Error Function (erf)
The error function erf(x) gives the probability that a normally distributed variable falls within a given range.
Includes examples and applications.
The Formula
Complementary: erfc(x) = 1 − erf(x)
The error function erf(x), introduced by mathematician J.W.L. Glaisher in the late 19th century, is a special mathematical function that arises naturally whenever you integrate the Gaussian (normal) bell curve. Its name comes from its original use in the theory of errors in physical measurements.
The function gives the probability that a normally distributed random variable falls within a symmetric interval around the mean. Specifically, if X is a standard normal variable, then the probability that X lies between −z and +z equals erf(z / √2).
Key values to remember: erf(0) = 0, erf(∞) = 1, and erf(1) ≈ 0.8427. The function is odd — erf(−x) = −erf(x) — and it is always between −1 and +1. Because there is no closed-form antiderivative for e^(−t²), the error function must be evaluated numerically or looked up in tables. Most scientific calculators and programming languages provide it directly.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| erf(x) | Value of the error function at x, ranging from −1 to +1 | dimensionless |
| x | Upper limit of integration (the input value) | dimensionless |
| t | Integration variable | dimensionless |
| π | Pi ≈ 3.14159 | dimensionless |
| erfc(x) | Complementary error function = 1 − erf(x) | dimensionless |
Example 1
A manufacturing process produces bolts with diameters that are normally distributed. What fraction of bolts falls within 1 standard deviation of the mean?
For 1 standard deviation: x = 1/√2 ≈ 0.7071
Probability = erf(0.7071) ≈ 0.6827
About 68.27% of bolts fall within 1 standard deviation of the mean — the famous "68-95-99.7 rule" value
Example 2
A diffusion process in a semiconductor has diffusion length L = 0.5 μm. What fraction of dopant atoms has diffused further than 1 μm? Use erfc.
The fraction beyond distance d in diffusion problems is given by erfc(d / (2L))
Argument: 1.0 / (2 × 0.5) = 1.0
erfc(1.0) = 1 − erf(1.0) = 1 − 0.8427 = 0.1573
About 15.73% of dopant atoms have diffused further than 1 μm — erfc is the standard tool in diffusion analysis
When to Use It
Use the error function when:
- Calculating probabilities for normally distributed data without a full statistical table
- Solving heat conduction and diffusion problems in physics and engineering
- Analyzing bit error rates in digital signal transmission
- Modeling option pricing and risk in quantitative finance
- Evaluating detection probabilities in radar, sonar, and communications systems