Normal Distribution
Bell curve PDF and CDF with shaded probabilities
About this calculator
The Normal Distribution Calculator draws the bell curve for any mean and standard deviation and shades the probability you choose — the left tail P(X ≤ x), the right tail P(X ≥ x), or the area between two bounds P(a ≤ X ≤ b). It also reports the density f(x), the z-score, and the percentile. Useful for statistics, z-scores, hypothesis testing, and exam grading.
How to use the Normal Distribution calculator
- Enter the values for your problem into the input fields.
- Read the result — it updates instantly as you type.
- Check the formula and the visual explanation to follow how the answer was found.
- Copy the page URL to share the exact calculation.
Common examples
- Standard normal N(0, 1): P(Z ≤ 1) = 0.8413 (84th percentile)
- Right tail P(Z ≥ 1.5) = 0.0668
- One-sigma rule: P(−1 ≤ Z ≤ 1) = 0.6827
- Two-sigma rule: P(−2 ≤ Z ≤ 2) = 0.9545
- IQ scores N(100, 15²): P(X ≤ 130) = 0.9772 (z = 2)
Frequently asked questions
How does the Normal Distribution calculator work?
Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.
When would I use the Normal Distribution calculator?
It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.
How accurate are the results?
Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.
Can I share a specific calculation with someone else?
Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.