Tools · epiphany

A distribution, in glass and light.

The bell curve is usually drawn as a thin line on a flat axis. Here it is a stained-glass window: a statistically rigorous distribution computed for correctness, then rendered as living leaded glass. Move the controls and the figure — and its math — change with you.

Drag the four sliders to reshape the distribution. Show Values labels the real numbers on the glass — the mean, median and mode, the standard-deviation axis, and the true probability in each band. Quantiles re-cuts the window into equal-volume sections (every pane the same share of the area). High Contrast renders it as white glass on black. Save Window downloads the current window as an image.

A real object, not a drawing

Under the glass is an actual distribution — a sinh-arcsinh normal, which is an ordinary bell curve when skewness and kurtosis are zero and a principled generalization of it otherwise. Every number you see is computed by integrating that density, not eyeballed to fit the art: the standard deviation, the skewness, the excess kurtosis, the mean, the median, the mode, and the probability mass inside each section are all measured from the object itself.

Try it: turn on Show Values, then add skewness. The mean holds its place while the mode slides off toward the peak and the median settles between them — the textbook ordering mode < median < mean, drawn rather than asserted.

What each control does

μ — mean: Slides the whole distribution left and right. The mean line stays anchored to it.
σ — standard deviation: The spread. The ±1σ and ±2σ section lines move with it; the realized standard deviation always equals this value.
γ₁ — skewness: Leans the distribution, lengthening one tail. The readout shows the true skewness of the resulting shape.
γ₂ — excess kurtosis: Sharpens the peak and fattens the tails (positive) or flattens it (negative). The readout shows the true excess kurtosis — zero is a true Normal.

Equal width, or equal volume?

By default the vertical sections sit at equal standard-deviation distances — the famous 68 / 95 / 99.7 split, where the central panes hold most of the mass and the outer ones almost none. Switch on Quantiles and the same window is re-cut so every section holds an equal share of the distribution instead: the panes crowd together where the glass is bright and fan out into the dim tails. Same distribution, two ways of dividing the light.