# Error function and the normal distribution

The error function erf(x) and the normal distribution Φ(x) are essentially the same function. The former is more common in math, the latter in statistics. I often have to convert between the two.

It’s a simple exercise to move between erf(x) and Φ(x), but it’s tedious and error-prone, especially when you throw in variations on these two functions such as their complements and inverses. Some time ago I got sufficiently frustrated to write up the various relationships in a LaTeX file for future reference. I was using this file yesterday and thought I should post it as a PDF file in case it could save someone else time and errors.

## 8 thoughts on “Error function and the normal distribution”

1. Blaise F Egan

2. jyotsna

That was very useful ! thanks for your post ! :)

3. Theodore

Greetings,

Thanks for the post. Shouldn’t the last term in the third equation in your pdf file be erf(x) and not erfc(x) ?

Regards

4. Yes, you are right. Thanks for pointing out the error, no pun intended. I’ve corrected the file.

5. Great that you are posting this. Found it through google by searching for “error function density normal” :)

6. Diego Alonso Cortez

Thank you sir!

7. Richard

many thx

8. A. B. Kaye

In your 5th equation from the bottom, should it not be:
\Phi (x) = 1/2 erfc (- \frac{x}{\sqrt{2}} (i.e., you are missing a minus sign)?