There are a few masters courses that could lead you to a career in data science, but it depends on your background. Most of the courses I found were specialisations, e.g. a masters in computer vision or HCI for those with a computer science background, or a masters in digital media or design and technology for those with an arts background. I had neither, so I thought I might have to do a conversion course into computer science or art and then specialise. However, when I read the article above, I realised I was more on the design side anyway (happier in Photoshop and Illustrator than a text editor) and that I needed to learn the computing and statistics side. This brought me to bioinformatics since I already have a background in biochemistry. The course will teach me programming, data mining and statistics. I also get to do three projects, which I am going to do in visualisation of life science data - which is why I got interested in this topic in the first place.
I think undergraduates have a greater opportunity to 'cobble' a course together tailored towards data science.
I found it useful to do a bit of research on anyone who does a visualisation that you like and see what their background is.