The Central Limit Theorem (CLT) says that if you repeatedly draw a sample of
n values from almost any population — flat, lopsided, or split into two
humps — and average each sample, the histogram of those averages approaches a bell
curve as n grows, no matter how strange the original population looked.
n independent values from the population, averages them, and adds one data point to the front histogram.σ/√n — bigger samples produce averages that cluster tighter around the true population mean, and the shape becomes normal even when the population wasn't.The CLT is why so many real-world measurements — heights, exam scores, measurement errors — look approximately normal: they're often the sum or average of many small, independent effects, and it barely matters what those individual effects look like.
Pick a lopsided, exponential or two-humped population, draw repeated samples of size n from it, and watch the 3D histogram of the sample averages sharpen into a bell curve — live, no matter how strange the population looks.
The teal row is the fixed population you're sampling from; the green row is the sampling distribution of the mean. As n grows, the green histogram narrows toward σ/√n and its shape converges on the theoretical normal curve overlay, regardless of the population's own shape.
Choose a population shape, raise the sample size n, and increase draw speed to accumulate sample means quickly. Compare the live mean and predicted standard deviation against the growing histogram, and reset whenever you change n to see the shape rebuild from scratch.
The CLT holds for almost any population with finite variance — it's why averages of noisy, skewed real-world measurements so often end up looking like a tidy bell curve.