# Probability for high z-scores

Where can I find a z-table (or app or whatever) that will give me the probability associated with very high z-scores (e.g. 12 or 15)?

• These typically aren't tabulated, in part because they are straightforward to compute: see stats.stackexchange.com/questions/7200/….
– whuber
Feb 1, 2019 at 21:29
• As you probably know, most printed tables don't go much beyond $\pm 3.5.$ Areas beyond values farther into the tails are very small, and the corresponding probabilities are of limited interest in most applications. First, because it is usually enough to know that such a probability is 'extremely close to 0', second, because normal models are often approximate and one does not trust the values far out in normal tails to be descriptive of reality. // Roughly one can say US men have hts in dist'd NORM(69", 3.5"). Strictly speaking this would imply a tiny probability below 0; neg hts impossible. Feb 1, 2019 at 21:30
• Such extreme tail calculations are nearly always nonsense anyway. Extreme tail probabilities are very sensitive to the (almost certainly false) model assumption, where even fairly modest deviations from the supposed distribution will lead to many orders of magnitude difference in the tail probability -- how could such calculations carry any value? It would be spurious precision. ... ctd Feb 2, 2019 at 1:18
• ctd... Even the 2-3 figure accuracy in an upper-tail approximation like $1-\Phi(x)\approx \phi(x)/x$ is ludicrously precise that far out when the inaccuracy caused by even slightly incorrect distributional and dependence assumptions looms much, much larger Feb 2, 2019 at 1:23

Various kinds of statistical software give exact values for far-tail standard normal probabilities. If $$Z \sim \mathsf{Norm}(0,1),$$ then R software gives values as shown below for $$P(Z > v).$$ (Ignore row numbers in brackets [ ].)

v = 1:7;  pr = 1-pnorm(v);  cbind(v, pr)
v           pr
[1,] 1 1.586553e-01
[2,] 2 2.275013e-02
[3,] 3 1.349898e-03
[4,] 4 3.167124e-05
[5,] 5 2.866516e-07
[6,] 6 9.865877e-10
[7,] 7 1.279865e-12


Values of $$P(Z > 10),$$ and so on are so near to 0 that R does not print them.

Notes: (1) Some of the most extreme values may be beyond the numerical accuracy of R. I cannot promise that all decimal places given for the last few entries are exactly accurate. (2) R software is available free of change: r-project.org. (3) Perhaps see this link. But notice that no authoritative references are given there concerning methods or accuracy. (For those see details in @whuber's link.)

• R will print the values of $\Pr(Z\gt z)$ up to the point where they completely underflow IEEE floats, which occurs where $z$ equals -qnorm(2^(-2^10 + 1 - 51)), approximately $38.46.$ Up to -qnorm(2^(-2^10 + 1)), or approximately $37.54,$ there should be nearly full precision in the result (around 14 to 15 decimal digits). This can be verified by computing using the partial fraction expansion of Mills Ratio: seven terms suffice for $z \gt 24.$ No software can give truly "exact" values, but some software will compute them to arbitrary precision.
– whuber
Feb 1, 2019 at 22:31