  The sensitivity of Impact Factors (IFs) to journal size causes systematic
bias in IF rankings, in a process akin to {\it stacking the cards}: A random
``journal'' of $n$ papers can attain a range of IF values that decreases
rapidly with size, as $\sim 1/\sqrt{n}$ . The Central Limit Theorem, which
underlies this effect, also allows us to correct it by standardizing citation
averages for scale {\it and} subject in a geometrically intuitive manner
analogous to calculating the $z$-score. We thus propose the $\Phi$ index, a
standardized scale- and subject-independent citation average. The $\Phi$ index
passes the ``random sample test'', a simple check for scale and subject
independence that we argue ought to be used for every citation indicator. We
present $\Phi$ index rankings for 12,173 journals using 2020 Journal Citation
Reports data. We show how scale standardization alone affects rankings,
demonstrate the additional effect of subject standardization for
monodisciplinary journals, and discuss how to treat multidisciplinary journals.
$\Phi$ index rankings offer a clear improvement over IF rankings. And because
the $\Phi$ index methodology is general, it can also be applied to compare
individual researchers, universities, or countries.
