## Saturday, 12 October 2013

### Error Function and Gaussian CDF/Quantile Calculator

The Error Function is very important in Statistics, as it is used to calculate the CDF (Cumulative Distribution Function) of the much used Gaussian distribution. It is also known as the Gauss Error Function, and is given by the formula

$\large erf(x)=\frac{2}{\sqrt{\pi}}\int_0^xexp(-t^2)dt$

A numerical implementation of this function occurs in the calculator below. Fill in the left entry only to calculate the Error function - pressing the calculate button will populate the right box with the result. Alternatively, fill in the right box only to calculate its inverse Error Function (values must lie between -1 and +1, as this is the range of the Error function). Pressing the calculate button will populate the left box with the result.

 $x$: $erf(x)$:

An implementation of the Gaussian CDF and Quantile function Calculator occurs below. Recall that the Gaussian or Normal distribution function (for mean $\mu$ and standard deviation $\sigma$) is:-
$\large N(t)=\frac{1}{\sigma\sqrt{2\pi}}exp(\frac{-(t-\mu)^2}{2\sigma^2})$

The mean and std (standard deviation) fields have to be filled in, as well as two out of the three fields which are labelled Lower Limit, Upper Limit and Probability. The lower limit field needs to contain either a number or the string -inf (for minus infinity). The upper limit field needs to contain either a number or the string inf (for plus infinity). The probability field must contain a number only.

 mean: std:

 Lower limit: Upper limit: Probablility:

Extra probability results pending...

 Plot of distribution ($f(x)$) values against $x$ values $f(x)$ $x$