Draw samples from populations and build sampling distributions for means, proportions, and variances with live animation.
One sample mean is a single noisy number; a thousand of them tell a full story. This tool lets you define a population (distribution shape and parameters), then repeatedly draws samples of a chosen size, computes the mean, proportion, or variance of each, and animates the growing histogram of those statistics: the sampling distribution. A theoretical curve is overlaid for comparison. The most common confusion: the spread of this histogram depends on the sample size, not on how many samples you draw.
A concrete input and expected output from the current implementation.
Input
Skewed population, mean 50, SD 10; statistic: sample mean; sample size 25; draw 1,000 samples
Expected output
A histogram of 1,000 sample means that is already roughly bell-shaped even though the population is skewed. It centers near 50, and its spread runs close to 2, matching the standard error 10/sqrt(25) = 10/5 = 2; the normal overlay with mean 50 and SD 2 tracks the histogram.
Each sample mean lands near the population mean with spread sigma/sqrt(n), and here sigma/sqrt(n) = 10/5 = 2. With n = 25, the central limit theorem has already made the distribution of means approximately normal even though the population is not.
You pick a population (distribution shape and parameters), the statistic to track (mean, proportion, or variance), and a sample size n. Each draw takes n random values from the population, computes the statistic once, and adds the result as one bar in a running histogram, so the sampling distribution assembles itself sample by sample while the population histogram stays on screen for contrast.
The tool also draws the theoretical sampling distribution: normal with mean mu and SD sigma/sqrt(n) for sample means, normal with mean p and SD sqrt(p(1-p)/n) for proportions, and chi-square with n-1 degrees of freedom for the variance when the population is normal. Drawing more samples makes the histogram converge toward the overlay; a lasting mismatch usually means n is too small for the approximation.
The central limit theorem is an asymptotic statement: the approximation improves as n grows but is exact only when the population itself is normal. Heavy skew or extreme outliers in the population push the needed n much higher. Try n = 100 or more, and check that you have drawn enough samples to see the true shape rather than noise.
No. The standard error of the mean, sigma/sqrt(n), is determined by the population's spread and the sample size n; the number of draws does not appear in the formula at all. Drawing 10,000 samples instead of 100 makes the histogram smoother and closer to the theoretical curve, but its width stays about the same.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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