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The main advantage of the 40Ar/39Ar method over conventional K-Ar dating is that it does not depend on any absolute abundance or concentration measurements, but only uses the relative ratios between five isotopes of the same element -argon- which can be measured with great precision on a noble gas mass spectrometer. The relative abundances of the argon isotopes are subject to a constant sum constraint, which imposes a covariant structure on the data: the relative amount of any of the five isotopes can always be obtained from that of the other four. Thus, the 40Ar/39Ar method is a classic example of a 'compositional data problem'. In addition to the constant sum constraint, covariances are introduced by a host of other processes, including data acquisition, blank correction, detector calibration, mass fractionation, decay correction, interference correction, atmospheric argon correction, interpolation of the irradiation parameter, and age calculation. The myriad of correlated errors arising during the data reduction are best handled by casting the 40Ar/39Ar data reduction protocol in a matrix form. The completely revised workflow presented in this paper is implemented in a new software platform, Ar-Ar_Redux, which takes raw mass spectrometer data as input and generates accurate 40Ar/39Ar ages and their (co-)variances as output. Ar-Ar_Redux accounts for all sources of analytical uncertainty, including those associated with decay constants and the air ratio. Knowing the covariance matrix of the ages removes the need to consider 'internal' and 'external' uncertainties separately when calculating (weighted) mean ages. Ar-Ar_Redux is built on the same principles as its sibling program in the U-Pb community (U-Pb_Redux), thus improving the intercomparability of the two methods with tangible benefits to the accuracy of the geologic time scale. The program can be downloaded free of charge from -geochron.com. 2b1af7f3a8