Central Limit Theorem

Overview

The Central Limit Theorem (CLT) is one of the most important results in statistics. It states that the sampling distribution of the sample mean approaches a normal distribution as sample size increases, regardless of the population's distribution.

Statement

For a population with mean $\mu$ and finite variance $\sigma^2$, the sampling distribution of $\bar{x}$:

$$\bar{x} \sim N\left(\mu, \frac{\sigma^2}{n}\right) \quad \text{as } n \to \infty$$

Or equivalently, the standardized mean:

$$Z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} \sim N(0, 1)$$

Conditions

1. Independence: Observations are independent
2. Sample size: $n$ is "large enough" (typically $n \geq 30$)
3. Finite variance: Population has finite mean and variance

How Large is "Large Enough"?

Population Shape	Recommended $n$
Normal	Any $n$
Symmetric	$n \geq 15$
Moderate skewness	$n \geq 30$
Highly skewed	$n \geq 50$ or more

Visual Demonstration

Population (any shape):
    The CLT says: For large n,
     /\         the sampling distribution
    /  \        of x̄ becomes:
___/    \___          ∩
                    ╱    ╲
                  ╱        ╲
                ─┴──────────┴─
                   Normal

Key Points

1. Works for any shape: Population can be skewed, uniform, bimodal, etc.
2. Larger n → better approximation: More samples mean closer to normal
3. Focus on $\bar{x}$, not $X$: Individual observations keep the population distribution
4. Foundation of inference: Enables z-tests and confidence intervals

Applications

Confidence Intervals

When $n$ is large:

$$\bar{x} \pm z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$$

Hypothesis Testing

Test statistic (large $n$):

$$Z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

This $Z$ follows $N(0,1)$ approximately by CLT.

Examples

Example 1: Non-Normal Population

Die rolls (uniform distribution, $\mu = 3.5$, $\sigma = 1.71$):

For $n = 40$ die rolls, find $P(\bar{x} > 3.7)$:

$$SE = \frac{1.71}{\sqrt{40}} = 0.27$$ $$Z = \frac{3.7 - 3.5}{0.27} = 0.74$$ $$P(Z > 0.74) = 1 - 0.7704 = 0.23$$

Example 2: Skewed Population (Incomes)

Income: \mu = \60{,}000$,$\sigma = $25{,}000$ (right-skewed)

For $n = 100$, find P(\bar{x} < \55{,}000)$:

$$SE = \frac{25000}{\sqrt{100}} = 2500$$ $$Z = \frac{55000 - 60000}{2500} = -2.0$$ $$P(Z < -2.0) = 0.0228$$

Example 3: Sum of Random Variables

By CLT, for large $n$, the sum $S_n = X_1 + X_2 + \cdots + X_n$:

$$S_n \sim N(n\mu, n\sigma^2) \quad \text{approximately}$$

Applied to insurance claims, total sales, etc.

Relationship to Other Concepts

Concept	CLT Connection
Standard Error	$SE = \sigma/\sqrt{n}$ comes from CLT
Confidence Intervals	Justified by CLT
Z-tests	Work because of CLT
Sample Size	Larger $n$ means CLT applies better

Common Misconceptions

Misconception	Reality
"Population becomes normal"	No, only $\bar{x}$'s distribution does
"Works for any $n$"	Need $n$ large enough
"Individual values are normal"	No, only sample means are
"Exact normality"	It's an approximation

Why It Matters

1. Enables inference: Most statistical procedures rely on CLT
2. Practical flexibility: Don't need to know population distribution
3. Universal application: Works for many types of data
4. Quality control: Basis for control charts
5. Polling/surveys: Justifies sample-based conclusions