p-value to Z-statistic for a KS test

Say you had a two tailed normal distribution A for some dataset, and you then measured the probability that some set of data points B were drawn from A, using a KS test.

Could someone explain to me the process by which you would convert the p-value returned by the KS test into a Z score? (No matter how outlandish or silly the reasoning for doing so)

I realise the first thing you would do is look at a Z table but I’m interested here in knowing how you go about doing the conversion explicitly.

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Here is my current understanding/interpretation of how to do this:

• I understand that the p-value gives you the area under a normal distribution from a position N$sigma $ away from the mean, outwards.

• If I understand correctly the Z value is the number of $sigma$ away from the mean you are at any position along the x axis in a 1D Normal distribution. Therefore, here, N = Z.

A pictorial representation of the above points:

• So my approach for getting Z given a p-value would be to integrate the normal equation and solve for the lower limit. Then dividing through by $sigma$ will leave you with Z. Is that correct?

• So integrate:

  $int_{a}^{b} frac{1}{sigma sqrt{2pi}}expleft(-frac{1}{2}left( frac{x-mu}{sigma} right)^2right) dx$

  and solve for a, given that the integral should equal the p-value.

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However, is there a simpler, better way to do this given that the KS test returns a d-statistic using the cumulative distribution from your data? Perhaps because of that there is a neat trick to make a simpler conversion that I cannot yet understand?

Many thanks for any help in this matter.