  Estimating the quality of a single-photon source is crucial for its use in
quantum technologies. The standard test for semiconductor sources is a value of
the second-order correlation function of the emitted field below $1/2$ at zero
time-delay. This criterion alone provides no information regarding the
amplitude of the single-photon contribution for general quantum states.
Addressing this question requires the knowledge of additional observables. We
derive an effective second-order correlation function, strongly connected to
the Mandel-$Q$ parameter and given in terms of both the second-order
correlation and the average photon number, that provides a lower bound on the
single-to-multi-photon projection ratio. Using both observables individually
allows for lower and upper bounds for the single-photon projection. Comparing
the tightness of our bounds with those in the literature, we find that relative
bounds may be better described using the average photon number, while absolute
bounds for low excitation states are tighter using the vacuum projection. Our
results show that estimating the quality of a single-photon source based on
additional information is very much dependent on what aspect of the quantum
state of light one is interested in.
