The bell curve in practice
The normal distribution describes many measurements that result from lots of small, independent influences: heights, test scores, measurement errors, the fill volume of bottles on a production line. It is fully defined by two numbers, the mean μ, which sets its center, and the standard deviation σ, which sets its width. This calculator gives the probability that a value falls below a limit, above another, between the two or outside them.
The formulas
- Convert each limit to a z-score: z = (x − μ) ÷ σ
- P(X < a) = Φ(z_a), the cumulative normal probability
- P(X > b) = 1 − Φ(z_b)
- P(a < X < b) = Φ(z_b) − Φ(z_a)
Φ has no simple closed form; the calculator evaluates it with the error function, to about 15 significant digits.
Worked example: filling bottles
A machine fills bottles with a mean of 500 mL and a standard deviation of 4 mL. The specification accepts anything between 492 and 508 mL.
- z-scores of the limits: (492 − 500) ÷ 4 = −2 and (508 − 500) ÷ 4 = +2.
- P(492 < X < 508) = Φ(2) − Φ(−2) = 95.45% of bottles are within specification.
- 2.275% are underfilled and 2.275% overfilled, so 4.55% are rejected.
- Tightening the process to σ = 2 mL would put the limits at ±4σ, and fewer than 1 bottle in 15,000 would be rejected.
The 68-95-99.7 rule
| Range | Share inside | Share outside |
|---|---|---|
| μ ± 1σ | 68.27% | 31.73% |
| μ ± 1.645σ | 90% | 10% |
| μ ± 1.96σ | 95% | 5% |
| μ ± 2σ | 95.45% | 4.55% |
| μ ± 2.576σ | 99% | 1% |
| μ ± 3σ | 99.73% | 0.27% |
Common uses
- Manufacturing: the share of parts within tolerance, and the basis of Six Sigma targets.
- Test scores: IQ tests are scaled to a mean of 100 and a standard deviation of 15, so 68% of people score between 85 and 115.
- Biology and medicine: reference ranges for lab results are often the central 95% of healthy people, which means 1 healthy person in 20 falls outside.
- Finance: risk models sometimes assume normal returns, although real market crashes happen far more often than the bell curve predicts.
Check that the model fits
The normal distribution is symmetric and its tails fall off very fast. Data with a hard lower limit (waiting times, prices), strong skew (incomes) or heavy tails (stock returns) can give very different probabilities, especially far from the mean. When in doubt, compare a histogram of your data with the bell curve, or use the z-score calculator to check individual values.