The central limit theorem

About this lesson

Statistics (AP-aligned) › Sampling distributions

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The central limit theorem

Warm-up

Three quick questions from earlier lessons.

Discover

Waiting times at a clinic are strongly right-skewed. You average the times of 50 random patients, many times over. What shape do these averages have?

  1. Approximately normal
  2. Right-skewed

Key idea

Averages tend to be normal.

Central limit theorem: for random samples of size nn from any population with finite SD, the sampling distribution of xˉ\bar{x} is approximately normal when nn is large (a common guide: n≥30n \ge 30, more for strong skew). It is about the distribution of sample means, not of the data.

Worked example

Three situations.

Any nn: xˉ\bar{x} is normal
Normal population.
xˉ\bar{x} approximately normal (CLT)
Skewed population, n = 60.
xˉ\bar{x} still skewed: CLT does not apply yet
Skewed population, n = 4.

Your turn, guided

Your turn: a population with σ=15\sigma = 15 and n=36n = 36. The SD of the (approximately normal) xˉ\bar{x}.

2.52.5
15/6.

Practice ★

  1. Household incomes are strongly right-skewed. What is the shape of the sampling distribution of x̄ for random samples of 52 households?
    1. Approximately normal, by the central limit theorem
    2. Right-skewed like the population
    3. Uniform
  1. A population is strongly skewed. Is the sampling distribution of x̄ for samples of size 3 approximately normal?
    1. Not necessarily: n = 3 is too small for the CLT to apply to a strongly skewed population
    2. Yes, sample means are always normal
    3. No, sample means are never normal

Watch out!

Practice ★★

  1. Which statement is the central limit theorem, for random samples of size 166?
    1. The distribution of sample means is approximately normal
    2. The data in a large sample are approximately normal
    3. The population becomes normal when n is large

Exit ticket

Three questions to check you have got it.

  1. Household incomes are strongly right-skewed. What is the shape of the sampling distribution of x̄ for random samples of 52 households?
    1. Approximately normal, by the central limit theorem
    2. Right-skewed like the population
    3. Uniform
  2. Household incomes are strongly right-skewed. What is the shape of the sampling distribution of x̄ for random samples of 73 households?
    1. Approximately normal, by the central limit theorem
    2. Right-skewed like the population
    3. Uniform
  3. Household incomes are strongly right-skewed. What is the shape of the sampling distribution of x̄ for random samples of 72 households?
    1. Approximately normal, by the central limit theorem
    2. Right-skewed like the population
    3. Uniform

Summary card

Central limit theorem

Large nn: xˉ\bar{x} approximately normal, whatever the population.

Watch out: it is about means, not about the data.

End of the lesson

Next lesson: Sampling distributions of differences Back to the chapter

Answers: The central limit theorem

A

  1. Approximately normal, by the central limit theorem

B

  1. Not necessarily: n = 3 is too small for the CLT to apply to a strongly skewed population

C

  1. The distribution of sample means is approximately normal

D

  1. Approximately normal, by the central limit theorem
  2. Approximately normal, by the central limit theorem
  3. Approximately normal, by the central limit theorem

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