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

z-score =NORMSINV(percentile) edited: thanks @AndroidMasterZ More on reddit.com

Where on earth do the z-scores on this table come from? they're values of the inverse of the cdf (quantile function) for a standard normal distribution https://en.wikipedia.org/wiki/Normal_distribution#Quantile_function Say I wanted to get the z-score for the 1.5th percentile. Is this doable? Sure, any decent stats package will have it. For example in R it would be qnorm(1.5/100) which gives -2.17009. You can even get it from Excel (e.g. see normsinv) Many numerical libraries offer either this function directly or the erf-1 function from which it can be obtained. Is there a formula somewhere for calculating these? Yes and no. If you want a simple closed form formula there isn't one. There are numerous approximations, but they're not really ideal for writing down easily; they're designed to be accurate as well as reasonably fast on computers but they're not designed for dealing with by humans; there are a bunch of approximations given in a number of papers. 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