The error function measures the probability that a normally distributed error lies within a given range – hence its name. It also solves the heat equation for a sudden temperature change and describes diffusion of particles.
How to use the Error Function Calculator
- Choose the function
- Input x
Results update as you type, and everything is calculated in your browser.
How it works
erf is computed from the standard normal distribution, erf(x) = 2Φ(x√2) − 1, using a double-precision algorithm. The inverses use an accurate inverse normal function.
Inputs explained
| Field | What to enter |
|---|---|
| Choose the function | Choose one: Error function erf(x), Complementary error function erfc(x), Inverse error function erf⁻¹(x), Inverse complementary erfc⁻¹(x) |
| Input x | A number |
What each result means
- erf(x) = erf([Input x])
- erfc(x) = erfc([Input x])
Example values
How the main result changes when Input x changes and everything else stays at the example values above:
| Input x | Result |
|---|---|
| 0.25 | 0.2763 |
| 0.375 | 0.4041 |
| 0.5 | 0.5205 |
| 0.625 | 0.6232 |
| 0.75 | 0.7112 |
| 1 | 0.8427 |
Formula
erf(x) = (2/√π) ∫₀ˣ e^(−t²) dt
erfc(x) = 1 − erf(x)
erfc⁻¹(x) = erf⁻¹(1 − x)
Worked example
Inputs — Choose the function: Error function erf(x); Input x: 0.5.
Results — Result = 0.5204998778; erf(x) = 0.5204998778; erfc(x) = 0.4795001222.
Key terms
- Error function
- erf(x) = (2/√π)∫₀ˣ e^(−t²) dt.
- Complementary error function
- erfc(x) = 1 − erf(x).
Frequently asked questions
What is erf(1)?
About 0.8427: 84.27% of a normal distribution with σ = 1/√2 lies within ±1.
What is the domain of erf⁻¹?
−1 < x < 1. For erfc⁻¹ it is 0 < x < 2.
How is erf related to the normal distribution?
Φ(z) = ½(1 + erf(z/√2)).
Is erf odd?
Yes: erf(−x) = −erf(x).