Central Limit Theorem

Statement of the CLT

The Central Limit Theorem (CLT) states:

For a random sample of size $n$ from any population with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of $\bar{x}$ is approximately Normal when $n$ is sufficiently large:

$\bar{x} \approx N\left(\mu, \frac{\sigma}{\sqrt{n}}\right) \text{ for large } n$

Why Is the CLT Important?

1. It works for any population shape — even skewed or bimodal!
2. It justifies using Normal-based methods for inference
3. It's the theoretical foundation for confidence intervals and hypothesis tests

How Large Is "Large Enough"?

The required sample size depends on the population shape:

| Population Shape | Required $n$ | |-----------------|-------------| | Normal | Any $n$ (already Normal!) | | Slightly skewed | $n \geq 15$ | | Strongly skewed | $n \geq 30$ | | Extremely skewed or with outliers | $n \geq 40+$ |

Rule of thumb: $n \geq 30$ is generally sufficient for the CLT to apply.

CLT for Sample Proportions

The CLT also applies to $\hat{p}$. When $np \geq 10$ and $n(1-p) \geq 10$:

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

Visual Understanding

As $n$ increases, the sampling distribution of $\bar{x}$:

1. Shape: Becomes more Normal (bell-shaped)
2. Center: Stays at $\mu$ (unbiased)
3. Spread: Decreases (proportional to $\frac{1}{\sqrt{n}}$)

Example

A population has $\mu = 100$ and $\sigma = 20$ (not Normal).

For samples of $n = 50$: $\bar{x} \approx N\left(100, \frac{20}{\sqrt{50}}\right) = N(100, 2.83)$

$P(\bar{x} > 105) = P\left(Z > \frac{105 - 100}{2.83}\right) = P(Z > 1.77) \approx 0.0384$

Common Misconception

The CLT says the sampling distribution becomes Normal. It does NOT say the population becomes Normal or the sample data becomes Normal.

AP Tip: On the AP exam, always check whether the CLT applies by verifying sample size conditions. State: "Since $n = 50 \geq 30$, the CLT tells us the sampling distribution of $\bar{x}$ is approximately Normal."