4.1 The Central Limit Theorem

What the Central Limit Theorem Says

The Central Limit Theorem (CLT) states that as sample size n increases, the sampling distribution of the sample mean x̄ becomes approximately Normal, regardless of the shape of the population distribution. This holds even if the original population is skewed or non-Normal, as long as n is sufficiently large (commonly n ≥ 30).

Mean and Standard Deviation of x̄

The sampling distribution of x̄ has mean μ_x̄ = μ (the population mean) and standard deviation σ_x̄ = σ/√n. As n increases, the spread of the sampling distribution decreases, meaning sample means cluster more tightly around the true population mean.

When Normality Already Holds

If the population itself is already Normally distributed, then the sampling distribution of x̄ is exactly Normal for any sample size, even small ones. The CLT is only needed to justify approximate Normality when the population shape is unknown or skewed and n is large.

Key terms

Central Limit Theorem (CLT)

As sample size n increases, the sampling distribution of the sample mean x̄ becomes approximately Normal, regardless of the population's shape.

Mean of sampling distribution of x̄

μ_x̄ = μ, the population mean — x̄ is an unbiased estimator of μ.

Standard deviation of sampling distribution of x̄

σ_x̄ = σ/√n — spread decreases as sample size increases.

When is CLT most useful?

When the population distribution is unknown or skewed and n is large (commonly n ≥ 30).

Sampling distribution if population is already Normal

x̄'s sampling distribution is exactly Normal for any sample size, no CLT needed.

Try an AP-style question

What does the Central Limit Theorem say?

1. Every sample mean equals the population mean once n is large
2. Large samples always come from populations that are Normal
3. For large n, x̄ is approximately Normal for any population shape
4. The population itself becomes Normal as the sample size increases