Error Function (erf) Calculator

About Error Function (erf) Calculator

The error function is related to the normal distribution CDF: Φ(x) = [1 + erf(x/√2)]/2. Used in statistics and diffusion problems.

How to use this calculator

1. Enter the value of \(x\) in the input box.
2. Choose whether you want a positive or negative input, since \(\operatorname{erf}(x)\) changes sign for negative numbers.
3. Click calculate to get the decimal value of \(\operatorname{erf}(x)\).
4. Read the result and, if needed, use the step-by-step explanation to see how the value is connected to the integral.

The formula explained

The formula \(\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt\) computes the scaled area under \(e^{-t^2}\) from 0 to \(x\). The factor \(\frac{2}{\sqrt{\pi}}\) is there so the output stays between \(-1\) and \(1\).

• x = the input value where you want to evaluate the error function
• t = the variable of integration inside the area formula
• \(\operatorname{erf}(x)\) = the error function value at \(x\)

Step by step method

1. Start with the input value \(x\), for example \(x=1\).
2. Evaluate the integral \(\int_0^1 e^{-t^2}dt\), which is the area under the curve from 0 to 1.
3. Multiply that area by \(\frac{2}{\sqrt{\pi}}\) to get \(\operatorname{erf}(1)\).

Worked example

Problem. Find the error function value for \(x=1\).

1. Use the definition \(\operatorname{erf}(1)=\frac{2}{\sqrt{\pi}}\int_0^1 e^{-t^2}\,dt\).
2. Approximate the integral numerically, which gives about \(0.7468\).
3. Multiply by \(\frac{2}{\sqrt{\pi}}\) to get \(\operatorname{erf}(1)\approx 0.8427\).

Tips and common mistakes

• For negative inputs, use \(\operatorname{erf}(-x)=-\operatorname{erf}(x)\), so you can reuse positive values.
• Do not confuse the error function with the normal distribution probability itself, even though they are closely related.

Frequently asked questions