How to use this calculator
Enter the mean and standard deviation, choose what you want to find, and type the value or values. The answer is the shaded area under the curve. To work backwards, choose x for a given percentile and enter the percentage of values that should fall below it.
Formulas
Worked example
Exam scores with mean 70 and standard deviation 10: what fraction score below 85? (the default)
- z = (85 − 70)/10 = 1.5.
- Φ(1.5) = 0.9332, so 93.3% of students score below 85.
- Inverse: the 95th percentile is x = 70 + 10 × 1.645 = 86.4.
Where engineers use it
Manufactured dimensions, material strengths and measurement errors are usually close to normal. Tolerance analysis asks what fraction of shafts will fall outside 25.00 ± 0.05 mm; quality control sets limits at μ ± 3σ, which 99.73% of parts should meet; design allowables use a low percentile of strength rather than the average.
Assumptions and limits
- The quantity really is normally distributed. Fatigue life, wind speeds and many skewed or bounded quantities are not.
- μ and σ are the population values. If they come from a small sample, the uncertainty in them matters too: see the confidence interval calculator.
Frequently asked questions
What is a z-score?
The number of standard deviations a value sits above (positive) or below (negative) the mean. It turns any normal distribution into the standard one with mean 0 and σ = 1, which is what z-tables list.
What is the 68–95–99.7 rule?
About 68.27% of values lie within 1σ of the mean, 95.45% within 2σ and 99.73% within 3σ. Choose area between with μ ± σ to see it.
Why does P(X = x) not appear?
For a continuous distribution the probability of any single exact value is zero. Only ranges have probability, so “less than” and “less than or equal” give the same answer.