Solved – How to interpolate for the normal distribution (or any other distribution)?

Many of my students ask this as well. The solution is a matter of opinion.

If you need many digits of precision and you are willing to pursue numerical techniques (of which interpolation is one), there is no need to reinvent the wheel. The programming language R is free to download, and it has a built-in function called pnorm for calculating normal distribution probabilities. Many websites have calculators for this purpose, too: see http://stattrek.com/online-calculator/normal.aspx

If you must stick with tables, my preference is to keep track of probabilities on either side of the true answer. A simpler option is to choose a nearby but conservative option (one that results in a slightly higher p-value), or you could just use the closest table entry.

This last paragraph is for amusement, and I would not recommend this approach unless I woke up tomorrow in the year 1900. If you are just really excited about interpolation, which some people are, you could use a linear approximation based on the two nearest values. In other words, if you have access to $p(Z < x_1)$ and $p(Z < x_2)$ for $x_1$ and $x_2$ on either side of $5/3$, you could use $$frac{(5/3-x_1)p(Z < x_1) + (x_2-5/3)p(Z < x_2)}{x_2 – x_1}.$$