An analysis is given of the statistical error in the measurement of the absorption coefficient when the experiment involves the counting of random events (as in nuclear physics). Optimum designs which minimise the statistical error are presented for a wide range of background counting rate (zero and 0.001 to 100 times the incident rate) and for various numbers of observations (2 to 30). Both constant absorber volume and constant intercepted beam area are considered. Some restrictions are introduced to facilitate the optimisation calculations, but they are likely to be in line with many experimental arrangements and alternative restrictions are examined, so the limitation imposed is not unduly severe. The smallest error is found to obtain under the restriction to equal counting times for the different absorber thicknesses with equal increments in thickness: nevertheless equal weighting of the individual observations, again with equal thickness increments, is preferred if a linearity check of the usual semi-log plot is required. At large background, the error for a given total counting time decreases as the number of observation points is increased, and errors smaller than those of the ideal designs with n = 2 can be achieved in both the constant absorber volume and constant intercepted beam area cases. This is true even at small background in the case of equal counting times with equal thickness increments at constant absorber volume, but the effect is more pronounced at large background. All the optimum thicknesses and counting times quoted are readily achievable in practice and, since they are not very strongly dependent on background, an imprecise initial measurement of the background will suffice for planning the experiment.