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We analyze the completeness of the MOSDEF survey, in which z ~ 2 galaxies were selected for rest-optical spectroscopy from well-studied HST extragalactic legacy fields down to a fixed rest-optical magnitude limit (H_AB = 24.5). The subset of z ~ 2 MOSDEF galaxies with high signal-to-noise (S/N) emission-line detections analyzed in previous work represents a small minority (<10%) of possible z ~ 2 MOSDEF targets. It is therefore crucial to understand how representative this high S/N subsample is, while also more fully exploiting the MOSDEF spectroscopic sample. Using spectral-energy-distribution (SED) models and rest-optical spectral stacking, we compare the MOSDEF z ~ 2 high S/N subsample with the full MOSDEF sample of z ~ 2 star-forming galaxies with redshifts, the latter representing an increase in sample size of more than a factor of three. We find that both samples have similar emission-line properties, in particular in terms of the magnitude of the offset from the local star-forming sequence on the [N II] BPT diagram. There are small differences in median host galaxy properties, including the stellar mass (M_*), star-formation rate (SFR) and specific SFR (sSFR), and UVJ colors; however, these offsets are minor considering the wide spread of the distributions. Using SED modeling, we also demonstrate that the sample of z ~ 2 star-forming galaxies observed by the MOSDEF survey is representative of the parent catalog of available such targets. We conclude that previous MOSDEF results on the evolution of star-forming galaxy emission-line properties were unbiased relative to the parent z ~ 2 galaxy population.