Normal Distribution Explorer

These percentages aren't rounded guesses — they fall directly out of the Gaussian integral, each one the area under e^(−z²/2) between ±1, ±2, and ±3 standard deviations, exactly what the error-function steps below compute.

Physicists borrow this scale for 'discovery' thresholds: the 2012 Higgs boson announcement required 5σ evidence, a one-sided false-positive rate of roughly 1 in 3.5 million, precisely because a real signal has to clear a bar that high before it's trusted over statistical noise.

The rule also explains why 'unlikely' events aren't actually rare in large datasets — with a million independent measurements you should expect about 63 to land beyond ±4σ purely by chance, while fewer than one would clear ±5σ. That gap is most of the reason the bar sits at five rather than four, the rest being a field that had already watched three-sigma results evaporate.

The bell curve plots probability density, not probability itself — so the height at any single point is not a probability. It compares values — twice the height means twice as likely per unit of x — but the chance of landing exactly there is zero. What matters is area: the shaded strip between x₁ and x₂ is the chance a value lands in that range, and the whole curve always encloses an area of exactly 1. Sliding μ moves the peak left or right; raising σ flattens and widens the curve, yet that total area never changes.

The curve was famous enough to be printed on cash: Germany's 10-Deutsche-Mark note, in circulation from 1991 to 2001, showed Carl Friedrich Gauss beside this very curve and its formula f(x). He refined it while modelling measurement errors in astronomy, which is where the curve earns its other name of the error function. Yet Abraham de Moivre had derived the shape back in 1733, decades before Gauss was born — a textbook case of Stigler's law, which jokes that nothing is named after its actual discoverer.