Professional Sports: Salary Distributions and Relationship to Team Size and League Revenues

Some casual fans of pro sports are actually much more interested in the business and psychology of the leagues than the action on the field. IF that's you, read on! I started wondering 1) what is the minimum you could make in each sport? What was the relationship between size of team in each sport, player salaries, and profitability? and 2) that if any of the figures in my table are glaringly wrong, please comment below and I will correct it (and my written conclusions if they're affected.)

Leag	Max/min	avg/med	rev, $B	sal/rev	rev/plyr	rev/pl / avg sal
NFL	76	1.28	18.0	0.10	10,613,208	9.68
MLB	56	2.92	9.56	0.40	10,988,506	2.51
NHL	21	1.04	5.2	0.42	7,065,217	2.36
NBA	44	1.97	10.0	0.34	22,222,222	2.96

Observations:

1. The smaller the team roster size, the better players get compensated (both in terms of minimum income, median, and average.) That is to say: if you're a player, the NBA is the best place to be.
   
   
2. To get an idea of the salary spreads in each sport, we can look at the max:min ratio. (Minimum salaries in each case are determined by the players union contract with the league.) A catchier name for this would beis the "J.P. Morgan index"*, is highest in football, then baseball and not far behind basketball, with the NHL the lowest.
   
   To get a Gini index (to see income inequality within the league) we'd need all the individual salary data. While that may be available, and I welcome you to find it and crunch the numbers yourself, for my level of curiosity a quick-and-dirty index of average salary divided by median will work. Despite the NFL having some outgroup super-earners giving it the highest J.P. Morgan index, baseball is the most unequal (suggesting the most high-earning outliers) at almost 3, followed by the NBA, NFL, and NHL. In the NHL the average and median are almost the same, and the top earner is an outlier among all the leagues at only $16M. (The J.P. Morgan index is also the lowest there.)
   
   As an additional thought, I wonder how closely real output (measured in goals, sacks, strike-outs etc.) correlates with pay - and does this explain why the NHL is so much more egalitarian in pay than the MLB or NFL? (Again there could be contract or business-specific reasons that I'm not aware of, rather than the nature of the sport.) It also raises the question of whether relative status has more of an effect in some sports, i.e. are there more positional effects in hockey?
   
   To explain this concretely: if you look at say, a brick-making factory, there will be people who are faster at making bricks, and people who are slower. If they're salaried, then you would expect rationally that someone who makes twice as many bricks as the median would get paid twice as much, and similarly someone who makes half as many bricks would get paid half as much.
   
   But that's not what we see - the curve is sigmoidal, flattening out at both ends, with the most productive people not making as much per unit as the people in the middle of the distribution, and the slowpokes getting more per unit. How does this make sense? The theory (as described by Robert Frank in Choosing the Right Pond) is that part of the compensation the fast workers are getting is status - they can strut around the factory with everyone knowing they're the best - and similarly to keep the humiliated slowpokes coming back day after day, you have to give them a premium to pay for the humiliation they suffer. So does a quarterback who throws for twice as many yards or scores twice as many touchdowns get twice as much? It's easy to tell just-so stories in other directions before we look at the data - yes, obviously status is a massive part of sports played for an audience, so the effect would be greater - or, people want to play sports professionally and the status of being a professional athlete more than offsets the humiliation of being the worst quarterback, paid commensurate with miserable stats. But again, I would need data at individual salary levels and have to compare stats (which differ for different positions), so if that's interesting enough to you to gather and crunch the numbers, I invite you to do it and post a link in the comments.
   
   
3. I was also curious about whether player salary or team payroll made a difference, and fortunately other people have been too and already crunched the numbers. Keep in mind that what matters to a franchise's owners is how much money they make, and winning or losing is just a means to that end (if it actually matters to the fan's expenditures at all - and the more loyal the fans, the less it matters - see more here.) But it turns out that baseball has the highest gap between payrolls, and it also makes the biggest difference to team performance; the NFL has the least gap, and it makes the least difference. According this article, the low-payroll teams actually made it to the Final Four more often than the high-payroll teams. (More data on MLB payroll-performance relationship here; yes payroll does help you win in MLB.
   
   
4. The most important thing is how much does each player produce? As someone involved in a league or franchise from a business standpoint, you might think of your sport this way: players are machines that produce revenue. In this analysis, using the average salary and number of players per league, as well as the overall revenue, we can see how much of the revenue is taken by salaries, and how efficient each league's players are at producing profit. Interestingly, here the NFL is in a league of its own. Its salaries are only about 10% of its revenue, compared to the other three which range 34-42%, and players in the NFL produce a share of the revenue 9.7 times their salaries, as opposed to the other three, who range from 2.36 to 2.96 times their salaries. Why is this? Does the NFL have better merchandising? More expensive tickets? Or it's just an inherently better TV sport so there's a bigger audience?
   
   (College football is obviously ideal in this regard. The individual player's share of revenue generated-to-salary ratio is nearly infinite. We might all be forgiven for dismissing the overwrought pleas not to pollute the moral purity of the sport by paying the players coming from the administrators of these programs, or the business office of the learning institutions that for some reason host them. It will be interesting to see, with college football consolidating essentially into two superleagues, how they'll keep the rankings and championships opaque in order to continue maximizing the number of bowls and profits therefrom.)

*Wasn't it J.P. Morgan who said the CEO should not make more than a certain multiple of what the lowest paid employee makes? I couldn't find the quote so if I'm misattributing it please correct me in the comments.