There is a widely held view on measurement inferences, that goes back to Stevens’s () theory of measurement scales and ‘permissible statistics’. This view defends the following prohibition: you should not make inferences from averages taken with ordinal scales (versus quantitative scales: interval or ratio). This prohibition is general—it applies to all ordinal scales—and it is sometimes endorsed without qualification. Adhering to it dramatically limits the research that the social and biomedical sciences can conduct. I provide a Bayesian analysis of this inferential problem, determining when measurements from ordinal scales can be used to confirm hypotheses about relative group averages. The prohibition, I conclude, cannot be upheld, even in a qualified sense. The beliefs needed to make average comparisons are less demanding than those appropriate for quantitative scales. I illustrate with the alleged paradigm ordinal scale, Mohs’ scale of mineral hardness, arguing that the literature has mischaracterized it.