We theoretically analyze the behavior of estimation errors in one-qubit state tomography with finite data, and propose a method for improving these estimation errors by using an adaptive design of experiment. First, we derive an explicit form of a function reproducing the behavior of the estimation errors for finite data by introducing two approximations: a Gaussian approximation of the multinomial distributions of outcomes, and linearizing the boundary. Second, in order to reduce estimation errors, we consider an estimation scheme adaptively updating measurements according to previously obtained outcomes and measurement settings. Updates are determined by the average-variance-optimality (A-optimality) criterion, known in the classical theory of experimental design and applied here to quantum state estimation. We compare numerically two adaptive and two nonadaptive schemes for finite data sets and show that the A-optimality criterion gives more precise estimates than standard quantum tomography. These are results of collaboration with Peter S. Turner and Mio Murao.