Among the loudest voices expressing shock over some of the massive deals signed on the first day of NBA free agency were NFL players. Ryan Wilson at CBS Sports has a good round-up. This example from DeAngelo Williams sums up the sentiment.

Call me a hater but these NBA deals are insane I have to google the players getting paid #nonamechecks #productiondoesntcount — DeAngelo Williams (@DeAngeloRB) July 1, 2016

Many sports fans — even many NBA fans, who have had plenty of exposure to predictions that the numbers were going to rise significantly — are feeling the same. It's not just about Chandler Parsons getting more than Andrew Luck annually, or zero-time All-Star Mike Conley being offered a deal larger than the biggest NFL contract ever. It's about low-production Solomon Hill getting $48 million guaranteed, or relative unknown E'Twaun Moore grabbing $34 million. It's about Matthew Dellavedova inking a $38 million offer sheet. It's about Luck signing for an NFL-record $87 million in guaranteed salary three days ago, and seven NBA players exceeding that in the first 24 hours of free agency.

So ... why do NBA players make so much more than NFL stars? Math, basically.

1. NBA players get a slightly bigger slice of league revenue than do NFL players.

Under the current labor deal, NBA players get roughly half of all league revenue before expenses. Given an estimated $6 billion in qualifying NBA league revenue, players are allotted roughly $3 billion in total. There are mechanisms in place to ensure that players get their share of basketball-related income guaranteed to them under the collective bargaining agreement.

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The NFL revenue split is more complex, with varying splits based on income source and some other carve-outs for franchisees (like stadium credits). But the upshot is that NFL players are collectively guaranteed just under half of league revenues. That comes out to about $6 billion or so, given projections that NFL revenue will surpass $13 billion next season.

So in total, thanks to labor deals reached between the leagues and unions, NBA players collectively earn about $3 billion per year and NFL players collectively earn about $6 billion per year.

2. There are almost four times as many NFL roster spots as NBA roster spots.

This is the big one. NFL players get twice as much total money as NBA players, but that money is spread over almost four times as many athletes. There are 32 NFL teams with 53 roster spots each, making for 1,696 NFL players at any given moment. There are 30 NBA teams with 15 roster spots each, making for 450 NBA players. (Many NBA teams carry just 13 or 14 players, as well. NFL teams do not typically go under 53.)

Basic math bears out how important this basic fact is to the issue at hand. With $6 billion spread over 1,696 players, the average annual NFL salary would be about $3.5 million. With $3 billion shared among 450 players, the average annual NBA salary is roughly $6.7 million. So just because of roster sizes, the average NBA salary is roughly double that of the average NFL salary.

3. The NBA's player maximums inflate the salaries of the middle class.

The NBA has a soft salary cap structure — you're more likely to see teams exceed the cap than sit under it. But there is a hard cap on individual player salaries. These figures are based on percentages of the cap and years of service. Veterans with a decade in the league can sign deals starting at 35 percent of the cap. So LeBron James, who's been around 13 seasons, can't sign for more than a starting salary of about $33 million this season, even though every team in the league would offer him $50 million or more if given the opportunity. If LeBron is worth $60 million per year but can only make $33 million, that other $27 million has to go somewhere.

The max salaries are lower for younger players: Kevin Durant has nine years in, so he can only sign for a starting salary that is 30 percent of the team cap. (That's why he's likely to sign a short deal; his max salary increases to 35 percent next year, once he has 10 years in the league.) Players signing their first contract after their rookie deal can only go up to 25 percent unless they meet strict criteria to exceed it (multiple All-NBA nods, an MVP, etc.). These are artificial constraints that depress the freedom of teams to give the best NBA players what they are truly worth on the court.

But that $3 billion has to be spent! So we end up with a feeding frenzy for middle-tier and lower-tier free agents, especially when the NBA team salary cap jumps up as it has in 2016 (and will again in 2017).

This is not the case in the NFL, where (for the most part) the stars are paid like stars and the lower-tier players are paid like lower-tier players.

3b. The NBA rookie scale is damn small.

The No. 1 pick in the NBA Draft, Ben Simmons, will make $4.9 million in 2016-17, or 5 percent of the team salary cap. In 2017-18 he'll make $5.1 million, or 4.7 percent of the projected $108 million team cap. And he's the No. 1 pick! Lower first-round picks are capped even tighter, all for the first four years of their careers. As with maximums for superstars, this means more money for the NBA middle class.

The NFL recently got its own rookie contracts under control. No. 1 pick Jared Goff will make only marginally more than Simmons over the next four years. In addition, because of the league's age minimums, NFL draft picks are in theory better prepared (physically, if nothing else) to produce early in their career as opposed to the NBA's 19-year-olds.

Why is this an issue? Say 35 total NBA draft picks (from both rounds in the current year plus stashed players from previous years) make opening night rosters. That's almost 8 percent of the entire league on small, fixed-rate contracts. Some of these players might not be productive, but teams will keep them on their rosters with hopes of developing their potential. That limitation on the available roster spots for veteran players inflates the dollar value for that middle class we discussed above.

This will likely change in the next NBA labor deal, possibly with rookie scale deals becoming percentages of the team salary cap instead of set dollar amounts. (There's talk the league could also expand roster sizes to add developmental slots.)

3c. The NBA's salary cap structure diminishes the role of non-guaranteed contracts.

A huge difference in NFL and NBA player salaries is that only a fraction of the typical NFL contract is guaranteed. Most NBA contracts are fully guaranteed, or include a player option. There are plenty of team options or non-guaranteed contracts out there. But because of the issues described above -- there's so much money to go around, and you can't pay the superstars more if you wanted to -- players and their agents have power and can wave away non-guarantees. That power doesn't exist for the NFL middle class. Only the biggest stars get huge signing bonuses (which equate to guarantees, effectively).

There are also some factors beyond the basic math that lead to larger NBA contracts than NFL contracts. Note that these likely have a much smaller impact than the above factors.

4. The NBA player pool is much smaller.

In the NFL, there are short athletes and tall athletes, fast athletes and slow athletes, svelte athletes and rotund athletes, quick athletes and strong athletes. Most if not all NFL players are incredible athletes in some fashion, at the so-called skill positions and on the lines. Even placekickers and punters: those are legit athletic skills! The important thing is that there is variety. If you're a small, quick sprinter type, with loads of hard work and some luck you could possibly make it to the NFL as a running back, defensive back or slot receiver. If you're a tall guy with a huge frame and immense work ethic, you could (with hard work and luck) make it to the NFL as an offensive lineman or defensive tackle. If you're a freak athlete who weighs 300 pounds, you could be a defensive end. Pretty much every male body type is found in the NFL. Height and muscle mass are seriously limiting factors, but there's hope for many.

That's not the case in the NBA. The average American male is 5'10. There are a couple guys shorter than that in the entire NBA. Those guys are some of the best athletes in the world. You need to be a 6-footer at minimum if you want to make the NBA. And even then, unless you're closing in on 7 feet, you need to be an excellent athlete or have some elite basketball skills. If you have a large, muscular frame, you'd better be at least 6'8 with some leaping ability or 6'10 with some touch. With only 450 roster spots, the NBA is very particular about what types of bodies make it. Consider Draymond Green, who hit the genetic lottery for height (6'7) and put in years and years (and years) of hard work to develop a strong body and excellent NBA skills ... and still didn't get drafted in the first round. The NBA is very particular. The NFL is less so.

5. NBA players last longer.

Due to injuries, churn and (increasingly) early retirements, the average NFL career is shorter than the average NBA career. This isn't absolute — European domestic leagues are littered with NBA washouts in their 20s — but as a general concept it bears out. According to the players' unions, the average NFL career is 3.2 years, and the average NBA career is 4.8 years.

Why does this matter? Salaries tend to escalate as players reach their athletic primes. In addition, the NBA salary structure emphasizes years of service in figuring players' maximum salaries.

This figures into the player pool issue above: because the NBA is so self-limiting in terms of the type of athletes who can make a roster, there's simply fewer options for teams, leading more fringe NBA players to stay in the league or return to it after spending time elsewhere. There's also the matter of minor leagues. There are non-NFL options for football players who wash out of the league. But there are far more non-NBA options (including the domestic D-League) for basketball players who fall off a roster. It's much easier to keep the NBA dream alive through your 20s than it would be to keep the NFL dream alive. Basketball is tough on bodies, but not nearly as much as the brutal sport of football.