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

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: stats.stackexchange.com/questions/77107/… – Andy Jun 4 '14 at 15:20
• How is your table organized? There are at least 3 or 4 different setups. – Glen_b -Reinstate Monica Jun 4 '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, 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 -Reinstate Monica Jun 4 '14 at 23:50

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!

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

> qnorm(.95)
 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 -Reinstate Monica Jul 24 '17 at 9:32