Sampling Distributions

What is a Sampling Distribution?

Statistic: Number calculated from sample (e.g., sample mean $\bar{x}$, sample proportion $\hat{p}$)

Sampling Distribution: Distribution of a statistic across all possible samples of size n

Key insight: Statistics vary from sample to sample (sampling variability). Sampling distribution describes this variability.

Example: Sampling Distribution of $\bar{x}$

Population: All students, μ = 70, σ = 10

Take many samples of n = 25:

• Sample 1: $\bar{x}_1$ = 72
• Sample 2: $\bar{x}_2$ = 68
• Sample 3: $\bar{x}_3$ = 71
• ...

Plot all sample means → Sampling distribution of $\bar{x}$

Properties of Sampling Distribution of $\bar{x}$

Center:

$\mu_{\bar{x}} = \mu$

Sample mean is unbiased estimator of population mean

Spread:

$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

Called standard error (SE)

Key: Larger sample → smaller standard error (more precise estimates)

Shape:

• If population normal → sampling distribution exactly normal
• If population not normal → approximately normal if n large enough (CLT)

Sampling Distribution of Sample Proportion

Population proportion: p
Sample proportion: $\hat{p} = \frac{\text{count}}{n}$

Center:

$\mu_{\hat{p}} = p$

Spread:

$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

Shape: Approximately normal if np ≥ 10 and n(1-p) ≥ 10

Example: Coin Flips

Fair coin (p = 0.5), n = 100 flips

Center: $\mu_{\hat{p}}$ = 0.5

Spread: $\sigma_{\hat{p}} = \sqrt{\frac{0.5(0.5)}{100}} = \sqrt{0.0025} = 0.05$

Shape: np = 50 ≥ 10, n(1-p) = 50 ≥ 10 → approximately normal

Interpretation: Sample proportions typically within 0.05 of true value 0.5

Bias vs Variability

Bias: Systematic over- or under-estimation

• $\mu_{statistic} \neq$ parameter

Variability: Spread of sampling distribution

• Measured by standard error

Ideal: Low bias AND low variability (unbiased with small SE)

Increase n:

• Doesn't reduce bias
• DOES reduce variability (SE decreases)

Standard Error

Standard Error (SE): Standard deviation of sampling distribution

For sample mean: $SE_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

For sample proportion: $SE_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

Key pattern: SE ∝ 1/√n

To cut SE in half, need 4× sample size

Using Sampling Distributions

Find probabilities about statistics:

Example: Population μ = 100, σ = 15. Sample n = 25.

P($\bar{x}$ > 105) = ?

$\bar{x} \sim N(100, 15/\sqrt{25}) = N(100, 3)$

Standardize: $z = \frac{105-100}{3} = 1.67$

P(Z > 1.67) ≈ 0.0475

Difference Between Two Means

Two independent samples:

$\mu_{\bar{x}_1 - \bar{x}_2} = \mu_1 - \mu_2$

$\sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$

Shape: Approximately normal if both samples meet conditions

Difference Between Two Proportions

$\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2$

$\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$

Conditions: Each sample meets np ≥ 10 and n(1-p) ≥ 10

Simulating Sampling Distributions

Steps:

1. Take sample of size n from population
2. Calculate statistic
3. Repeat many times
4. Plot distribution of statistics

Result: Empirical approximation of theoretical sampling distribution

Common Misconceptions

❌ Confusing population distribution with sampling distribution
❌ Thinking larger sample reduces bias (only reduces variability)
❌ Forgetting √n in denominator of SE
❌ Using σ instead of σ/√n for $\bar{x}$

Quick Reference

Sampling Distribution of $\bar{x}$:

• Center: μ
• Spread: σ/√n
• Shape: Normal (if population normal or n large)

Sampling Distribution of $\hat{p}$:

• Center: p
• Spread: √(p(1-p)/n)
• Shape: Normal (if np ≥ 10 and n(1-p) ≥ 10)

Remember: Statistics vary from sample to sample. Sampling distribution describes this variability!