StatReason

Sampling Distribution Simulator

See directly, through simulation, how much a sample mean bounces around from one sample to the next, and how that variability shrinks as sample size grows, the core intuition behind every standard error formula on this site.

What this answers

This tool answers "if I repeatedly drew samples of this size from this population, how much would the sample mean actually bounce around?" turning the abstract idea of a standard error into a visible, simulated distribution of real sample means.

How it is calculated

This calculator draws a normal population with your chosen mean and standard deviation, takes a simulated sample of your chosen size, computes that sample's mean, and repeats the whole process many times using a seeded, fully reproducible pseudorandom generator. The spread of those simulated sample means (the observed standard error) is then compared directly to the theoretical formula, the population standard deviation divided by the square root of the sample size.

Worked example

For a population with a standard deviation of 10 and a sample size of 25, the theoretical standard error is 10 divided by the square root of 25, exactly 2. Simulating 2,000 samples of size 25 from that population and measuring how much their means actually vary gives an observed standard error very close to that same value of 2, a direct empirical confirmation of the formula rather than just trusting it.

Assumption audit

Calculated from your data: the mean and standard deviation of the simulated sample means across every repetition, compared directly to the theoretical standard error formula.
Evidence to review: how closely the observed and theoretical standard errors match; with enough repetitions they should be very close, and a large mismatch would suggest a setup or interpretation issue rather than the formula itself being wrong.
You must verify: that your real-world population is reasonably well-approximated by a normal distribution if you want to extend this simulation's intuition to your actual data; this tool illustrates the concept using a normal model specifically, not a general proof for every possible population shape.

What this result does not mean

This simulation illustrates how the Central Limit Theorem behaves for a normal population; it does not prove the theorem holds for every possible population shape, though in practice the sample mean's distribution tends toward normal for a wide range of underlying shapes as sample size grows.

Limitations

This calculator simulates from a normal population model only at launch; simulating from a different underlying shape (skewed, uniform, or another named distribution) is not available from this specific engine.