Bell curve: generator and the empirical rule
A bell curve is the shape of the normal distribution. Change its centre and spread, then step through the empirical rule band by band.
Outside that band: 31.73%, split evenly, 15.87% in each tail.
The 68-95-99.7 rule
For any normal distribution, the share of values within a given number of standard deviations of the mean is always the same, whatever the units. That is the empirical rule:
- 68.27% within 1 standard deviation (μ ± σ)
- 95.45% within 2 standard deviations (μ ± 2σ)
- 99.73% within 3 standard deviations (μ ± 3σ)
Drag the band slider above and watch the percentage change; it never depends on the mean or standard deviation you typed, only on how many σ wide the band is. That is why statisticians convert everything to z-scores.
Example
Suppose exam scores are roughly normal with mean 70 and SD 10. About 68% of students scored between 60 and 80, about 95% between 50 and 90, and only about 0.15% scored above 100 (the upper half of the 0.27% outside ±3σ).
When the rule breaks
The rule only holds for bell-shaped data. For skewed data, such as incomes, Chebyshev's inequality gives the guarantee that works for any shape: at least 1 − 1/k² of values lie within k standard deviations, so at least 75% within 2σ and 89% within 3σ.
Where do bell curves come from in the first place? The Galton board on the home page and the central limit theorem simulator show a bell emerging from pure chance.
Still not clicking? Ask the tutor
An AI tutor that sees the numbers you entered above and explains them. It can make mistakes, so check anything important against the worked steps.
Questions students ask
What is the 68-95-99.7 rule?
For a normal distribution, about 68.27% of values lie within one standard deviation of the mean, 95.45% within two, and 99.73% within three. It is also called the empirical rule or the three-sigma rule.
What does a bell curve tell you?
Where values cluster (the mean, at the peak), how spread out they are (the standard deviation sets the width), and how rare a value is: the further into the tails, the less area and the less likely.
What does "grading on a curve" mean?
It means grades are assigned by where a score sits relative to the class distribution, often using the mean and standard deviation, rather than by fixed percentage cut-offs.
Is every bell-shaped curve normal?
No. The t distribution and the logistic distribution are also bell-shaped but have heavier tails. Normal has a precise formula: f(x) = e^(−(x−μ)²/2σ²) / (σ√2π).