Simulated data were used to assess the accuracy of determinations of Km and Vmax of the Michaelis-Menten equation using exhaustive Monte Carlo modelling of error variances in data sets. Curves were fitted using a direct least squares fit and the unweighted double reciprocal plot (commonly but incorrectly called the Lineweaver-Burk plot). At least 10 pairs of data points are needed in an experiment for Km to have an error that is approximately normally distributed: 12 were used routinely in this study. Most published estimates of Km are not based upon a sufficient number of data points. Evenly spaced data points were compared with substrate concentrations spaced logarithmically and inversely proportionally spaced data points. Data sets were spread below the Km around the Km and above the Km. If the substrate concentrations are all below the Km, the unweighted double reciprocal plot method is unreliable. Km estimates made using unweighted double reciprocal plots were consistently inferior to estimates made using the direct least squares method. Estimates of Km using the direct least squares fit are normally distributed under a wide range of experimental error conditions. Errors can be independent of substrate concentration, have a constant relative error, or be inversely proportional to the velocity (rate). Substrate concentrations are best evenly spaced; inversely proportionally spaced and logarithmically spaced substrates lead to skewed estimates of Km. The majority of Km values published are based on the statistically unsatisfactory unweighted double reciprocal plot. If similar data are analysed using the double reciprocal "Lineweaver-Burk" plot and the least squares methods, the least squares method can give useful results when the other does not.