Normal Distribution Calculator
Find normal distribution probabilities below, above or between values, and the value at a given cumulative probability.
Your inputs
Results
P(X < x)
0.97724987
- P(X > x)
- 0.02275013
- z-score
- 2
✓Exact calculation — rounded only for display
What this calculator does
Many measurements are approximately normally distributed. This tool computes areas under the bell curve for any mean and standard deviation.
Formula and method
z = (x − μ) ÷ σ; P(X < x) = Φ(z), with Φ computed from a high-precision complementary error function; inverse via Acklam’s algorithm refined by Halley’s method.
Worked example
Calculated by this tool with its default inputs:
Inputs
- Mean (μ): 100
- Standard deviation (σ): 15
- Find: Probability below / above x
- x (or x₁): 130
Results
- P(X < x): 0.97724987
- P(X > x): 0.02275013
- z-score: 2
How to read the results
About 68 % of values lie within one standard deviation and 95 % within about 1.96.
Assumptions and limitations
- Real data may not be normal, especially in the tails.
Questions
What is the 95th percentile of IQ?
With mean 100 and SD 15, about 124.7.
Methodology and verification
- Result type
- Exact mathematical calculation (rounded only for display)
- Formula version
- 1.0
- Automated verification
- 4 reference and validation test cases, checked against independently computed values. The site cannot be built if any of them fail.
- Editorial review
- Awaiting human editorial review. The explanation was drafted with AI assistance; the formula and examples are verified by the automated tests above.
Privacy
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