Don't be misguided by the beauty of mathematics, if the data tells you otherwise
I was trained as a mathematician and it was only last year, when I attended the Royal Statistical Society conference and met many statisticians that I understood how different the two groups are.
In mathematics you often start with some axioms, things you assume to be true, and these axioms are then the basis from which new theory is derived. In statistics or more general in science you start with a theory, or better a hypothesis and try to disprove it. And if you can’t disprove it, you accept it until you have other evidence. Or to phrase it like Karl R. Popper: you can only be proven wrong.
Now, why do I mention this? I have met many mathematicians who talk about the beauty of mathematics and I agree, a mathematical concept, theorem or proof can indeed be beautiful. However, when you work in applied mathematics and particular when you use mathematics to build models, there is a danger that you stick to the beautiful idea and ignore reality. Remember the financial crisis?
For example, it might be handy to assume that your data follow a normal distribution, e.g. to make the calculations easier. However, if the data tells you otherwise then be bold and ruthless and change your model. As strange as it might sound, it is has to be your aim to prove a model doesn’t work in order to use it successfully.
Remember Pythagoras? He believed in beautiful integers and the realisation that the square root of two was not a fraction of two integers caused a big crisis.
I would argue that we need mathematics to do statistics and statistics to do science. The developments over the last 350 years really demonstrate the success the scientific method. Of course some ideas had to go: the earth can no longer be regarded as the centre our solar system - instead it appears more like a little pale blue dot.
Diggle and Chetwynd, from Lancaster University, published a nice little book that gives a good introduction into statistics and of the scientific method. Two quotes of the book stuck in my mind (pages 1&2):
A scientific theory cannot be proved in the rigours sense of a mathematical theorem. But it can be falsified, meaning that we can conceive of an experimental or observational study that would show the theory to be false. … The American physicist Richard Feynman memorable said that ‘theory’ was just a fancy name for a guess. If observation is inconsistent with theory then the theory, however elegant, has to go. Nature cannot be fooled.