Central Limit Theorem: Intro Stats Chapter 6

Why this chapter matters

Inference depends on how sample statistics vary across repeated samples, and the CLT explains why normal-based methods often work.

What you will learn

Differentiate a population distribution from a sampling distribution.
Compute standard errors for sample means and proportions under assumptions.
Use CLT conditions to justify approximate normal inference.

Understand the core ideas

The central limit theorem explains why many inference methods rely on normal approximations for sample statistics. A sampling distribution is the distribution of a statistic, such as sample mean or sample proportion, over repeated samples of the same size. This is different from the distribution of individual population values. For sample means, as sample size increases, the sampling distribution tends toward normal shape under broad conditions. Its center is the population mean, and its spread is the standard error, often written as sigma divided by square root of n.

The same logic extends to sample proportions when success-failure counts are sufficiently large. Increasing sample size reduces standard error, so estimates become more precise. This is the practical reason larger studies often produce narrower confidence intervals. The theorem does not say raw data become normal, and it does not remove bias from poor sampling. It only describes the behavior of statistics under repeated random sampling. Using it well means checking conditions before applying normal or t-based procedures.

Key terms

Sampling distribution: Distribution of a statistic across repeated samples of fixed size.
Standard error: Standard deviation of a statistic's sampling distribution.
Central limit theorem: Result stating that many sample statistics become approximately normal as sample size increases.
Sample proportion: Statistic p-hat, equal to sample successes divided by sample size.

Sampling distribution of mean

Population mean is 50, population standard deviation is 20, and sample size is 100.

Set sampling distribution center at 50.
Compute standard error: 20 / $\sqrt{100}$ = 2.
Standardize threshold 54: z = (54 - 50) / 2 = 2.
Use normal table upper tail at z = 2, approximately 0.0228.

Result: Under the model, probability that sample mean exceeds 54 is about 2.28 percent.

A common misconception

Claim: The CLT means the original population distribution becomes normal.

Correction: The theorem describes the sampling distribution of statistics, not a change in the underlying population.

Lessons in this chapter

Sampling distributionsDescribe the long-run distribution of a statistic across repeated samples.
CLT for sample meansApply normal approximation to x-bar with known or estimated spread behavior. Read the full guide →
Sampling distribution of p-hatUse success-failure checks and standard error for proportions.
Interpreting standard errorConnect sample size changes to precision changes.

Study task

Compare standard errors for sample sizes 25, 100, and 400 from the same population standard deviation, and explain the pattern.

Chapter checkpoint

How does doubling sample size affect standard error for a mean?

Standard error scales by 1 over square root of n, so doubling n multiplies standard error by 1/ $\sqrt{2}$ , about 0.707.

Central Limit Theorem