The population (what you're sampling from)
Distribution of the sample mean (the dashed curve is the CLT prediction)
Samples drawn
0
Mean of means
—
SD of means
—
CLT predicts SE = σ/√n
—
Pick any population — even a wildly skewed or two-humped one — draw samples, and watch the distribution of the sample mean turn into a normal bell curve.
The Central Limit Theorem (CLT) says that if you take many random samples of size n from any population with a finite mean and variance, the distribution of the sample means will be approximately normal — regardless of the population's own shape — provided n is reasonably large. The approximation gets better as n grows.
Averaging more values cancels out extremes, so sample means bounce around less. The standard deviation of the sample mean — the standard error — is σ/√n. Because of the √n, quadrupling the sample size halves the spread of the means. Watch the bell get narrower as you raise n.
It's the engine behind most everyday statistics. It's why confidence intervals and t-tests work even when the underlying data isn't normal — because the tests rely on the distribution of the mean, not the raw data, and the CLT makes that mean normal.
A common rule of thumb is n ≥ 30, but it depends on how skewed the population is. For a roughly symmetric population, even n = 5–10 looks normal; for a heavily skewed one you may need n = 50+. Use the Skewed population here and slide n to see for yourself where the bell becomes convincing.