Error Function Erf(x) & Complimentary Error Function Erfc(x)

Note: The error function and complimentary function used in communications are not exactly the same as the ones typically used in statistics. The relationship between the two is given at the bottom of the page.

In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics, materials science, and partial differential equations.

In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics, materials science, and partial differential equations. - Wikipedia

The Gaussian probability density function with mean = 0 and variance =1 is

The error function Erf(x) is defined as:

Note that Erf(0) = 0.5, and that Erf∞)=1.

The complimentary error function Erfc(x) is defined as:

The following graph illustrates the region of the normal curve that is being integrated.

For large values of x (>3), the complimentary error function can be approximated by:

The error in the approximation is about -2% for x=3, and -1% for x=4, and gets progressively better with larger values of x.

An even closer approximation (about 10x better) is:

In standard statistics texts, the error function is typically defined as (note lower case “e”):

The relationship between erfc(x) and Erfc(x) is as follows:

The following chart illustrates how the Erfc(x) function and the closer approximation formulas converge for larger values of “x.” The chart was generated using the Analysis ToolPak Add-In in Microsoft Excel for the Erfc(x) function (and the modification for the communications version as shown above) and entering the approximation formula in adjacent cells.

Webmaster: Kirt Blattenberger, BSEE, UVM 1989