The normal distribution, visualized
The normal distribution — the bell curve — is the most important distribution in statistics and machine learning. It's set by just two numbers: the mean (where it's centered) and the standard deviation (how wide it spreads). It models measurement noise, averages, and the outputs of the Central Limit Theorem.
Slide the mean to move the bell sideways and the standard deviation to make it tall-and-narrow or short-and-wide. The shaded bands show the 68–95–99.7 rule: about 68% of values fall within one standard deviation of the mean, and about 95% within two.
Drag the mean and the spread
Shaded band μ ± σ
-1 → 1
≈ 68% of all data falls in here
μ ± 2σ
-2 → 2
≈ 95% of all data falls in here
Free · runs entirely in your browser · nothing to install
How to use it
- Slide the mean μ and watch the whole bell shift left or right.
- Slide the standard deviation σ and watch it get taller and thinner, or shorter and wider.
- Read the shaded bands to see the 68% (μ ± σ) and 95% (μ ± 2σ) ranges.
What you'll take away
- What the mean and standard deviation each control.
- The 68–95–99.7 rule and why '3-sigma events' are rare.
- Why standardizing (z-scores) rescales any normal to a common shape.
Want to actually build this?
This demo is one moment inside a full Math to Machine lesson — predict, build, and explain the concept, with an AI tutor that gives hints, not answers. The first five lessons are free.
FAQ
- What is the normal distribution?
- It's the symmetric bell-shaped curve described entirely by a mean and a standard deviation. Values near the mean are most likely, and the chance drops off smoothly and symmetrically on both sides.
- What is the 68-95-99.7 rule?
- For any normal distribution, about 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. It's a quick way to judge how unusual a value is.