Central Limit Theorem Simulator

Math Probability • Continuous Probability Distributions

Written by STEM Calculators Team Published February 14, 2026 Updated February 24, 2026

Simulate the Central Limit Theorem by sampling means from a non-normal parent distribution and watching the histogram approach a normal curve.

Parent distribution

Uniform lower \(a\) Uniform upper \(b\)

For Uniform\([0,1]\): \(\mu=0.5\), \(\sigma^2=1/12\), and \(\mathrm{SE}=\sigma/\sqrt{n}=1/\sqrt{12n}\).

Sample size \(n\) Number of means (simulations) Histogram bins Random seed (optional)

Tip: Increase \(n\) to see faster convergence to a normal shape. Increase simulations to reduce noise in the histogram.

Ready

Histogram of sample means (CLT)

Animation ready

The animation adds simulated means in batches so you can watch the histogram evolve. The overlay is the normal approximation \(N(\mu,\sigma^2/n)\).

Choose a parent distribution and parameters, then click “Calculate”.

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

Theory

Central Limit Theorem (CLT): the big idea

The Central Limit Theorem explains why normal distributions appear everywhere in statistics. Suppose \(X_1,X_2,\dots,X_n\) are independent and identically distributed (i.i.d.) with finite mean \(\mu\) and finite variance \(\sigma^2\). Define the sample mean:

\[ \bar X=\frac{1}{n}\sum_{i=1}^{n}X_i. \]

Then, as \(n\) becomes large, the distribution of \(\bar X\) becomes approximately normal — even when the parent distribution is skewed or otherwise non-normal. A standard CLT statement is:

\[ \frac{\bar X-\mu}{\sigma/\sqrt{n}} \;\Rightarrow\; N(0,1)\quad \text{as } n\to\infty, \]

In words: the standardized mean converges in distribution to the standard normal.

Sampling distribution of the mean: \(\mu\), variance, and standard error

Regardless of the parent distribution’s shape (as long as \(\mu\) and \(\sigma^2\) exist), the mean and variance of \(\bar X\) are:

\[ E[\bar X]=\mu,\qquad \mathrm{Var}(\bar X)=\frac{\sigma^2}{n}. \]

The standard error (typical spread of \(\bar X\)) is:

\[ \mathrm{SE}=\sqrt{\mathrm{Var}(\bar X)}=\frac{\sigma}{\sqrt{n}}. \]

The simulator overlays the normal approximation \(N(\mu,\sigma^2/n)\) on top of the simulated histogram of means. As \(n\) increases, the histogram typically becomes more bell-shaped and concentrates around \(\mu\).

Parent distribution	\(\mu\)	\(\sigma^2\)	\(\mathrm{SE}=\sigma/\sqrt{n}\)
Uniform\([a,b]\)	\((a+b)/2\)	\((b-a)^2/12\)	\((b-a)/(\sqrt{12}\sqrt{n})\)
Exponential \(\mathrm{Exp}(\lambda)\)	\(1/\lambda\)	\(1/\lambda^2\)	\(1/(\lambda\sqrt{n})\)
Bernoulli \(\mathrm{Bern}(p)\)	\(p\)	\(p(1-p)\)	\(\sqrt{p(1-p)}/\sqrt{n}\)

How the simulator works (Monte Carlo view)

The simulator uses a Monte Carlo approach:

Choose a parent distribution for \(X\).
Pick \(n\) (the number of i.i.d. draws that go into one sample mean).
Repeat this experiment \(m\) times to create \(m\) sample means \(\bar X_1,\dots,\bar X_m\).
Plot a histogram of those means and overlay the normal curve \(N(\mu,\sigma^2/n)\).

The animation adds the means in batches so you can watch the histogram “settle” as more simulations accumulate. Increasing \(m\) reduces Monte Carlo noise; increasing \(n\) improves the normal approximation.

You can use a random seed to reproduce the same simulated histogram (helpful for teaching and screenshots).

University note: rate of convergence and Berry–Esseen (idea)

The CLT guarantees convergence, but it does not say how fast. A classic refinement is a Berry–Esseen type bound: the maximum difference between the standardized mean’s CDF and the standard normal CDF is typically on the order of \(1/\sqrt{n}\), and depends on the parent distribution’s tail behavior (a third absolute moment term).

\[ \sup_x \left|P\!\left(\frac{\bar X-\mu}{\sigma/\sqrt{n}}\le x\right)-\Phi(x)\right| \;\lesssim\; \frac{C}{\sqrt{n}}. \]

Practical takeaway: heavy skew or heavy tails often require larger \(n\) before the mean looks “nearly normal.” If the parent variance is infinite, the classical CLT may fail (you can get non-normal stable limits instead).

Advanced extension: multivariate CLT (random vectors)

In many university-level problems, observations are vectors \(X_i\in\mathbb{R}^d\). Under suitable conditions, the vector sample mean \(\bar X\) approaches a multivariate normal distribution:

\[ \sqrt{n}\,(\bar X-\mu)\;\Rightarrow\;N_d(0,\Sigma), \]

where \(\Sigma\) is the covariance matrix of the parent vector. This supports normal approximations in higher dimensions (and connects to asymptotic confidence regions and many university-level applications).