Statistical.NormDist

Calculates the value of the Normal distribution (also known as Gaussian distribution) for a given input x. It supports both the probability density function (PDF) and the cumulative distribution function (CDF), depending on the cumulative parameter.

Syntax

Statistical.NormDist(
    x as number,
    optional mean as number,
    optional std as number,
    optional accumulative as logical
) as number

Parameters

• x: The value for which the normal distribution will be evaluated.
• mean (optional): The mean ($\mu$) of the distribution. Defaults to 0 if not provided.
• standard deviation (optional): The standard deviation ($\sigma$) of the distribution. Defaults to 1 if not provided.
• cumulative (optional): Logical value indicating whether to return the cumulative distribution (true) or the probability density (false). Defaults to true.

Remarks

• When cumulative = false, the function returns the probability density at point x using the formula:
  • $\varphi(z)=\frac{1}{\sqrt{2\pi}} \exp(-\frac{z^2}{2})​$
  • where $z = \frac{x-\mu}{\sigma}$ is the number of standard deviations from mean.
• When cumulative = true, the function returns the cumulative probability up to point $x$ using the formula:
  • $\phi(z)=\frac{1}{2} + \frac{1}{\sqrt{2 \pi}} \int_{0}^{z}{e^{-t^{2}/2}dt}$.
• The integral part is calculated by Gaussian Quadrature, which uses a 24-point Legendre-Gauss approximation for high accuracy.
  • $\frac{1}{\sqrt{2\pi}} \int_{0}^{z}{e^{-t^{2}/2}dt} = \frac{z}{4} \sqrt{\frac{2}{\pi}} \sum_{i=1}^{24}{w_{i} \exp(-\frac{z^{2}(t_{i}+1)^2}{8})}$
  • where $w_{i}$ and $t_{i}$ are parameters provided by a Gaussian Quadrature table for 24-point approximation
• This function is useful for statistical modeling, hypothesis testing, and data normalization.

Return Value

Returns the normal cumulative probability up to a given $x$ by default. If cumulative = false, returns the normal probability density at point $x$. If neither x or y are given, returns the standard normal CDF up to a given $x$ (which will be treated as the Z-score), or returns the standard normal PDF at $x$ if cumulative is false.

Examples

Example 1: Calculating the cumulative probability for a value of $x$ in a normal distribution with provided mean and standard deviation.

Statistical.NormDist(100, 80, 10)

Result

0.97724986805182079

Example 2: Calculating the normal PDF for given mean and standard deviation.

Statistical.NormDist(100, 80, 10, false)

Result

0.0539909665131881

Example 3: In order to calculate the standard normal CDF, just don't input any mean nor standard deviation.

Statistical.NormDist(1.96)

Result

0.97500210485177974

Example 4: Calculating the standard normal PDF.

Statistical.NormDist(1.96, null, null, false)

Result

0.058440944333451476

Credits

• Gaussian Quadrature Weights and Abscissae
  • Author: Mike "Pomax" Kamermans
  • Published at: June 5th, 2011