We review the statistical method cited in the report of the radiocarbon dating of the Shroud of Turin. Following strict analytical protocol, we find the Shroud data to be heterogeneous, while data from three control samples show no heterogeneity. We consider two potential sources for the Shroud data heterogeneity. The first, an approximate linear dependence of the dates on the original sample locations suggests a variation in the carbon isotopic composition. The second, differences in the cleaning protocols of the three laboratories may have given rise to differences in residual contamination. We suggest experiments to test the two competing hypotheses.