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:
(x19)
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:
(x18)
by using the following definite integral:
(x17)
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:
(x10)
The relation between the normalized gaussion distribution and the error function equals:
(x20)
A series approximation for small value of x of this function is given by:
(x11)
while an approximate expression for large values of x can be obtained from:
(x12)

The Complementary Error function

The complementary error function equals one minus the error function yielding:
(x13)
which, combined with the series expansion of the error function listed above, provides approximate expressions for small and large values of x:
(x14)
(x15)

General

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
Normalized gaussian with s = 1 (black curve), error function (red curve) and complementary error function (green curve)

gaussian.xls - gausslog.gif
Gaussian (upper) and the complementary error function on a semi-logarithmic scale. The constant A is chosen to equal one and s = 1

© Bart J. Van Zeghbroeck, 1998