistic
Mean	μ (mu)	x̄ (x-bar)
Standard Deviation	σ (sigma)	s
Proportion	p	p̂ (p-hat)
Variance	σ²	s²

Remember: Greek letters (μ, σ, p) = POPULATION parameters
Roman letters (x̄, s, p̂) = SAMPLE statistics

2. Sampling Distributions

Sampling Distribution: The probability distribution of a sample statistic (like x̄ or p̂) based on all possible samples of size n from the population.

Important Distinction

Population distribution: Distribution of individual values in the population
Sample distribution: Distribution of individual values in ONE sample
Sampling distribution: Distribution of a statistic across ALL possible samples

Sampling Variability: Different random samples produce different statistics. This is normal and expected!

3. The Central Limit Theorem (CLT)

Central Limit Theorem: For a population with mean μ and standard deviation σ, the sampling distribution of x̄ from samples of size n will be approximately normal with:

Mean: μₓ̄ = μ
Standard deviation (SE): σₓ̄ = σ/√n

This approximation improves as n increases, and is generally good for n ≥ 30.

CLT Conditions

Random sampling: Samples randomly selected from population
Independence: Individual observations independent (10n rule for sampling without replacement)
Sample size: Either n ≥ 30 OR population is already normal

Why CLT is Amazing: Regardless of population shape (skewed, bimodal, uniform), the distribution of sample means will be approximately normal if n is large enough!

Properties of Sampling Distribution of x̄

Shape: Approximately normal (when n ≥ 30 or population is normal)

x̄ ~ N(μ, σ/√n)

Center:

μₓ̄ = μ

Mean of sampling distribution equals population mean

Spread (Standard Error):

σₓ̄ = σ / √n

Standard error decreases as sample size increases

z-score for Sample Mean:

z = (x̄ - μ) / (σ/√n)

4. Standard Error

Standard Error (SE): The standard deviation of a sampling distribution. Measures the typical distance between a sample statistic and the population parameter.

Standard Error vs. Standard Deviation

Aspect	Standard Deviation (σ)	Standard Error (SE)
What it measures	Variability of individuals	Variability of sample means
Affected by n?	No (population SD is fixed)	Yes (SE = σ/√n)
Interpretation	How spread out the data is	Precision of estimates

Sample Size Effect: To cut SE in half, you must multiply n by 4 (because of √n)

Standard Error Formulas

For Sample Means:

SE = σ / √n

For Sample Proportions:

SE = √(p(1-p) / n)

5. Sampling Distribution of Proportions

Sample Proportion: p̂ = x/n, where x = number of "successes" and n = sample size

Properties of Sampling Distribution of p̂

Center:

μₚ̂ = p

Spread:

σₚ̂ = √(p(1-p) / n)

Shape: Approximately normal when success-failure condition is met

Success-Failure Condition

For normal approximation to be appropriate:
np ≥ 10 AND n(1-p) ≥ 10

z-score for Sample Proportion:

z = (p̂ - p) / √(p(1-p)/n)

SE for proportions is largest when p = 0.5 (maximum variability)

6. Step-by-Step Procedures

Finding Probability about Sample Mean

Check CLT conditions
- Random sample?
- n ≥ 30 OR population normal?
Find sampling distribution
- μₓ̄ = μ
- σₓ̄ = σ/√n
Calculate z-score
- z = (x̄ - μ) / (σ/√n)
Find probability
- Use z-table or calculator

Finding Probability about Sample Proportion

Check success-failure condition
- np ≥ 10?
- n(1-p) ≥ 10?
Find sampling distribution
- μₚ̂ = p
- σₚ̂ = √(p(1-p)/n)
Calculate z-score
- z = (p̂ - p) / √(p(1-p)/n)
Find probability
- Use z-table or calculator

7. Example Problems

Example 1: CLT Application

Problem: A population has μ = 100, σ = 20. For n = 64, find P(x̄ > 105).

Solution:

Check: n = 64 ≥ 30
μₓ̄ = 100, σₓ̄ = 20/8 = 2.5
z = (105-100)/2.5 = 2.0
P(z > 2.0) = 1 - 0.9772 = 0.0228

Answer: 2.28% chance

Example 2: Sample Proportions

Problem: If p = 0.40 and n = 200, find P(p̂ < 0.35).

Solution:

Check: np = 80 ≥ 10 , n(1-p) = 120 ≥ 10
SE = √(0.40 × 0.60 / 200) = √0.0012 ≈ 0.0346
z = (0.35 - 0.40) / 0.0346 ≈ -1.45
P(z < -1.45) ≈ 0.0735

Answer: About 7.35% chance

Example 3: Required Sample Size

Problem: To achieve SE ≤ 2 when σ = 20, what n is needed?

Solution:

Want: σ/√n ≤ 2
20/√n ≤ 2
√n ≥ 10
n ≥ 100

Answer: Need at least n = 100

8. Common Mistakes to Avoid

Confusing σ and SE
- σ = standard deviation of individuals
- SE = standard deviation of sample means
- Always use SE (not σ) when working with sample means
Forgetting to check CLT conditions
- Always verify n ≥ 30 or population normal
- For proportions, check np ≥ 10 AND n(1-p) ≥ 10
Using wrong formula for SE
- For means: SE = σ/√n
- For proportions: SE = √(p(1-p)/n)
- Don't mix them up!
Not squaring when solving for n
- If √n = 10, then n = 100 (not 10!)
Thinking larger samples change μₓ̄
- μₓ̄ = μ regardless of sample size
- Sample size affects SE, not the center

9. Quick Decision Guide

Which Distribution Should I Use?

If you're finding probability about...	Use...
One individual value (x)	Population distribution with σ
Sample mean (x̄)	Sampling distribution with SE = σ/√n
Sample proportion (p̂)	Sampling distribution with SE = √(p(1-p)/n)

When Can I Use Normal Distribution?

For...	Conditions
Individual values	Population must be normal
Sample means	n ≥ 30 OR population normal
Sample proportions	np ≥ 10 AND n(1-p) ≥ 10

10. Formula Sheet

All Key Formulas

Sample Proportion:

p̂ = x / n

Mean of Sampling Distribution:

μₓ̄ = μ

μₚ̂ = p

Standard Error:

SE(x̄) = σ / √n

SE(p̂) = √(p(1-p) / n)

z-scores:

z = (x̄ - μ) / (σ/√n)

z = (p̂ - p) / √(p(1-p)/n)

Sample Size for Desired SE:

n = (σ / SE)²

(For sample means)

11. Study Tips

Understand the "why": Don't just memorize formulas—understand why CLT works
Master notation: Keep parameters (Greek) vs. statistics (Roman) straight
Always check conditions: Before applying CLT, verify n ≥ 30 or normal population
Draw pictures: Sketch normal curves to visualize what you're finding
Practice SE calculations: Get comfortable with σ/√n and √(p(1-p)/n)
Compare SD vs. SE: Remember SE measures precision of estimates, not data spread
Work many examples: Sampling distributions are abstract—practice builds intuition

Ready to Test Your Knowledge?

Use this study guide to prepare, then take the Module 5 quiz!

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Module 5: Sampling Distributions | Safaa Dabagh | Free Statistics Course

Module 5 Study Guide

1. Parameters vs. Statistics

Notation

2. Sampling Distributions

Important Distinction

3. The Central Limit Theorem (CLT)

CLT Conditions

Properties of Sampling Distribution of x̄

4. Standard Error

Standard Error vs. Standard Deviation

Standard Error Formulas

5. Sampling Distribution of Proportions

Properties of Sampling Distribution of p̂

Success-Failure Condition

6. Step-by-Step Procedures

Finding Probability about Sample Mean

Finding Probability about Sample Proportion

7. Example Problems

Example 1: CLT Application

Example 2: Sample Proportions

Example 3: Required Sample Size

8. Common Mistakes to Avoid

9. Quick Decision Guide

Which Distribution Should I Use?

When Can I Use Normal Distribution?

10. Formula Sheet

All Key Formulas

11. Study Tips

Ready to Test Your Knowledge?