Okay blackbootz I'm going on no sleep and a cross-country flight but here goes nothing. At least there's some coffee in front of me. Maybe? I'm not sure what your reference point is for "Six Sigma management philosophy".
I don't consider it a management philosophy so much as a problem solving approach that can be used across disciplines, industries, research, and transaction issues (how business is done, infosys, etc.). It's very statistics driven, and has a central tenet of improving process capabilities and reducing variation. There are some issues with how that is measured, but that's a more technical topic. It is a management approach in the way that problems are handled, as DMAIC and DMADV are foundations of Six Sigma.
How does this tie in to lean?
Well, lean is all about reducing waste and non-value added activities. These two topics have a lot of overlap, which is why "Lean Six Sigma" is a catch phrase right now. Lean tools and Six sigma tools can be used with each other, such as Kaizen Events which can be used for larger, Six Sigma projects. Ultimately these two approaches combine for a similar purpose. Let me know if you have questions on any of this, or follow-ups. That's a really high-level overview and I'm unsure if it will be covered in an economics statistics course (I'm assuming it's a course within your major and not, say, an engineering stats course).
This stuff has a firm grip on certain industries, such as semiconductors and electronics manufacturing in general. One of the gold standards for lean manufacturing is probably Boeing. Seriously, watch that video. It looks so simple and yet it's not at all.