Central Limit Theorem Calculator

Demonstrate the Central Limit Theorem. Specify a population distribution and sample size to see how the sampling distribution of the mean approaches normality.

What Is the CLT?
Formula
Conditions
FAQ

What Is the Central Limit Theorem?

The Central Limit Theorem (CLT) is one of the most important results in statistics. It states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's original distribution. This holds for any population with a finite variance.

The CLT explains why the normal distribution appears so frequently in nature and why many statistical methods (confidence intervals, hypothesis tests) work reliably even with non-normal populations.

Formula

X̄ ~ N(μ, σ²/n) as n → ∞

Standard Error = σ / √n

Conditions

Independence: Samples must be independent (random sampling, or n < 10% of population).
Sample size: n ≥ 30 is the common rule of thumb. For symmetric populations, smaller n suffices.
Finite variance: The population must have a finite mean and variance.

FAQ

Why is n=30 the rule of thumb?

For moderately skewed populations, n=30 is usually sufficient for the sampling distribution to be approximately normal. For heavily skewed or heavy-tailed distributions, larger samples may be needed.

Does CLT apply to proportions?

Yes. The sample proportion p̂ is approximately normal with mean p and SE=√(p(1-p)/n) when np≥10 and n(1-p)≥10.