Central Limit Theorem, explained.
The central limit theorem gives conditions under which a suitably centred and scaled sum or mean approaches a normal distribution as sample size grows.
Why it happens
Generate 200–1,000 independent sample means from n fair die or unit-mean exponential observations. Within each sample a fixed block of 30 random draws preserves the seed as n changes. Compare histogram, observed standard deviation and σ/√n.
The central limit theorem gives conditions under which a suitably centred and scaled sum or mean approaches a normal distribution as sample size grows.
Read the result
The histogram contains averages, not individual observations. Increasing observations per mean changes their spread, while collecting more means makes the sampling distribution easier to see.
A worked example
Averages of a skewed source
Begin with exponential observations and one observation per mean. The collected values retain a strongly right-skewed source shape.
Increase observations per mean to thirty while preserving the seed, and compare the histogram and observed standard deviation.
Collect additional means to see the shape more clearly. The theoretical standard deviation is one divided by the square root of thirty.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Generate 200–1,000 independent sample means from n fair die or unit-mean exponential observations. Within each sample a fixed block of 30 random draws preserves the seed as n changes. Compare histogram, observed standard deviation and σ/√n.
Where this idea is useful
A practical use
Averages can have a more regular sampling distribution than individual observations, helping explain some statistical approximations.
A common misconception
“The central limit theorem makes all large datasets normal.”
It concerns appropriately standardised sums or means under conditions. The original source distribution can remain skewed regardless of how many observations are collected.
What this explanation leaves out
- These sources are independent with finite variance. The approximation may be poor for small n and does not make raw data normal. The last histogram bin includes overflow; this is not a universal justification for normal inference.
Is thirty always enough for a normal approximation?
No. Required sample size depends on the source distribution and the accuracy needed. Dependence or infinite variance can invalidate familiar versions of the theorem.
Does your proposed approximation concern raw observations, an average, or something else entirely?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.