In the documentation of METAL, a software for meta-analyzing GWAS results, Φ is in the formulas.

Intermediate Statistics $Z_i = \Phi^{−1}(P_i/2) * \textrm{sign}(\Delta_i$) Overall P-value


where $P_i$ is the overall p-value.

What does $\Phi$ mean? Doesn't seems like Phi coefficient.

  • $\begingroup$ Seems like conversion between z score and p-value. stats.stackexchange.com/questions/425561/… $\endgroup$
    – John Smith
    Mar 23, 2023 at 6:14
  • 4
    $\begingroup$ it's the cumulative distribution function of the standard normal random variable. $\endgroup$
    – utobi
    Mar 23, 2023 at 6:26
  • 2
    $\begingroup$ Because you already know the name of this symbol is "Phi," your question is easy to answer just by searching our site. $\endgroup$
    – whuber
    Mar 23, 2023 at 13:33
  • 4
    $\begingroup$ P=2Φ(|−Z|) may have a typographical error (the "$-$" inside the "$|\,|$" serves no purpose) and might have been intended to be $p=2\Phi(-|Z|)$ $\endgroup$
    – Henry
    Mar 23, 2023 at 14:40
  • $\begingroup$ This question is very clear. $\endgroup$
    – BigMistake
    Oct 7, 2023 at 4:04

2 Answers 2


The Phyrexian or Greek letter Φ is being used to describe the CDF (cumulative distribution function) of the standard normal.

The lowercase Greek letter φ is being used to describe the PDF (probability density function) of the standard normal distribution.

And as noted in the other answer as well (I copy it to make this answer more compleated): Φ⁻¹ is the inverse of the CDF, which is also known as the quantile function of the standard normal.

  • $\begingroup$ "Phyrexian"? I only know that term from M:tG... $\endgroup$ Mar 24, 2023 at 6:22
  • $\begingroup$ @StephanKolassa yes Phyrexian, it is not an active language on the plain of earth, just like ancient Greek, but the letters from the alphabet and symbols are still being used. $\endgroup$ Mar 24, 2023 at 8:09
  • $\begingroup$ I would understand "Phoenician", since the Greek alphabet was derived from the Phoenician one... Otherwise, I would love to learn more about Phyrexian outside M:tG. Any pointers? $\endgroup$ Mar 24, 2023 at 14:42
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    $\begingroup$ @StephanKolassa I believe that the contact of Phyrexians with earth has, at the moment, only happened via cards and spells. We could say that there is a non-zero probability that the resemblance of the Phyrexian 'ə' and the ancient Greek capital 'phi', both being written as 'Φ', might indicate that in earlier times there has been some contact. Although, the use of combining a circle 'o' with a bar '|' is not that much spectacular and it is extremely likely that the resemblance between the two is a coincidence. $\endgroup$ Mar 24, 2023 at 15:18
  • $\begingroup$ Until further data is present, I will assign an uninformative prior to it, with a fifty-fifty probability for the two possibilities. $\endgroup$ Mar 24, 2023 at 15:25

As the previos answer noticed, $\Phi$ is just a short notation to denote the CDF of random variable which has standard normal distribution. But as I see, in METAL documentation you need $\Phi^{-1}$ (normal quantile function) so these links to Wiki can help you. https://en.wikipedia.org/wiki/Error_function#Related_functions https://en.wikipedia.org/wiki/Probit


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