See how sampling distributions converge to a bell curve regardless of the original population shape. Pick any distribution: uniform, skewed, or bimodal. Then draw random samples and watch the magic happen.
Choose a population. Each has a very different shape, but sample means will always converge.
Bell-shaped, symmetric. Mean and median coincide. Most natural measurements follow this.
μ=50.7 σ=14.3
Every value equally likely. Flat histogram, no center tendency.
μ=50.0 σ=29.4
Steep peak near zero with a long right tail. Models wait times, income distributions.
μ=24.1 σ=19.2
Two distinct peaks at ~30 and ~70. Represents two mixed sub-populations.
μ=50.6 σ=21.6
Adjust sample size and draw samples to build the sampling distribution.
Larger n → narrower sampling distribution (SE = σ/√n)
Histogram of the sample means x̄ drawn so far (bar height = how many means fall in that bin). Red dashed curve = predicted normal (CLT). SE = σ/√n = 5.36
Click "Draw 1 Sample" to start
Population (Uniform)
N = 500
Sample Means (n=30)
Draw samples →
0 means collected
CLT predicts: SE = σ/√n. Watch it converge.
Sample size
n: values per draw
Samples drawn
N: total draws
Empirical mean of means
x̄̄: average sample mean
Empirical SD of means
Observed spread of x̄
Theoretical SE
σ/√n: CLT prediction
Metrics will appear as you draw samples.
Confidence Intervals
Because sample means become approximately normal, we can build confidence intervals even when the population is not normal. The intervals are approximations whose accuracy improves as n grows.
Hypothesis Testing
t-tests, z-tests, and ANOVA rely on approximate normality of sample means. The CLT is why this approximation becomes good for large n.
The n ≥ 30 Rule of Thumb
For mildly non-normal distributions, n = 30 often gives a reasonably normal sampling distribution. But it is a heuristic, not a law: heavy tails or extreme skew can need hundreds of observations, and distributions without finite variance never converge at all.