How this calculator works
The value is standardized with z=(x−μ)/σ. Density uses the normal PDF; cumulative probability uses a deterministic error-function approximation.
Formula
z=(x−μ)/σ; f(x)=exp(−z²/2)/(σ√(2π))Standardizes x and evaluates normal density plus an approximation to the cumulative distribution.
Worked example
At the mean is a verified test case used by the calculation engine.
Assumptions and limitations
- Standard deviation must be positive.
- The modeled variable is normally distributed.
- The cumulative probability is a numerical approximation.
Methodology & sources
This calculator uses deterministic, versioned calculation logic. The formula and verified examples above are part of the calculation definition used by CalcuMint. Technical calculation version: 1.
- CalcuMint deterministic statistics engine
Frequently asked questions
What is the CDF result?
It is the approximate probability P(X≤x) under the specified normal distribution.
What is the upper-tail probability?
It is 1−CDF, the approximate probability above x.