For a large enough sample, the sampling distribution of the mean is approximately normal, whatever the population shape.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why it matters
It means normal-based inference works even for badly skewed populations, which is most real data.
How large is large
A common guideline is n at least 30. A more skewed population needs a larger sample.
When it is not needed
If the population is already normal, the sampling distribution is normal for any sample size.
It says nothing about the data
The theorem describes sample means, not individual values. A larger sample does not make the population normal.
Centre and spread
The sampling distribution is centred on the population mean with standard error sigma ÷ √n.
The central limit theorem
For a large enough sample, the sampling distribution of the mean is approximately normal, whatever the shape of the population. That is a remarkable claim and it is why the normal distribution is everywhere.
How large is large enough
For roughly symmetric populations, quite small samples suffice. For strongly skewed ones, more are needed; n ≥ 30 is the usual rule of thumb. It is a guideline, not a theorem.
It is about the statistic, not the data
The theorem says the distribution of sample means becomes normal. The population and any individual sample can stay as skewed as they like. This distinction is central and routinely lost.
Why it makes inference possible
Knowing the sampling distribution is what lets you compute how unusual an observed statistic is. Without the central limit theorem, inference would require knowing the population shape, which you never do.
Step 2: Try It Yourself
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Step 3: Watch an Example
One step at a time.
Watch Sana Justify Normality
A strongly skewed population has mean 40 and standard deviation 12. Sana takes samples of 36.
- Step 1
The population is skewed, so individual values are not normal.
Step 4: Your Turn
Practice makes it stick.
The Guideline
Problem 1 of 2
What sample size is commonly cited as large enough for the theorem?
The Standard Error
Problem 2 of 2
Sigma 12 and n = 36. What is the standard error?
Apply the Theorem
1 of 8
The population is already normal, n = 5. Is the sampling distribution normal? 1 yes, 0 no.
2 of 8
The population is skewed, n = 50. Approximately normal? 1 yes, 0 no.
3 of 8
Does the theorem make the population normal? 1 yes, 0 no.
4 of 8
Sigma 20, n = 100. Standard error?
5 of 8
Population mean 40. Where is the sampling distribution centred?
6 of 8
A more skewed population needs a larger or smaller n? 1 larger, 2 smaller.
7 of 8
Which statements does the theorem support?
8 of 8
Sigma 9, n = 9. Standard error?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Sigma 24 and n = 36. What is the standard error?
Question 2 of 2
What does the Central Limit Theorem describe?
What You Learned
- For large enough n, sample means are approximately normal regardless of the population.
- A guideline is n at least 30, larger for more skewed populations.
- The theorem describes sample means, never individual values.