Central Limit Theorem

What is the Central Limit Theorem?

Central Limit Theorem (CLT): For large enough n, sampling distribution of $\bar{x}$ is approximately normal, regardless of population shape

This is remarkable! Population can be skewed, uniform, bimodal, anything → sampling distribution still approximately normal

Formal Statement

If:

• Take random samples of size n
• From ANY population with mean μ and standard deviation σ
• n is "sufficiently large"

Then: $\bar{x}$ is approximately distributed as:

$\bar{x} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)$

How Large is "Large Enough"?

Rule of thumb:

• Population roughly normal → n ≥ 10 okay
• Population moderately skewed → n ≥ 30
• Population heavily skewed or has outliers → n ≥ 40+

Key: More skewed population → need larger n

In practice: n = 30 often cited as general threshold

Why CLT Matters

Problem: Real populations rarely normal

Solution: CLT lets us use normal distribution for inference anyway (if n large enough)

Applications:

• Confidence intervals for means
• Hypothesis tests for means
• Control charts in quality control

Power: Works for ANY population distribution!

Example: Uniform Population

Population: Uniform on [0, 10]

• μ = 5
• σ = 10/√12 ≈ 2.89
• Shape: Rectangular (not normal!)

Sample n = 30:

$\bar{x} \sim N\left(5, \frac{2.89}{\sqrt{30}}\right) \approx N(5, 0.528)$

Even though population uniform, $\bar{x}$ approximately normal!

Visualizing CLT

Population: Right-skewed (e.g., salaries)

Sampling distributions for different n:

• n = 2: Still skewed
• n = 5: Less skewed
• n = 10: Nearly symmetric
• n = 30: Very close to normal

Pattern: As n increases, sampling distribution becomes more normal

CLT for Proportions

Also applies to sample proportions!

If: np ≥ 10 and n(1-p) ≥ 10

Then: $\hat{p}$ approximately normal:

$\hat{p} \sim N\left(p, \sqrt{\frac{p(1-p)}{n}}\right)$

This is why binomial → normal for large n!

Using CLT in Practice

Example: Battery life μ = 50 hours, σ = 8 hours (unknown distribution). Sample n = 40.

P($\bar{x}$ > 52) = ?

By CLT (n = 40 ≥ 30):

$\bar{x} \sim N(50, 8/\sqrt{40}) = N(50, 1.265)$

$z = \frac{52-50}{1.265} \approx 1.58$

P(Z > 1.58) ≈ 0.057

CLT vs Normal Population

If population already normal:

• Sampling distribution of $\bar{x}$ is exactly normal for any n
• Don't need CLT

If population not normal:

• Need CLT to justify normal approximation
• Require "large enough" n

Standard Error from CLT

Population σ usually unknown

Use sample standard deviation s:

$SE = \frac{s}{\sqrt{n}}$

For large n, s ≈ σ, so:

$\bar{x} \sim N\left(\mu, \frac{s}{\sqrt{n}}\right)$

This is basis for t-procedures!

Implications of CLT

1. Large samples good: Overcome non-normality

2. Can make inferences: Use normal probabilities for $\bar{x}$

3. Justifies methods: Confidence intervals, hypothesis tests work even if population not normal

4. Sample size planning: Can determine n needed for desired precision

CLT Limitations

Doesn't apply if:

• Sample not random
• Population infinite variance (very rare)
• n too small relative to skewness

Doesn't fix:

• Bias in sampling
• Non-random samples
• Measurement errors

Remember: CLT is about shape of sampling distribution, not about making bad samples good!

Historical Context

Discovered: 18th century
Pierre-Simon Laplace: Proved for binomial (1812)
Lindeberg-Lévy: General version (1920s)

One of most important theorems in statistics!

Sum vs Mean

CLT applies to both:

Sum: $S = \sum X_i$
$S \sim N(n\mu, \sigma\sqrt{n})$

Mean: $\bar{X} = S/n$
$\bar{X} \sim N(\mu, \sigma/\sqrt{n})$

Relationship: $\bar{X} = S/n$, so properties related

Checking Conditions

Before using CLT:

1. Random sample? (Independence)
2. 10% condition? (If sampling without replacement)
3. Large enough n? (Depends on population shape)

If all met → Proceed with normal approximation

Common Mistakes

❌ Using CLT with small n from skewed population
❌ Thinking CLT makes population normal (it doesn't!)
❌ Forgetting to divide by √n
❌ Applying CLT when sampling not random

Quick Reference

CLT: For large n, $\bar{x} \sim N(\mu, \sigma/\sqrt{n})$ regardless of population shape

Conditions:

• Random sample
• n ≥ 30 (general rule)
• 10% condition if without replacement

Power: Works for ANY population distribution!

Remember: CLT is the foundation for much of statistical inference. It's why we can use normal-based methods even when populations aren't normal!