How can I find a Z score from a p-value?

Question

How can I find a Z score from a p-value? I know how to look up the p-value from a Z score using a Normal distribution table, but I don't know how to calculate it. For example, a question says the alpha equals 5 percent. From this, I see in my handout that the Z score is calculated to be 1.65. How do I determine this? Thank you.

Seems to be related to this question: http://stats.stackexchange.com/questions/77107/from-z-score-to-p-value-how-to-do-this-for-this-hypothesis-set — Andy, Jun 04 '14 at 15:20
How is your table organized? There are at least 3 or 4 different setups. — Glen_b, Jun 04 '14 at 23:41
You can use the same table as for finding p-values, but you look for the probability value in the *body* of the table, and then read off the Z that gives that value. If you can convey adequately what is tabulated in your tables and how it's organized (there are images of [tables on line](https://www.google.com/search?q=normal+distribution+tables&tbm=isch), if you find one that's effectively identical to yours, a link would suffice), then I'll try to come back with more details. — Glen_b, Jun 04 '14 at 23:50

score 16 · Answer 1 · edited Apr 13 '17 at 12:44

Actually, $z_{.95}=1.6\underline{\mathbf{4}}$; your handout is LIES! (I'm being facetiously hyperbolic because the author seems to have rounded up somewhat improperly. It's not really a big deal.)

In r, the qnorm function converts probabilities (akin to distribution quantiles) to z-scores. Thus:

> qnorm(.95)
[1] 1.644854

The documation for qnorm lists the following reference for the algorithm, in case you want it. If you prefer not to use R, John Walker's calculator works through JavaScript-enabled web browsers. He also offers some equations that could be rearranged to do this by hand. You may also wish to check "How to deal with Z-score greater than 3?"

^{Reference

Wichura, M. J. (1988) Algorithm AS 241: The percentage points of the normal distribution. Applied Statistics, 37, 477–484.}

It perhaps makes sense to round up (or more generally away from the median) if you want your resulting significance levels to be conservative. — Glen_b, Jul 24 '17 at 09:32

score 2 · Accepted Answer · answered Jun 04 '14 at 16:23

Typically the tables for $p$-values for the $Z$ distribution are arranged with values of $Z$ defining the row and column headers, and the body of the table consists of $p$-values.

If you are given a $Z$ value, you go to the corresponding row and column to look in the table. However, you can do the reverse of this, right? Given $p$ (or $\alpha$), you can find this value in the table and then look at the row and column headers to get $Z$.

Ta da!

How can I find a Z score from a p-value?

2 Answers2