Central Limit Theorem
Explore how sample means converge to a normal distribution
Statement
Let $X_1, X_2, \dots, X_n$ be i.i.d. random variables with mean $\mu$ and finite standard deviation $\sigma$. Then as $n \to \infty$:
$$\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} N(0, 1)$$
In Plain English
No matter the shape of the population, the distribution of sample means will be approximately normal for large enough sample sizes, centered at μ with standard error σ/√n.
Interactive Simulation
Population Mean (μ)
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Population Std (σ)
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Mean of X̄
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Empirical SE
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Theoretical SE (σ/√n)
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Ratio (≈ 1.00)
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How It Works
- Choose a population: Select any distribution — skewed, flat, bimodal, or discrete
- Set sample size (n): This is how many observations are in each random sample
- Draw many samples: We draw thousands of samples and compute the mean of each one
- Watch the magic: The distribution of those sample means forms a bell curve — that's the CLT!
Key Insights
Sample size matters: Larger n → better normal approximation AND narrower spread (SE = σ/√n)
Symmetry helps: Symmetric populations (like Uniform) converge faster than highly skewed ones (like Exponential)
It works for discrete too: Even dice rolls produce normally-distributed means!
When CLT fails: The CLT requires finite variance. Heavy-tailed distributions like the Cauchy distribution do not satisfy this — their sample means never converge to normal.
Why It Matters
| Application | How CLT helps |
|---|---|
| Confidence Intervals | Build intervals using the normal distribution |
| Hypothesis Testing | z-tests and t-tests rely on normality of X̄ |
| Quality Control | Control charts assume sample means are normal |
| Polling & Surveys | Margin of error calculations |