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Concept Breakdowns

Central Limit Theorem and Sampling Distributions

The Central Limit Theorem and sampling distributions form the backbone of inferential statistics, essential for AP Statistics, college stats courses, and the GRE Quantitative section. Understanding how sample means behave — and why they approximate a normal distribution — unlocks confidence intervals, hypothesis tests, and virtually every statistical inference procedure.

Interactive Deck

5 Cards
1
Front

What is the Central Limit Theorem?

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1
Back

Central Limit Theorem (CLT): As sample size increases, the distribution of sample means approaches a normal distribution, regardless of population shape — typically when n ≥ 30.

2
Front

What is a sampling distribution?

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2
Back

Sampling distribution: The probability distribution of a statistic (e.g., sample mean) across all possible samples of size n drawn from a population.

3
Front

Standard error of the mean formula

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3
Back

Standard Error (SE) = σ/√n

  • σ = population standard deviation
  • n = sample size
  • SE decreases as n increases
4
Locked

When does the CLT apply?

5
Locked

How does sample size affect sampling distribution?

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Frequently Asked Questions

What is the difference between a population distribution and a sampling distribution?

Population distribution describes all individual values in a population, while a sampling distribution describes the distribution of a statistic computed from repeated samples.

  • Population distribution: values of individuals
  • Sampling distribution: values of sample statistics

Why is the Central Limit Theorem important in statistics?

The CLT is the foundation of inferential statistics. It guarantees that, for large enough samples, the distribution of sample means is approximately normal — enabling the use of z-tests, t-tests, and confidence intervals even when the population is not normal.

How large must a sample be for the CLT to apply?

A sample size of n ≥ 30 is the standard rule of thumb for the CLT to produce an approximately normal sampling distribution. However, larger samples are needed for heavily skewed or multi-modal populations.