You really can't do it with pen and paper. It requires approximating an integral for which there is no anti derivative. More on reddit.com

Yes, there most definitely is. The function you want is the normal cumulative distribution function (cdf) NORMSDIST if you already have the z-score. If you don't want to convert to z-score first, you can also use NORMDIST with the info you used to find the z-score. More on reddit.com

There's little value out that far, since essentially nothing is close enough to normal for those numbers to be accurate. For what purpose do you need these values? when you say "percentiles" are you actually looking for the cdf, the ccdf (survivor function), the two-tailed area - outside (-6,6) and (-12,12) - or what? The upper tail areas (ccdf) are: > pnorm(c(6,12),lower.tail=FALSE) [1] 9.865876e-10 1.776482e-33 More on reddit.com

Z-score is suitable when you have a distribution that is unimodal and close to symmetric, just like a normal distribution. Taking a cursory glance at the percentile numbers, these distributions appear to deviate from the symmetry requirement. You should consider plotting these distribution. It may be that the two types of test scores you wish to relate are qualitatively different in their distribution, in which case the Z score will reflect that difference, something you may not want. But it seems to me you already have the answer to what you seek in these percentile tables. Let’s say one test score is X. You look up in the table what percentile that corresponds to, say P. You go to the table for the other type of test score, and look what test score corresponds to percentile P, say Y. So now you can argue that at least relative the population of test takers, X corresponds to Y. Of course, the key issue here is that the two populations of test takers may be quite different. Perhaps people taking one test already are self-selected along some axis, and hence are better/worse at verbal reasoning than the people taking the other test. The ideal data would be to have identical populations take the two tests, something that perhaps is approximately true, but remains to be proven. If I was tasked with criticizing your approach, that’s where I would push. More on reddit.com