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The EUV Imaging Spectrometer (EIS) instrument on board the Hinode satellitehas access to some of the best coronal density diagnostics and the highsensitivity of the instrument now allows electron number density, N_e,measurements to an unprecedented precision of up to +/-5 % in active regions.This paper gives a thorough overview of data analysis issues for the bestdiagnostics of Fe XII and Fe XIII and assesses the accuracy of themeasurements. Two density diagnostics each from Fe XII (186.88/195.12 and196.64/195.12) and Fe XIII (196.54/202.04 and 203.82/202.04) are analysed intwo active region data-sets from 2007 May 3 and 6 that yield densities in therange 8.5 < log N_e < 11.0. The densities are derived using v5.2 of the CHIANTIatomic database. The Fe XII and Fe XIII diagnostics show broadly the same trendin density across the active region, consistent with their similar temperaturesof formation. However the high precision of the EIS measurements demonstratessignificant discrepancies of up to 0.5 dex in derived log N_e values, with FeXII always giving higher densities than Fe XIII. The discrepancies may partlybe due to real physical differences between the emitting regions of the twoplasmas, but the dominant factor lies in the atomic models of the two ions. Twospecific problems are identified for Fe XII 196.64 and Fe XIII 203.82: theformer is found to be under-estimated in strength by the CHIANTI atomic model,while the high density limit of the 203.82/202.04 is suggested to be inaccuratein the CHIANTI atomic model. The small grating tilt of the EIS instrument isfound to be very significant when deriving densities from emission linesseparated by more than a few angstroms.

High precision density measurements in the solar corona: I. Analysis methods and results for Fe XII and Fe XIII