All Subjects



AP Stats



Unit 5

5.3 The Central Limit Theorem

3 min readjune 4, 2020


Harrison Burnside


Josh Argo

Brianna Bukowski


The Central Limit Theorem is often tested on free response questions dealing with quantitative data (means). Now, you cannot assume that the sample is normally distributed. The question must either explicitly state so, or you have to follow the central limit theorem.
The Central Limit Theorem states that if n (the sample size) is large, the sampling distribution is normal. The larger n is, the more normal the sample is.
So, what is large? If n is greater or equal to 30, it is considered large. Thus, if the same size is greater than or equal to 30, it is approximately normally distributed.
Also, the samples must all be independent of each other. One way to assume this is if the sample size is 10% or less of the population.
Another aspect to check is that the sample is a simple random sample.
So, for a quantitative sample to be normally distributed according to the central limit theorem, it must:
  • Be independent
  • Be a random sample
  • Have an n that is greater than 30


Use the Central Limit Theorem when calculating the probability about a mean or average.
For example, if a question asks for the probability about the mean size of fish, you cannot assume that the sample of fish, say in a pond, is greater than 30 unless the problem states so.
In this question, reasoning might include that since the sample of fish is 40, which is greater than 30, it is approximately normal. This is because according to the central limit theorem, if n is large, then the sample is approximately normally distributed. Additionally, you must state that the samples are independent of each other.  This is true for both quantitative and categorical data (and the statement is the same). 
You must explicitly state that you assumed the sample is normally distributed because of the central limit theorem.
The picture below shows what happens as you increase your sample size. The smallest triangle is a sample size of 5 and the tallest is a sample size of 100.

Source: SurgeForce Demo

Bigger is Better!

With sampling distributions, the larger the sample size, the less spread the curve is going to have. This is because the larger a sample is, the more the sampling distribution is going to hone in on the true population parameter
Think of this...if you flip a coin 6 times, the proportion of heads you get is likely to be greater than or less than 0.5 (the data is somewhat skewed). However, once you flip the coin 50, 100, 1000 times, it is very unlikely that the proportion of heads is going to be much different from the true population proportion, which we know is 0.5.  
Due to this concept, the larger the sample size, the better.   A large sample size allows us to hone in on the population parameter (either 𝝁 or 𝝆), which is EXACTLY what we are after when using a sampling distribution.

Was this guide helpful?

🔍 Are you ready for college apps?
Take this quiz and find out!
Start Quiz
FREE AP stats Survival Pack + Cram Chart PDF
Sign up now for instant access to 2 amazing downloads to help you get a 5
Browse Study Guides By Unit
Big Reviews: Finals & Exam Prep
Free Response Questions (FRQs)
Unit 1: Exploring One-Variable Data
Unit 2: Exploring Two-Variable Data
Unit 3: Collecting Data
Unit 4: Probability, Random Variables, and Probability Distributions
Unit 6: Inference for Categorical Data: Proportions
Unit 7: Inference for Qualitative Data: Means
Unit 8: Inference for Categorical Data: Chi-Square
Unit 9: Inference for Quantitative Data: Slopes
Join us on Discord
Thousands of students are studying with us for the AP Statistics exam.
join now
Play this on HyperTyper
Practice your typing skills while reading The Central Limit Theorem
Start Game
💪🏽 Are you ready for the Stats exam?
Take this quiz for a progress check on what you’ve learned this year and get a personalized study plan to grab that 5!
Hours Logo
Studying with Hours = the ultimate focus mode
Start a free study session
📱 Stressed or struggling and need to talk to someone?
Talk to a trained counselor for free. It's 100% anonymous.
Text FIVEABLE to 741741 to get started.
© 2021 Fiveable, Inc.