Statistics is both an art and a science.  The science of statistics is what we teach in the classroom: hypothesis tests, confidence intervals, sampling distributions, and the like.  It tends to be mathematical, and for a given set of assumptions about the nature of a problem, it typically gives clean, concise answers. Without the science of statistics, the entire discipline, and hence all other disciplines that use statistics, would be sheer guesswork.

Students armed with a good understanding of the basic elements of statistical science go out into the world ready to take on real-life problems.  But eventually one problem (often the first) they face doesn't quite meet the assumptions they learned in class.  The samples aren't quite independent or even random, the data are not really normal, the variances are not all equal, the model doesn't fit,  and the list goes on.  If they are lucky, they can remember how to handle one or maybe two of these violated assumptions (transformations, generalized linear models, repeated measures analysis, or some such), but handling more than that in one problem was never covered in class!

This is where the art of statistics comes in.  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 requires an understanding of and appreciation for the added difficulty often involved in altering an analysis to suit the needs of a failed assumption (which may, in turn create a new violation of a previously-satisfied assumption).  It therefore obviously requires a knowledge of a broad range of methods from which appropriate choices can be made.  Statisticians like to refer to this as their "toolkit", for which the saying applies, "If all you own is a hammer, then everything looks like a nail."  The more tools you have, the more problems you can solve, provided you know when to use them all.  You certainly don't want to use a hammer when a glasscutter is needed...

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.  A statistician needs to recognize the problem before s/he can develop a solution, and that's not as easy as it sounds! Communication  is a vital skill that needs to be developed in the context of statistical thinking.  Being able to take in information delivered in nonstatistical terms by a nonstatistician (sometimes with errors!), convert that into statistical thought, and reply with probing questions, delivered again in nonstatistical terms,  is cricial to the practice of statistics.  A person can't learn this unless s/he has had the chance to try it, repeatedly, and you can't learn that in the classroom.

At Kansas State, most classes in statistics teach the science of statistics, but some also teach the art of statistics. I try to mix in the art with the science wherever possible (most often in response to some of the good questions posed by students).  There are also many opportunities for students to gain practical consulting experience -- over half of our faculty are involved in university-supported statistical consulting, and we welcome the inclusion of students seeking practical experience.  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.