Normal Distribution
Tail probabilities under a (possibly non-normal) bell curve. Adjust the four moments and watch the percentages move.
The normal distribution is fully described by two numbers — its mean (μ) and standard deviation (σ) — and tail probabilities follow from those alone. Real data rarely match exactly: distributions skew, and they have heavier or lighter tails than the normal predicts. This visualization extends the standard display with two extra shape parameters — skewness (γ₁) and excess kurtosis (γ₂) — using the Gram–Charlier / Edgeworth expansion, so you can see how tail probabilities shift when those moments deviate from zero.
x = —
x₂ = —
P = —
under N(μ, σ²): —
Or invert: find x for P =
Click on the curve to place a marker.
⚠ Density goes negative in part of the displayed range — the Gram–Charlier expansion is invalid for these moments. Probabilities are still computed from the closed-form CDF.
μ, σ: shift and scale the curve. Tail probabilities don't change — μ and σ just relabel the x-axis.
γ₁: skewness. Positive ⇒ longer right tail and lower-tail probabilities shrink.
γ₂: excess kurtosis. Positive ⇒ heavier tails (extreme events more likely than the normal predicts) and a sharper peak.
The dashed purple curve shows N(μ, σ²) as a reference whenever the distribution is non-normal.
γ₁: skewness. Positive ⇒ longer right tail and lower-tail probabilities shrink.
γ₂: excess kurtosis. Positive ⇒ heavier tails (extreme events more likely than the normal predicts) and a sharper peak.
The dashed purple curve shows N(μ, σ²) as a reference whenever the distribution is non-normal.
Citation
Persson, B. N. (2026). Normal Distribution [Interactive visualization]. https://bjorn-persson.github.io/visualizations/normal-distribution/
@misc{Persson2026normaldistribution,
author = {Björn N. Persson},
year = {2026},
title = {Normal Distribution},
note = {Interactive visualization},
url = {https://bjorn-persson.github.io/visualizations/normal-distribution/}}