Normal distribution calculator
Enter a mean, standard deviation and interval bounds. The calculator returns three areas, standardises both bounds and draws the normal curve.
Your numbers stay in this browser. No account or upload to the server is needed.
How do I use this page?
- Enter mean μ and a positive standard deviation σ.
- Enter lower bound a and upper bound b.
- Read P(X≤a), P(a≤X≤b), P(X≥b) and review the Z substitution.
Method and worked example
Each bound is standardised with z=(x−μ)/σ, converting the problem to the standard normal distribution with mean 0 and standard deviation 1.
The interval probability is a difference of cumulative probabilities: P(a≤X≤b)=Φ(zb)−Φ(za). It is an area under the model, not a forecast of one observation.
Areas use numerical Simpson integration. A normal model should be justified by the process, measurement assumptions or a check of the data.
Example: μ=100, σ=15, a=85 and b=115 standardise to −1 and 1, covering about 68.27% of the distribution.