The central limit theorem, explained with examples
The central limit theorem (CLT) explains why the normal distribution appears all over statistics. It says that averages of large samples are approximately normal, whatever the shape of the population they come from.
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Sampling distributions first
Imagine taking a random sample of 40 students, recording their mean sleep time, and then doing it again with a new sample, and again, thousands of times. The sample means would vary. The distribution of all those possible sample means is called the sampling distribution of the sample mean. The CLT describes its shape, centre and spread.
What the CLT says
If you take random samples of size n from a population with mean μ and standard deviation σ, then when n is large enough, the sampling distribution of the sample mean x̄ is approximately normal, with
- mean μx̄ = μ, the same as the population mean, and
- standard deviation σx̄ = σ/√n, often called the standard error.
The remarkable part is that this holds even when the population is not normal, such as a skewed distribution of incomes or a uniform distribution of dice rolls. The larger the sample, the closer to normal the sampling distribution becomes.
How large is large enough?
In AP Statistics, the common rule of thumb is n ≥ 30. If the population itself is normal, the sampling distribution of x̄ is exactly normal for any sample size, so you do not need the CLT. If the population is strongly skewed or has outliers, you may need a sample larger than 30 for the approximation to be good. When n is small and the population shape is unknown, look at the sample data: no strong skewness or outliers suggests the normal approximation is reasonable.
The conditions
- Random: the data come from a random sample or randomised experiment.
- 10% condition: when sampling without replacement, n ≤ 10% of the population, so that σ/√n is accurate.
- Normal/large sample: the population is normal, or n ≥ 30 by the CLT.
On free-response questions, name the CLT explicitly when you use it: “Since n = 50 ≥ 30, the sampling distribution of x̄ is approximately normal by the central limit theorem.”
Worked examples
Example 1: the standard error
Commute times have mean 30 minutes and standard deviation 12 minutes. Find the mean and standard deviation of the sampling distribution of x̄ for samples of 36 commuters.
- μx̄ = 30 minutes.
- σx̄ = 12/√36 = 2 minutes.
Example 2: a probability about a mean
Using the same population, find the probability that the mean commute time of a random sample of 36 commuters is more than 33 minutes.
- The population is likely skewed, but n = 36 ≥ 30, so by the CLT x̄ is approximately normal with mean 30 and SD 2.
- z = (33 − 30)/2 = 1.5, so P(x̄ > 33) ≈ P(Z > 1.5) ≈ 0.0668.
Example 3: individual versus mean
For the same population, why can we not find P(one commuter takes more than 33 minutes) with a normal model?
- The CLT is about sample means, not individual values. Individual commute times follow the population's own distribution, which may be skewed, so a normal calculation would not be justified.
- This difference between individuals and averages is one of the most tested ideas in the topic.
Example 4: the effect of sample size
How does the standard deviation of x̄ change if the sample size increases from 25 to 100?
- σ/√25 = σ/5 and σ/√100 = σ/10.
- It is halved. Quadrupling n halves the spread of the sample mean.
The CLT for proportions
The same idea applies to sample proportions. For random samples of size n from a population with proportion p, the sampling distribution of p̂ has mean p and standard deviation √(p(1 − p)/n), and it is approximately normal when the large counts condition holds: np ≥ 10 and n(1 − p) ≥ 10.
Example 5: a sample proportion
40% of a city's residents cycle to work. In a random sample of 150 residents, find the probability that more than 45% cycle.
- np = 60 and n(1 − p) = 90, both at least 10, so p̂ is approximately normal.
- SD = √(0.4 × 0.6/150) = 0.04. z = (0.45 − 0.40)/0.04 = 1.25.
- P(p̂ > 0.45) ≈ 0.106.
Why the CLT matters
Almost every confidence interval and significance test in AP Statistics relies on the sampling distribution being approximately normal. The CLT is what allows us to use z and t procedures on real-world data that is rarely perfectly normal. Without it, inference for means from skewed populations would need much more complicated methods.
Seeing it for yourself
A good way to understand the CLT is with a simulation. Roll a single die many times: the results are uniform, not bell-shaped. Now roll five dice at a time and record the mean of each group. The distribution of those means is already mound-shaped and centred on 3.5. With 30 dice per group it looks almost perfectly normal, and much narrower. Many online applets show this in seconds, and AP questions sometimes ask you to describe exactly this kind of simulation.
Common mistakes
- Saying the CLT makes the population normal; it is the sampling distribution that becomes normal.
- Using the CLT for a single observation instead of a sample mean.
- Forgetting to divide σ by √n.
- Thinking larger samples change the mean of the sampling distribution; they only reduce its spread.
- Not stating the CLT by name when justifying normality on a free-response question.
Practice questions
Try each one before opening the answer.
Q1. A population has μ = 100 and σ = 20. Find σx̄ for n = 64.
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Q2. Is the sampling distribution of x̄ approximately normal for n = 10 from a strongly skewed population?
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Q3. A population has μ = 50 and σ = 10. For n = 100, find P(x̄ < 48).
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Q4. What happens to the shape of the sampling distribution as n increases?
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Frequently asked questions
Is the central limit theorem on the AP Statistics exam?
Yes. It is in Unit 5, sampling distributions, and it is used to justify normality in inference throughout Units 6 to 8.
Is the CLT in A level Maths?
No, not in A level Maths. It appears in some A level Further Maths statistics options.
Key takeaways
- The CLT is about the sampling distribution of a mean, not individual values.
- Mean of x̄ is μ; its standard deviation is σ/√n.
- With n ≥ 30, x̄ is approximately normal whatever the population shape.
- For proportions, use the large counts condition.
- Name the CLT when you use it to justify a normal model.