This simulator lets you draw repeatable random observations from a normal, binomial or uniform model and watch a frequency histogram build up draw by draw. Change the distribution and its parameters, then compare what the sample shows with the theoretical mean and variance the model predicts.
• A real-time 3D scene with 4 inspectable parts (Random draw source, Frequency histogram, Model reference and Current observation), 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: Distribution (Normal, Binomial, Uniform); Normal mean (-2-2); Normal standard deviation (0.5-2); Binomial trials (2-20); Binomial success probability (0.1-0.9); 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 (Generated observations; Theoretical mean; Theoretical variance; Observed mean; Observed population variance; Outside displayed histogram). • An Experiments tab with 2 guided presets (skewed binomial and wider normal) 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.
Each draw is an independent, seeded random observation, so the same seed reproduces the same run. A normal model is set by its mean and standard deviation; a binomial model counts successes in a fixed number of trials with a chosen success probability; the uniform option samples evenly across a fixed interval from -2 to 2.
The theoretical values come from closed-form results: a normal has mean mu and variance sigma squared, a binomial has mean np and variance np(1-p), and the uniform on [-2, 2] has mean 0 and variance 4/3. As draws accumulate, the observed mean and observed population variance readouts move toward those targets, while early in a run they wander - which is the point of watching it grow.
The histogram shows counts, not probability density, and the overlaid model curve is scaled independently for readability, so heights of the bars and curve should not be compared numerically. For the normal model the histogram covers the mean plus or minus four standard deviations, and any draws beyond that window are counted in a separate outside-the-histogram readout rather than silently dropped.
Each run stops after 500 draws. A binomial with a small success probability produces a visibly skewed histogram, a reminder that not every distribution is bell shaped, and a short run will always look rougher than the theory - the simulator is designed so you can see both effects side by side.
No. Only the normal family is bell shaped. The simulator lets you switch to a binomial, which can be strongly skewed when the success probability is far from one half, and to a uniform, which is flat. The shape depends on the underlying model and its parameters.
Because the histogram is built from a finite number of random draws. Sample statistics fluctuate around the theoretical values and converge toward them as more draws accumulate, so the observed mean and variance readouts differ from the model values early in a run.
For n independent trials with success probability p, the variance is np(1-p), because each Bernoulli trial contributes p(1-p) and independent variances add. The simulator displays this as the theoretical variance next to the observed population variance.
The repeatable seed makes each experiment reproducible: the same seed and parameters always generate the same sequence of draws, so you can compare two settings fairly or return to an interesting run.