How are the values in a z-table computed? Is it possible to compute confidence without looking up a z-table?
The simplest answer is that one either uses numerical quadrature techniques (Simpson's, for instance) on the PDF, or approximations such as those found in Abramowitz and Stegun. Personally however, since one merely needs a few digits of accuracy for the "error function" that is involved in computing z-scores, you might be interested the simple approximations given here by S. Winitzki.
A z-table gives you values of the cumulative distribution function for the standard (i.e., mean = 0, standard deviation = 1) normal distribution. The integral needs to be estimated numerically. See here for a discussion.
Of course, to answer your question literally, you could use some other lookup system other than a table, such as the
pnorm function in R.
> pnorm(2)  0.9772499
pnorm also provides the following reference on how to calculate the cdf of a normal distribution:
- Cody, W. D. (1993) Algorithm 715: SPECFUN – A portable FORTRAN package of special function routines and test drivers. ACM Transactions on Mathematical Software 19, 22–32.
I don't know how to post formulas here, so search google on "normal distribution function" or see, e.g., here.
Excel has pretty good functions for the normal distribution, so long as you aren't way out in the tail (where they fall apart).