Key Takeaways: Sampling Distributions and the Central Limit Theorem

One-Sentence Summary

The Central Limit Theorem guarantees that the sampling distribution of the mean is approximately normal — regardless of the population's shape — with mean $\mu$ and standard error $\sigma / \sqrt{n}$, making it the single theorem that bridges probability to inference and enables confidence intervals, hypothesis tests, and every other tool you'll learn in the rest of this course.

Core Concepts at a Glance

Three Distributions: Don't Confuse Them

The key insight: individual values can follow any distribution, but their averages follow a normal distribution. This is the CLT.

The Central Limit Theorem: Three Guarantees

$$\bar{X} \stackrel{\text{approx.}}{\sim} N\!\left(\mu, \frac{\sigma^2}{n}\right)$$

When Is $n$ "Large Enough"?

Rule of thumb: $n \geq 30$ works for most practical situations.

Standard Error Formulas

For the Mean

$$\boxed{\text{SE}_{\bar{x}} = \frac{\sigma}{\sqrt{n}}}$$

When $\sigma$ is unknown (the usual case), estimate it:

$$\widehat{\text{SE}}_{\bar{x}} = \frac{s}{\sqrt{n}}$$

For a Proportion

$$\boxed{\text{SE}_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}}$$

CLT conditions for proportions: $np \geq 10$ and $n(1-p) \geq 10$

Standardization

$$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \qquad \text{or} \qquad z = \frac{\hat{p} - p}{\sqrt{p(1-p)/n}}$$

The Diminishing Returns of Larger Samples

$$\text{SE} = \frac{\sigma}{\sqrt{n}}$$

The rule: To halve the standard error, quadruple the sample size.

      SE
      ▲
  σ   │ ╲
      │   ╲
      │     ╲__
      │        ╲___
      │            ╲_____
      │                  ╲__________
  0   ┼───────────────────────────── n
      0                      ►
        Steep drop          Diminishing returns
        at first            at large n

Law of Large Numbers vs. CLT

Both explain why statistics works: the LLN says sample means aim for the truth; the CLT says they're predictably distributed around the truth.

CLT Conditions and Cautions

Requirements

1. Random sample (or random assignment)
2. Independence (10% condition: $n \leq 0.10 \times N$)
3. Sufficiently large $n$ (depends on population shape)

What the CLT Does NOT Fix

Worked Example Summary: Sam's Daria Analysis

Python Quick Reference

import numpy as np
from scipy import stats

# --- Standard error calculations ---
sigma = 18    # Population SD (or use sample SD s)
n = 100       # Sample size
se = sigma / np.sqrt(n)

# Standard error for a proportion
p = 0.35
se_p = np.sqrt(p * (1 - p) / n)

# --- CLT probabilities ---
# P(x_bar > value)
mu = 126
prob = 1 - stats.norm.cdf(130, loc=mu, scale=se)

# --- CLT simulation ---
population = np.random.exponential(scale=10, size=100_000)
sample_means = [np.random.choice(population, size=30).mean()
                for _ in range(10_000)]
# sample_means will be approximately normal!

# --- Verify CLT conditions for proportions ---
print(f"np = {n * p:.1f} >= 10? {n * p >= 10}")
print(f"n(1-p) = {n * (1-p):.1f} >= 10? {n * (1-p) >= 10}")

Common Misconceptions

The One Thing to Remember

If you forget everything else from this chapter, remember this:

The Central Limit Theorem is why statistics works. It guarantees that sample means are approximately normally distributed — regardless of the population's shape — with a spread that shrinks as $\sigma / \sqrt{n}$. This means every sample mean you compute comes from a known, predictable distribution. That predictability is what makes confidence intervals possible (Chapter 12), hypothesis tests valid (Chapter 13), and A/B testing reliable (Chapter 16). Without the CLT, we'd compute sample means and have no idea how close they were to the truth. With it, we know the pattern of uncertainty — and that knowledge transforms data from a pile of numbers into a basis for decisions.

Key Terms