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.

9 thoughts on “Error function and the normal distribution

  1. 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) ?


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

  3. 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)?

  4. After bashing my head against this conversion for several hours and ending up with some misplaced factor or negative sign, it was a genuine relief and morale-booster to come across your post and statement that you too have got thoroughly fed up with constant rederivations! I’ve finally got it sorted.

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