Gaussian, Error and Complementary Error function

Table of Contents - Glossary - Active Figures - Equation Sheet - Study Aids - 1 2 3 4 5 6 7 8 9
In this section:
  1. The Gaussian function
  2. The Error function
  3. The Complementary Error function

The Gaussian function

The Gaussian function (also refered to as bell-shaped or "bell" curve) is of the following form: where s is refered to as the spread or standard deviation and A is a constant. The function can be normalized so that the integral from minus infinity to plus infinity equals one yielding the normalized Gaussian: by using the following definite integral: The gaussian function goes to zero at plus and minus infinity while all the derivatives of any order evaluated at x = 0 are zero.

The Error function

The error function equals twice the integral of a normalized gaussian function between 0 and x/2: The relation between the normalized gaussion distribution and the error function equals: A series approximation for small value of x of this function is given by: while an approximate expression for large values of x can be obtained from:

The Complementary Error function

The complementary error function equals one minus the error function yielding: which, combined with the series expansion of the error function listed above, provides approximate expressions for small and large values of x:


The gaussian function, error function and complementary error function are frequently used in probability theory since the normalized gaussian curve represents the probability distribution with standard deviation s relative to the average of a random distribution. The error function represents the probability that the parameter of interest is within a range between -x/2 and x/2, while the complementary error function provides the probability that the parameter is outside that range. All three functions are shown in the figure below:

gaussian.xls - gausslin.gif

gaussian.xls - gausslog.gif

© Bart J. Van Zeghbroeck, 1998