Electron magnetic chiral dichroism (EMCD) in a transmission electron microscope is an element-specific magnetic characterization technique and is extremely powerful for understanding magnetism of materials at the nanoscale. However, quantitative EMCD remains a challenge. In the present paper, we have highlighted and overcome major difficulties associated with the technique. For example, the experimentally observed low dichroic signal and imbalance between the L3 and L2 edge have been explained based on the oscillatory nature of electron propagation through the crystal thickness and specific momentum resolved signal detection, respectively. With this advancement in understanding, site-specific quantitative EMCD has been accomplished in epitaxial thin films of two important ferrimagnetic spinel oxides, NiFe2O4 (NFO) and CoFe2O4 (CFO), with varying degree of cation mixing and A site cation defects. A simple model based on phenomenological absorption has been developed for different site-specific signal contributions for the inverse spinel structure. The experimental moment values for NFO and CFO obtained using EMCD are in good agreement with first principle based theoretical calculations and the results strengthen the promise of utilizing EMCD as a routine nanoscale magnetic characterization technique. © 2015 American Physical Society.