Need help creating a formula for music chart

I need help with a formula for a music related website I am building. On

the site there will be a 'popularity chart' that is used to determine how

popular a band/artist is on the website. There are four pieces of criteria

that I want to use to determine this. Number of Fans, Unique daily page

views (artists have their own Myspace style web page), Unique daily song

plays and artist review score. The first three are self-explanatory, but to

elaborate on artist review score, for every artist that creates a profile

on the website, up to 1000 randomly chosen fans will be prompted to

'review' any given artists music on a simple 1-5 scale. 1 meaning they

don't like the artists music and 5 meaning they love it. Now the problem I

am having is determining how to equally weigh (25% per criterion) these

four categories into my formula since 3 of the determinants are open ended

(i.e. a band could have 1,000,000 page views, song plays or friends) but

only 1000 artist review votes (which will come to an average artist review

score). I would appreciate any help that someone can offer in regards to

creating an optimal formula to most fairly determine an artists popularity

using these four factors. Please keep in mind, I would like all four categories factor

in as 25% weight in determining an artists chart ranking.

Thanks,

Brandon

Re: Need help creating a formula for music chart

It shouldn't be a problem bebebe1 if you follow HallsofIvy's advice carefully.

You rescale FIRST, then average SECOND. So first of all each category is rescaled to a 0 to 1 scale separately. So if there are 3 million plays in total, and one artist gets 89000 plays their 'play' score is 0.029. There are different ways to perform this rescaling, you can do it linearly or nonlinearly. Probably linear rescaling is easier but not necessarily "optimal" as you desired.

And then as you figured out, the typical rescaled scores are going to be incongruous if you rescale in certain ways. The higher review scores might dominate because they range from 0.2 to 1, whereas the 'play' and 'views' scores range from 0 to 1.

To make each category contribute on average 25% to the overall score you can simply use a weighted average. How to do this?

After normalizing the category scores, next add up the scores from all artists for a category. Call these totals n1, n2, n3, n4. Suppose you have three artists, A, B, C. Their respective sets of normalized scores are (a1,a2,a3,a4), (b1,b2,b3,b4), (c1,c2,c3,c4) for the four respective categories. Then,

n1 = a1+b1+c1.

Add them together to get a grand sum N=n1+n2+n3+n4. Then the weight for category 1 is w1=n1/N, for category 2 the weight is w2=n2/N, etc. Now go back to each artist and calculate their oveall rating as the weighted average, so for artist 'A' we get,

A's score = (a1*w1+a2*w2+a3*w3+a4*w4)/4

where a1 is the normalized score for artist 'A' in category 1, etc.

Note that the typical ai*wi/N=0.25, which I believe is what you wanted.

Another solution might be to use a geometric average. If the overall score is the product of all normalized scores (geometric average), that might be fairer because all artists will get the same treatment: the higher review numbers will weight the total score product disproportionately, but then that'd be the same for all artists. Otherwise use the weighted average as previously described.

Whatever normalization and averaging scheme you adopt you can later check how it fairs. Use statistical tests to gauge the sensitivity of the final overall rating on each of the four categories.

The wider problem is that you have no meta-criteria for fairness. But you can figure that out later and adjust the weights you use for each of the four criteria. If you want artist review scores to be more important then your earlier proposed scaling and non-weighted average might be ok, but who knows? You need some other criteria to account for fairness. Then do the suggested sensitivity tests. Basically just calculate correlations between each category score and the overall scores. If one category is more highly correlated with overall rating then you have a bias. But you need a large sample size to make sure the difference in correlations for each category are significant.