•Wages at destination are used as a synthetic measure of migrants’ quality.•Wage distributions for low- and high-educated migrants to the US largely overlap.•Overlap coexists with substantial and significant differences in means.•Relying more on education for selecting immigrants could fail to improve quality. Many destination countries consider implementing points-based migration systems as a way to improve migrants’ quality, but our understanding of the actual effects of selective policies is limited. We use data from the ACS 2001–2017 to analyze the overlap in the wage distribution of low- and high-educated recent migrants from different origins after controlling for other observable characteristics. When we randomly match a high- with a low-educated immigrant from the same country, more than one-quarter of time the low-educated immigrant has a higher hourly wage, notwithstanding a statistically significant difference in the mean wage of the two groups for most origins. For 98 out of 114 countries, this synthetic measure of the overlap in the two wage distributions stands above the corresponding figure for natives. We also find that at least 82% of the variance in log wages for migrants with a given number of years of schooling is due to differences within rather than across countries. This suggests that heavily relying on education to select immigrants might fail to markedly improve their quality.