How to manually create a Z-table

Question

Z-tables are commonly found online. However, I am writing a precision program for this, and so I would like to find out how to calculate my own percentage values.

Answer 1

Z-tables are just values of the CDF $$ F(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x} e^{-y^{2}/2}dy $$ at various points $x$. If you want to generate a Z-table, pick some $x$ points and evaluate that integral (using a numerical method of your choice).

As mentioned in the comments to your question, MATLAB (and other numerical software) already do this for you. Probably best not to reinvent the wheel.

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Addendum: As @Ian mentioned, we should truncate the integral since it is on an unbounded domain. That is, we should look to compute instead

$$ F(x)\approx\frac{1}{\sqrt{2\pi}}\int_{w(x)}^{x}e^{-y^{2}/2}dy. $$ The question is thus, how do we pick $w(x)$? Let's say that you are only interested in computing the integral up to $\epsilon$ accuracy. Therefore, it might be reasonable to pick $w(x)$ such that $$ F(w(x))=\frac{1}{2}\text{erfc}(-w(x)/\sqrt{2})\leq\epsilon. $$ A well-known bound for the erf function is $$ \text{erfc}(x)\leq e^{-x^{2}}. $$ Plugging this into the above, $$ F(w(x))\leq\frac{1}{2}e^{-w(x)^{2}/2} $$ and thus it follows that $$ w(x)\leq-\sqrt{-2\log(2\epsilon)}. $$

For example, if $\epsilon=10^{-6}$, $w(x)\leq-5.1230$ (approximately). This makes sense, as 5-sigma events should not make much of a contribution in computation.

This bound is very conservative, however, and you could/should come up with a tighter one, or one that takes into account relative instead of absolute error (also mentioned in comments).