How this calculator works
- Step 1Standardisez = (x − μ) ÷ σTurn each value into a z-score.
- Step 2AreaΦ(z₂) − Φ(z₁)Probability between the two.
- Step 3Percentileμ + σ Φ⁻¹(p)Value at a chosen share.
A normal distribution is the familiar bell curve. Any normal variable can be turned into the standard normal with z-scores, so one table, or one function, serves them all.
The probability between two values is the area under the curve between them. Going the other way, a percentile is the value with a given share of the area below it.
Worked example: OpenStax Example 6.9
- P
Φ((2.75 − 2) ÷ 0.5) − Φ((1.8 − 2) ÷ 0.5)
- z of 1.8(1.8 − 2) ÷ 0.5−0.4
- z of 2.75(2.75 − 2) ÷ 0.51.5
- ProbabilityΦ(1.5) − Φ(−0.4)0.5886
- 25th percentile2 + 0.5 × Φ⁻¹(0.25)1.66 h
OpenStax gives 0.5886 and 1.66 hours.
Read the result
About 68% of values lie within one standard deviation of the mean: 0.6827 for N(100, 15).
- Value → probability0.3446Share of N(63, 5) above 65 (Example 6.8).
- Probability → value69.41The 90th percentile of the same scores.
Assumptions and limits
- Bell-shaped dataCheck a histogram first.
- Skewed dataIncomes and waiting times are often not normal.
- ContinuousFor counts, consider a continuity correction.
- Open rangesUse ±1e9 for no limit.
Related statistics tools
The normal curve turns a mean and spread into probabilities.
To place a single value on the curve, use the z-score calculator. The standard deviation calculator finds the spread from raw data, and the percentile calculator works from the data itself.
Common mistakes
Mistake: Using the variance instead of the standard deviation.
Instead: σ is the square root of the variance.
Mistake: Reading the 90th percentile as 90% of the mean.
Instead: It is the value with 90% of the data below it.
Checking the method
A separate script recomputes every probability and percentile with Python’s NormalDist and reproduces OpenStax Examples 6.8 to 6.12.
Read how we test our calculators, or browse more maths calculators.