Here is a question I just posted on stack overflow about weighted variance and weighted standard deviation:

http://stackoverflow.com/questions/16761965/weighted-variance-and-weighted-standard-deviation-in-c

I vaguely remember studying this in college, we had one lecture where the professor talked about variance when the samples are weighted. The very first question in my mind was whether:

variance( { 1, 2, 3, 3 } )

would equal the weighted variance:

variance( {1, 2, 3 }, { 1, 1, 2 } )

I can't remember the answer. Unfortunately, nearly every interesting topic in mathematics has an initial "what if" that goes with it. But for whatever reason, a lot of the math discussions on the web lack the answers to these. It's like people are regurgitating the formulas they learned by rote memorization, instead of providing the derivations.

I don't honestly expect this question to be answered, but if you look at my stack overflow profile, so many of my questions are like this, that I'm having trouble programming on a day to day basis. I'm hitting way too many showstoppers that take hours or days or my time, where I have to learn the entire context of the problem space I'm working in and then derive my own answer.

FYI this is for a simple 3D visualization where I'm trying to align a list of objects by their centroids and masses, basically minimizing the variance of the distribution so they are "close together". There are a number of ways to do this, I thought this would be an easy method, but finding the original formula turned out to be more challenging than applying it.

Hence this feeling that most of the work I do is a waste of time or could have been avoided if some small issue wasn't holding me up constantly.