This simulator draws repeated independent samples from a skewed or binary parent population, averages each sample, and drops the mean into a growing histogram. Change the parent population and the number of observations per sample and compare the raw observations with the distribution of means.
• A real-time 3D scene with 4 inspectable parts (Parent population tray, One random sample, Averaging collector and Sample-mean histogram), with home view, focus-selected-part, auto-rotate, expand, instrument-cover and hide-labels scene tools, plus a model response curve beneath the scene. • Experiment controls: Parent population (Exponential · mean1, Bernoulli · p=0.2, Uniform[0,1]); Observations per sample (1-30); Repeatable random seed (1-99); show animated explanatory markers; pause/resume, 0.1 s and 1 s single-step buttons, four playback speeds and a restart button. • A Curves & measurements tab with a parameter-comparison chart, a live-measurements chart, the model equations and snapshot readouts (Sample means collected; Parent mean; Theoretical SD of sample means; Mean of sample means; Observed SD of sample means; Means outside displayed histogram). • An Experiments tab with 2 guided presets (single draws and thirty per average) and a Model verification bench that runs independent fresh models, plus a timestamped event log and a copyable trial report. • A Learn & assess tab with guided lessons, a knowledge-check quiz with reset and a written model-scope statement linking to a technical reference.
Pick an exponential, a Bernoulli with p = 0.2, or a uniform parent on [0, 1], then choose how many observations go into each average. Every histogram entry is one sample mean, not one raw observation. The lab reports the parent mean, the theoretical standard deviation of the sample mean (sigma over the square root of n), and the observed mean and standard deviation of the collected means.
As n grows the means cluster more tightly around the parent mean and the histogram looks more bell shaped, even when the parent is strongly skewed or only takes the values 0 and 1.
The theorem concerns the distribution of suitably standardized sample means, not the raw observations: increasing n does not make the original data normal. Quadrupling n halves the standard error because it scales as 1 over the square root of n.
Each run stops after 300 sample means, and a finite histogram need not look perfectly normal, especially for small n or rare Bernoulli outcomes. The lab uses independent, identically distributed draws with finite variance and makes no claim that every real population satisfies those conditions. A separate readout counts means falling outside the displayed histogram range.
No. The central limit theorem describes how the distribution of sample means approaches a normal shape; the underlying observations keep their own distribution. The simulator shows both the raw draws and the means so the difference is visible.
It is cut in half. The standard deviation of the sample mean is sigma divided by the square root of n, so four times as many observations per sample gives one half the spread of the means.
With few observations per sample the average still inherits much of the parent distribution's shape, especially for a skewed exponential or a rare-event Bernoulli. Increase the observations per sample to watch it become more symmetric.
No. The theorem needs independent, identically distributed observations with finite variance. The three parents in the lab meet that, but the lab makes no claim about arbitrary real-world data.