Home / Shared Math Details / Distribution Functions / Standard Normal Distribution
Standard Normal Distribution¶
This page defines the standard normal functions Quantum XL reuses across tools: the density, the cumulative distribution \(\Phi\), its inverse \(\Phi^{-1}\), and the z-score standardization.
Notation¶
| Term | Description |
|---|---|
| \(\varphi\) | standard normal probability density |
| \(\Phi\) | standard normal cumulative distribution function |
| \(\Phi^{-1}\) | inverse of \(\Phi\) (the quantile function) |
| \(z_P\) | \(\Phi^{-1}(P)\), the value with \(\Phi(z_P) = P\) |
| \(\bar{x}\) | sample mean |
| \(s\) | sample standard deviation (n − 1 denominator) |
Standardization¶
A value is compared to a normal model through its z-score:
\[ z = \frac{x - \bar{x}}{s} \]
Density, distribution, and inverse¶
\[ \varphi(z) = \frac{1}{\sqrt{2\pi}}\,e^{-z^2/2}
\qquad
\Phi(z) = \int_{-\infty}^{z}\varphi(u)\,du
\qquad
z_P = \Phi^{-1}(P) \]
Quantum XL evaluates \(\Phi\) and \(\Phi^{-1}\) numerically: the C# analysis tools use the Math.NET Numerics implementation, and the C++ engine uses the Boost.Math implementation.
Used by¶
- Anderson-Darling Test: \(F(x) = \Phi\!\left((x - \bar{x})/s\right)\).
- Tolerance Interval: \(z_P\) and the density \(\varphi\) in the normal tolerance factors.
- Summary Statistics: the Shapiro-Wilk and Kolmogorov-Smirnov normality tests.
See Also¶
References¶
- Abramowitz, M., and Stegun, I. A. (1964). Handbook of Mathematical Functions. National Bureau of Standards.
- NIST/SEMATECH e-Handbook of Statistical Methods, section on the normal distribution.