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We often get reference tables to match z-scores with equivalent percentiles.

Can anyone help me calculate percentiles from z-scores? What's the formula?

Thank you.

bobmcpop
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  • Did you see [this](http://davidmlane.com/hyperstat/A79567.html) page? – LE Rogerson Nov 17 '16 at 12:06
  • The reason why you get given [tables](http://stats.stackexchange.com/questions/235869/percentile-from-z-score) instead of a formula is that there isn't a closed-form formula. The usual way to calculate it using computers is via [calling functions](http://stats.stackexchange.com/questions/56065/is-there-a-quick-way-to-convert-z-scores-into-percentile-score) that use one of several kinds of approximations, but these tend to be optimized for fast, accurate calculation on a computer, and are not especially suitable for working on a calculator or by hand. – Glen_b Nov 17 '16 at 23:59
  • Thank you that is very helpful, and the links too. I forgot to mention that I can do this in stats software, but a computer-programmer (using Java) had asked me how to implement it using a formula. I didn't know if there were equivalent functions in Java and I didn't know how our stats software did it. The best I could offer him was using an approximation of the curve (using exponentiated polynomial), which I assume can be used as a very simple approximation? – bobmcpop Nov 18 '16 at 14:04
  • I think "exact" duplicate is off, but the desired answer is the same. The primary answer on that page has a lot of useful information for this question. I was a bit surprised that the state of the art was based on a paper from 1969 because I have seen unexpected changes to these values in the last ten years. – Tavrock Nov 30 '16 at 21:32

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The following formula can be found in the Handbook of Mathematical Functions

$$\hat{Z}\left ( x \right )=\left ( b_1t+b_2t^2+b_3t^3+b_4t^4+b_5t^5 \right )+\varepsilon \left ( x \right )$$

Where $t={1}/{\left (1+px \right)}$, $\left | \varepsilon (x) \right |< 7.5\times 10^{-8}$, $p=.2316419$, $b_1=.319381530$, $b_2=-.356563782$, $b_3=1.781477937$, $b_4=-1.821255978$, and $b_5=1.330274429$.

Most modern software that allows the user to calculate the percentile from the $Z_{score}$ provide more accurate results than this old approximation.

Tavrock
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