Inferential Statistics • Lesson 5
Central Limit Theorem
Understand why the distribution of sample means tends to become approximately normal as the sample size becomes sufficiently large.
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What Does the Central Limit Theorem Say?
The Central Limit Theorem says that when independent observations are sampled from a population with finite mean and variance, the sampling distribution of the sample mean becomes approximately normal as the sample size becomes sufficiently large.
Learn by Seeing
Sample means → approximately normal
Population (any shape)
↓ sample size n large
Sampling distribution of X̄
↓
Approximately NormalThe theorem is about the sampling distribution of the mean — not the raw data.
Population · any shape
Sampling distribution
Interactive Activity
What does the Central Limit Theorem primarily describe?
Hint: It is about the distribution of a statistic, not individual observations.
Statistics Analyst Mindset
The theorem that makes data usable
Real populations are rarely normal. The CLT is what rescues you — it says that, provided the sampling is done properly and the samples are large enough, the distribution of the sample mean behaves as if the population were normal. That is the license behind most of applied inference.
Sample
Draw observations.
Repeat
Same size, many times.
Collect
One mean per sample.
Trust
The distribution is normal.
Lesson Summary
What you learned
Sample means become approximately normal.
No need for a normal population.
Independence and finite variance.
No universal cutoff — depends on the shape.
Foundation for CI, tests and standard error.
Population → samples → means → distribution.