Error Function
Connect Gaussian area to erf, its complement and its derivative.
Free theory articleConnect Gaussian area to erf, its complement and its derivative.
Concept & interpretation
The Error Function, commonly denoted as erf(x), is a special function of a sigmoid shape that occurs in probability, statistics, and partial differential equations. It is fundamentally defined by the integral of the Gaussian function over the interval from 0 to x. Note that while erf(x) shares a sigmoid shape with the Cumulative Distribution Function (CDF) of a normal distribution, they differ in scale and domain center; a standard normal CDF integrates from negative infinity to x, whereas erf(x) integrates from 0 to x.
Experiment
In the interactive module at https://riano.app/module/error-function, you can visualize the Error Function, its derivative, and the Gaussian integral.
- Select the Probe x control to evaluate at different points.
- Set the probe to x = 0. Notice the Gaussian curve peaks at 1, but erf(0) is exactly 0 and the Gaussian integral is 0.
- Change the probe to x = 1. Observe the shaded Gaussian integral region expanding from 0 to 1, and erf(1) reading approximately 0.8427.
- Change the probe to x = -2. The Gaussian integral becomes negative, reflecting the odd symmetry of erf(x) where erf(-x) = -erf(x).