Central Limit Theorem

Unit 5 · Sampling Distributions

What AP Stats asks here

The Central Limit Theorem says the sampling distribution of $\bar{x}$ is approximately normal once $n$ is large enough, regardless of the population's shape. It is the engine that lets normal-based inference work in practice. The recurring AP trap: claiming the CLT makes the raw data normal — it does not. The CLT is about the sample mean, not about individual observations.

\bar{x}

Center

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

Standard error

\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}}

Approximate distribution

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

When does the normality kick in

Population normal

\text{any } n

Mild skew or symmetric

n \ge 30

Severe skew

n \ge 50 \text{ or larger}

AP Tip: The SE shrinks like 1/√n. Quadrupling the sample size halves the SE. Doubling it only divides the SE by √2 — diminishing returns set in quickly.

Caution: The CLT is about the sampling distribution of x̄, not the data. Heights of adults may or may not be normal; that's a biological fact, not a CLT result.

Type 1

Compute the standard error of $\bar{x}$

Divide the population SD by $\sqrt{n}$ . Then standardize the sample mean using $\sigma_{\bar{x}}$ (not $\sigma$ ) to find probabilities.

Example 1

Nails from a factory have

\mu = 50

mm and

\sigma = 2

mm. For an SRS of

n = 100

, what is

P(\bar{x} > 50.4)

? (A) ≈ 0.421 (B) ≈ 0.20 (C) ≈ 0.0228 (D) ≈ 0.50

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Type 2

When CLT applies

Normal population: any $n$ works. Symmetric or mildly skewed: $n \ge 30$ is the standard threshold. Severely skewed: ask for larger $n$ .

Example 2

For which scenario is the sampling distribution of

\bar{x}

safely treated as approximately normal? (A) Severely right-skewed population (e.g., income);

n = 25

. (B) Approximately normal population;

n = 8

— any n works because the population is normal. (C) Bimodal population;

n = 10

. (D) Heavy-tailed population;

n = 15

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Type 3

CLT is about $\bar{x}$ , not raw data

Misuse: 'The CLT means heights are normal.' The CLT says sample means become normal — not the raw observations.

Example 3

A student claims: 'Because of the CLT, heights of US adults are normally distributed.' Which critique is correct? (A) The claim is correct. (B) The CLT is about the sampling distribution of

\bar{x}

, not the distribution of raw data. The heights may or may not be normal — that's a biological fact, not a consequence of the CLT. (C) The CLT applies only when

n = 100

. (D) The CLT applies only when the data are normal to begin with.

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Central Limit Theorem

What AP Stats asks here

Sampling distribution of xˉ\bar{x}xˉ

When does the normality kick in

Compute the standard error of xˉ\bar{x}xˉ

When CLT applies

CLT is about xˉ\bar{x}xˉ, not raw data

Sampling distribution of $\bar{x}$

Compute the standard error of $\bar{x}$

CLT is about $\bar{x}$ , not raw data