Pre-print, Working paper: Traditional indices of bi-dimensional inequality and polarization were developed for cardinal variables and cannot be used to quantify dispersion in ordinal measures of socioeconomic status and health. This paper develops two approaches to the measurement of inequality and bi-polarization using only ordinal information. An empirical illustration is given for 24 European Union countries in 2004-2006 and 2011. Results suggest that inequalities and bi-polarization in income and health are especially large in Estonia and Portugal, and that inequalities have significantly increased in recent years in Austria, Belgium, Finland, Germany, and the Netherlands, whereas bi-polarization significantly decreased in France, Portugal, and the UK.