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One of the largest sources of uncertainty in estimates of global temperature change is that associated with the correction of systematic errors in sea surface temperature (SST) measurements. Despite recent work to quantify and reduce these errors throughout the historical record, differences between analyses remain larger than can be explained by the estimated uncertainties. We revisited the method used to estimate systematic errors and their uncertainties in version 3 of the Met Office Hadley Centre SST data set, HadSST. Using comparisons with oceanographic temperature profiles, we make estimates of biases associated with engine room measurements and insulated buckets and constrain the ranges of two of the more uncertain parameters in the bias estimation: the timing of the transition from uninsulated to insulated buckets in the middle twentieth century and the estimated fractions of different measurement methods used. Here, we present HadSST.4.0.0.0, based on release 3.0.0 and 3.0.1 of the International Comprehensive Ocean-Atmosphere Data Set supplemented by drifting buoy measurements from the Copernicus Marine Environmental Monitoring Service. HadSST.4.0.0.0 comprises a 200-member ensemble in which uncertain parameters in the SST bias scheme are varied to generate a range of adjustments. The evolution of global average SST in the new data set is similar to that in other SST data sets, and the difference between data sets is reduced during the middle twentieth century. However, the changes also highlight a discrepancy in the global-average difference between adjusted SST and marine air temperature in the early 1990s and hence between HadSST.4.0.0.0 and, the National Oceanic and Atmospheric Administration SST data set, ERSSTv5.