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A star inherits the gas-phase metallicity at its birth. Set $y_e=y/(1+\nu )$; the preceding calculation gives $G(Z)=G_0{e}^{-Z/y_e}$ and $dS/dZ=G(Z)/y$. If the initial mass function is fixed, let $\kappa$ be the number of the counted stars per unit formed stellar mass; this includes any restriction to surviving long-lived tracers. The metallicity distribution in a leaky box is thereforeThe total eventual formed stellar mass is $G_0/(1+\nu )$, so $N_{\infty }=\kappa G_0/(1+\nu )$ and the equivalent normalized metallicity distribution function isIf $N(Z)$ means a cumulative count rather than a count per stellar metallicity interval, integration yieldsAt a finite observation time these expressions end at the current gas-phase metallicity $Z_f$ and the normalized distribution divides by $1-e^{-Z_f/y_e}$. A distribution per logarithmic stellar metallicity instead multiplies $dN/dZ$ by $(\ln 10)Z$; it is not the same shape as the distribution per linear stellar metallicity.

Assume the stellar yield is common to the compared stellar populations and compare their mass-weighted stellar metallicities. Inverting the metallicity distribution in a leaky box result givesFor example, with $y=Z_{\odot }$, a solar-metallicity massive elliptical galaxy requires $\nu \simeq 0$, whereas $⟨Z{⟩}_{\ast }<0.01Z_{\odot }$ in a dwarf galaxy requires $\nu >99$. A supersolar stellar metallicity requires a correspondingly supersolar stellar yield: outflow alone cannot raise $⟨Z{⟩}_{\ast }$ above $y$. If $y=2Z_{\odot }$, a massive elliptical galaxy with $⟨Z{⟩}_{\ast }$ between $Z_{\odot }$ and $2Z_{\odot }$ has $0\le \nu \le 1$, while the same low-metallicity dwarf galaxy needs $\nu >199$.

For a yield of order solar, the required mass loading runs from order zero or unity in massive spheroids to order one hundred or more in faint dwarfs. Without a specified stellar yield, the robust comparison isThis explains the sense of the galaxy mass--metallicity relation; galactic gas inflow, incomplete depletion, and changes in the initial mass function can also change observed stellar metallicity.