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AP®StatisticsCollege BoardStudy GuidesUnit 5: Sampling DistributionsSampling DistributionsThe Central Limit Theorem

The Central Limit Theorem (College Board AP® Statistics): Study Guide

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on

Central Limit theorem

What is the Central Limit theorem?

  • The Central Limit theorem states that

    • if a population is not normally distributed

      • but has a population mean mu

      • and population standard deviation sigma

    • and a large enough random sample of size n is taken

      • where n greater or equal than 30

      • and sample values are independent of each other

    • then the sampling distribution for sample means:

      • is approximately normally distributed

      • with mean mu

      • and standard deviation fraction numerator sigma over denominator square root of n end fraction

  • The approximation gets better as the sample size, n, increases

Examiner Tips and Tricks

The mean, mu, and the standard deviation, fraction numerator sigma over denominator square root of n end fraction, are given in the exam under 'Sampling distributions for means', in the row called 'For one population'.

How do I use the Central Limit theorem?

  • You can use the Central Limit theorem to calculate probabilities involving sample means, x with bar on top, taken from any population distribution

    • including population distributions that

      • are heavily skewed

      • are completely uniform (horizontal)

      • have multiple peaks

    • as long as the sample size is at least 30

Skewed, uniform or multi-peaked population distributions.
Skewed, uniform or multi-peaked population distributions

  • Probabilities using the Central Limit theorem will be estimates

    • as sample means follow an approximate normal distribution

    • Its standardized z-statistic is approximately fraction numerator x with bar on top minus mu over denominator fraction numerator sigma over denominator square root of n end fraction end fraction

      • mu and sigma are usually given in the question

  • If either mu or sigma are not given in the question then you may have to find them from the context or from other parts of the question

Examiner Tips and Tricks

In Central Limit theorem questions, always show that you have checked the sample size satisfies n greater or equal than 30!

What happens if the population is normally distributed?

  • If the population is normally distributed, then the Central Limit theorem is not needed

    • The sampling distribution for sample means will be exactly normally distributed

      • not approximately

    • It will have a mean of muand a standard deviation of fraction numerator sigma over denominator square root of n end fraction, where n is any sample size

      • not just n greater or equal than 30

Worked Example

The number of minutes that it takes to wait for a bus at a particular bus stop is distributed evenly between 0 minutes and 50 minutes. Any particular number of minutes that a person is required to wait between these two times is equally likely to occur. The standard deviation of waiting times is 14.4 minutes.

(a) If a person travels on the bus from this bus stop 40 times, estimate the probability that the mean time they have to wait exceeds 26 minutes.

Answer:

This a probability question about the mean of a sample (a sample of 40 waiting times)

Waiting times are not normally distributed (they are evenly distributed)

This means the Central Limit theorem is required (check n greater or equal than 30)

The sample size is n equals 40 which satisfies n greater or equal than 30

The Central Limit theorem states that sample means follow an approximate normal distribution with mean mu and standard deviation fraction numerator sigma over denominator square root of n end fraction

The question gives sigma but not mu, so work this out from the context

The mean waiting time between 0 and 50 minutes, if all times are equally likely, is 25 minutes

mu equals 25

Use the standard deviation of 14.4 minutes and n equals 40 to find fraction numerator sigma over denominator square root of n end fraction

fraction numerator sigma over denominator square root of n end fraction equals fraction numerator 14.4 over denominator square root of 40 end fraction equals 2.2768399...

Find the probability the sample mean exceeds 26, straight P open parentheses top enclose X greater than 26 close parentheses

The z-score is

fraction numerator 26 minus 25 over denominator 2.2768399... end fraction equals 0.4392...

Using 0.44 from the tables gives 0.6700 as the probability of being less than z

Subtract this from 1 to find the probability of exceeding z

1 - 0.6700

The probability that the mean waiting time exceeds 26 minutes is approximately 0.3300

(b) If, instead, the person travels on the bus from this bus stop 15 times, explain whether or not the method used in part (a) is still appropriate.

Answer:

The method used in part (a) involves the Central Limit theorem

The Central Limit theorem requires that n greater or equal than 30

However the sample size here is n equals 15 which is less than 30

So the method in part (a) is not appropriate

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    Previous:Sampling Distributions for Sample MeansNext:Sampling Distributions for Differences in Sample Means
    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.