How it works
Standardize both bounds to z=(x−mean)/sigma, then subtract the normal cumulative probabilities using a polynomial approximation.
Input definitions
- Mean
- Enter the value in the labeled unit.
- Standard deviation
- Enter the value in the labeled unit.
- Lower bound
- Enter the value in the labeled unit.
- Upper bound
- Enter the value in the labeled unit.
A worked example
- Mean
- 0
- Standard deviation
- 1
- Lower bound
- -1
- Upper bound
- 1
Assumptions & limits
Normally distributed variable, positive standard deviation and finite bounds. This is not a statistical test.
Sources & review
Maintained by AnsRio. Methods and examples use automated checks; this does not imply independent professional certification.
Method updated 2026-10-17. Content checked against the stated method and examples. Our verification approach.
Common questions
Is the normal interval probability exact?
It uses a numerical CDF approximation, with absolute error about 10⁻⁷ per bound. Extreme-tail relative accuracy is lower. A positive standard deviation and ordered finite bounds are required.
Why can statistical results differ between tools?
Denominator, interpolation and boundary conventions can differ. Use the method and assumptions on this page to compare the same definition; displayed decimals may also be rounded.
How does the Normal distribution interval probability tool work?
Standardize both bounds to z=(x−mean)/sigma, then subtract the normal cumulative probabilities using a polynomial approximation.
What are the assumptions and limits of Normal distribution interval probability?
Normally distributed variable, positive standard deviation and finite bounds. This is not a statistical test.