Central Limit Theorem
In short: Central limit theorem states that when the sample size is large enough, the distribution of sample means is approximately normal, whatever the shape of the population the samples came from.
AP StatisticsStatisticsAP Statistics Unit 2
What is Central Limit Theorem?
The central limit theorem explains why the bell curve appears everywhere in inference. Even if individual values are skewed or lumpy, averages of many values pile up symmetrically around the population mean.
If a random sample of size \(n\) is drawn from a population with mean \(\mu\) and standard deviation \(\sigma\), then for large \(n\) the sample mean \(\bar{x}\) is approximately normal with mean \(\mu\) and standard deviation \(\frac{\sigma}{\sqrt{n}}\). A common guideline is \(n \ge 30\); a strongly skewed population may need more. If the population itself is normal, \(\bar{x}\) is normal for every \(n\).
How to spot it
- A population is skewed or unknown, yet the problem asks about the probability of a sample average.
- The sample size is described as large, such as 40, 64 or 100.
- The wording asks for the sampling distribution of x-bar, or a probability that x-bar exceeds a value.
- A confidence interval or test for a mean is justified by saying the sample is large enough.
The method
- Check that the sample is random and that the observations are independent (sample under 10% of the population).
- Check the size: \(n \ge 30\), or the population is roughly normal, so the central limit theorem applies.
- State the sampling distribution: mean \(\mu\) and standard deviation \(\frac{\sigma}{\sqrt{n}}\).
- Convert to a \(z\)-score and use the normal curve to find the requested probability.
Central Limit Theorem: solved example
Problem
Delivery times for a pizza shop are right-skewed with mean 40 minutes and standard deviation 25 minutes. A random sample of 100 deliveries is taken. What is the probability that the sample mean exceeds 45 minutes?
Solution
Sample size 100 is large, so \(\bar{x}\) is approximately normal with mean 40 and standard deviation \(\frac{25}{\sqrt{100}} = 2.5\).
\(z = \frac{45 – 40}{2.5} = 2\). From the standard normal table, \(P(Z > 2) = 1 – 0.9772 =\) 0.0228.
A sample average above 45 minutes happens about 2.3% of the time.
Try it
Question 1
A population has \(\mu = 120\) and \(\sigma = 16\). For \(n = 64\), what are the mean and standard deviation of \(\bar{x}\)?
Show the answer
Mean 120; standard deviation \(\frac{16}{\sqrt{64}} = 2\).
Question 2
If the population is exactly normal, is \(n \ge 30\) needed to say \(\bar{x}\) is normal?
Show the answer
No. Means of samples from a normal population are normal for any sample size, so the theorem is only needed for non-normal populations.
Central Limit Theorem vs. Law of Large Numbers
The law of large numbers says the sample mean heads toward \(\mu\) as \(n\) grows. The central limit theorem describes the shape and spread of the distribution of \(\bar{x}\) for a fixed large \(n\): approximately normal with standard deviation \(\frac{\sigma}{\sqrt{n}}\). One is about the destination, the other about the scatter around it.
Keep going
Lessons
Worksheets and quizzes
- Sampling Distributions and the Central Limit Theorem – Worksheet
- Sampling Distributions and the Central Limit Theorem – Quiz
Related words
Whole dictionary: A–Z · AP Statistics · Statistics