The inverse of the CDF is often called the quantile function.
Software solutions. If you are using software there is usually a way to get quantile functions.
For example in R, the inverse of $\Phi$ is
qnorm (with default $\mu = 0$ and $\sigma = 1$), and in Minitab
it's the command
INVCdf followed by subcommand
NORM 0 1. Other software packages have their own syntax.
In R, the answer to your specific question would be obtained as
Just to check on this, the R code for the standard normal CDF
pnorm, and the statement
Here's how it looks in Minitab:
MTB > invcdf .8;
SUBC> norm 0 1.
Inverse Cumulative Distribution Function
Normal with mean = 0 and standard deviation = 1
P( X <= x ) x
Approximations from printed tables. However, you asked about printed tables. Suppose it's a straightforward CDF table. Then you look around
in the $body$ of the table to find the entry nearest to .8.
In the table I'm looking at, I find the entry .7995 corresponding
to z-value .84 in the $margins$ of the table. So without
interpolation .84 is as close as I can get. Linear interpolation
between entries .7995 and .8023 (corresponding to $z = .85$)
would get me a little closer: $.84 + 0.01(5/28) = 0.8418,$ which
is wrong in the fourth place.
But that's about as accurate an answer as you'll get from a
Notes: (1) I know of printed tables that give probabilities to five places rather than four, but all the ones I have seen give
z-values only to two places. (Perhaps see the five-place table
from the NIST online handbook.)
(2) As software is ever more widely used, I suppose printed
tables will disappear at some point. I know of a few recent probability
and statistics books that have no normal tables. So maybe I'm
not totally wasting your time by showing some software answers.
(3) Not to be too fussy, but the usual notation for the standard
normal CDF is $\Phi$, not $\phi$. In TeX, you get it by
$\Phi$, instead of
$\phi$. Sometimes $\phi$ is used for
the standard normal PDF. (I have edited your