= Cumulative and birth stellar metallicities
{title2=$Z_{\rm birth}(\delta)=\frac{d}{d\delta}[\delta\overline Z_*(\delta)]$}
In a well-mixed <closed-box model of galactic chemical evolution>, gas consumed into locked stellar mass supplies a natural cumulative mass coordinate $\delta$. The mean <stellar metallicity> is $\overline Z_*(\delta)=\delta^{-1}\int_0^\delta Z_{\rm birth}(s)\,ds$, giving the displayed derivative relation. If the birth abundance increases linearly, $Z_{\rm birth}=b\delta$, its cumulative mean is half the current maximum. More generally the maximum of any nonnegative abundance distribution cannot lie below its mean. A fixed <initial mass function> and the same survival selection are required to identify this mass-weighted distribution with a star-count distribution.
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