The art is driven by experience and knowledge regarding which assumptions are most crucial to the performance of a statistical method and which can be relaxed without too much harm. It therefore obviously requires a knowledge of a broad range of methods from which appropriate choices can be made. The more tools you have, the more problems you can solve, provided you know when to use them all. Unlike the science of statistics, the art is difficult to teach. Like so many other things in life it takes practice, patience, and an attentiveness to the things that aren't necessarily taught.
Communication is a vital skill which needs to be developed in the context of statistical thinking.
At Kansas State , most classes in statistics teach the science of statistics, but some also teach the art of statistics. There is also a student-run consulting lab, and professional summer internships are always encouraged. Finally, many graduate students who come to K-State do some teaching. This can help to cultivate the process of communicating with nonstatisticians. The opportunities to learn both the science and the art of statistics, as well as the communication between the two, are limited only by the student's efforts to pursue them.