Normal Distribution Probability Calculator
By Santaji GadepatilinUpdated October 27, 2026
calculated
calculated
calculated
Type in any field to change it. Press × on a value to calculate that field instead.
How to use this calculator
Enter the mean and standard deviation, and the lower and upper limits. You get the probability below the upper limit, above it, and between the two limits.
Formulas
P(X < b) = Φ((b − μ) ÷ σ)
P(a < X < b) = Φ((b − μ) ÷ σ) − Φ((a − μ) ÷ σ)
Worked example — IQ scores (μ = 100, σ = 15)
- P(85 < IQ < 115) = 68.3%
- P(IQ > 115) = 15.9%
- P(IQ > 130) = 2.28%
Manufacturing example
Bolts have a mean length of 50.0 mm with σ = 0.1 mm and a tolerance of 49.8–50.2 mm. The share within tolerance is 95.4%, so about 4.6% are rejected.
Frequently asked questions
What is the normal distribution?
The symmetric bell-shaped curve that describes many natural measurements, such as heights, test scores and measurement errors.
What is the probability within one standard deviation?
About 68.3%. Within two SDs about 95.4%, and within three about 99.7%.
Can I find a value from a probability?
Yes. Press × on the upper limit and type the probability below it.