As we near the end of the first trimester of the season, we’re already setting up for an epic conference finals in the playoffs: the Rockets, by most standards, have been outplaying the Warriors, who will be without Stephen Curry for a while now. Meanwhile, the Cavaliers are playing well again, and Isaiah Thomas hasn’t even returned. Either the Raptors or Celtics will provide some great competition. Never write off an entire season before it starts — I’m sill excited to see how this one ends.

The Big Penguin’s big improvement

The Detroit Pistons, despite a rough past couple weeks, are still one of the most improved teams this season, and it’s primarily because of their center Andre Drummond. When their coach Stan Van Gundy first came aboard, everyone was predicting Drummond would finally have his breakout season thanks to the coach, who aided in Dwight Howard’s development in a similar system. It didn’t happen immediately, but it finally happened.

It’s not especially notable when any player improves, and Drummond is still younger, but it’s the ways in which he’s improved. Usually you see players take more shots and offer a more varied offensive game, or perhaps they reinvent themselves with a new position or role. But Drummond has vastly improved his free throw percentage and his assist rate, two stats that are remarkably stable over a player’s entire career.

Let’s start with the free throw percentage. As much as people think free throws are a skill-based activity you can improve at, there are shockingly few cases in NBA history where a player has made substantial improvements over the course of just one year. Karl Malone is one famous example. Over his first two seasons he shot 54.8 percent from the line, and in his third season he hit 70 percent and got to 76 percent over the rest of his career — but even that jump isn’t large enough over one year.

Only four other players in NBA history have had a jump of 20 percent with at least 100 free throws for the season and 250 for all previous seasons: Michael Smith, Chris Webber, Adonal Foyle, and Kent Bazemore. Smith didn’t play much longer, so we can’t stay if that improvement was permanent. Foyle regressed a bit, but at least he was no longer a sub-50 percent shooter. Bazemore is a surprising case: for the first part of his career he shot the same percentage as Kendrick Perkins, and now he’s got a second season north of 80 percent. Chris Webber is a another positive case and role model for Drummond: he went from shooting like Shaquille O’Neal to someone who regularly hit over 70 percent of his foul shots.

As for the assists, there are few good comparisons. No one else with significant minutes tripled his best mark for assist rate in a season. The closest was Greg Ostertag (with only 1107 minutes in the season), who went from being a player with an embarrassingly low rate to something approaching normality for a center. If you look at a rate 2.5 times larger, then you get another list of four players: Nicolas Batum, Alex Len, Wayne Cooper, and Kelvin Cato. Three of those players are center whose rates were so low that it was easier for the rates to bounce around — and they all regressed. Batum’s the best example, as he legitimately transformed from a typical wing to a point forward. Drummond, however, is in his own club of improving in both assists and free throws, and it’s erased two of his biggest liabilities and has significantly changed some of his impact numbers. And it was done by fixing two of the hardest to change skills in basketball.

(If you look at other players with a 12 percentage point jump in assist rate you get a longer list — Deron Williams, James Harden, Nic Batum, Sleepy Floyd, Lester Conner, Michael Adams, T.J. Ford, Dean Meminger, Jordan Farmar, Muggsy Bogues, Mike Holton, and Draymond Green — but it’s a bit different for a high-assist player to add those percentage points than a low-assist player.)

MVP candidacy: LeBron James

LeBron James has single-handedly pulled the Cavaliers out of the danger zone, and he’s being rightfully considered a strong candidate in the MVP race. I think we’re all taking that greatness for granted because it’s awe-inspiring to see him perform at this level given his age and historic minutes load. He’s closing in on 30,000 career points and he’s 33 years old — yet he’s still playing at near a peak level. He’s aging like a fine wine, and only Michael Jordan and Karl Malone have won it at his age or older. I’m not entirely confident he’ll keep playing this well, but the fact that he’s producing like this for over a quarter of the season is notable.

Stephen Curry’s ankle strikes again

The biggest weakness for the Golden State Warriors reared its ugly head again: Stephen Curry is injured again, and he’ll be out for at least two weeks until he’s reevaluated again. (Yes, their biggest weakness is injury danger, not something on the basketball court.) This is the same ankle that endangered his career years ago, and that’s probably why they’re being so cautious.

Given Golden State’s lofty position in the league, this is more of an opportunity for the team than a challenge. They can hand the keys over the Kevin Durant, and they’ll get to see what the team is like without Curry. So far they’ve won games against the Pistons, Hornets, and Blazers, and Durant indeed has been the one increasing his role. The team should have plenty of scoring power though with him and Klay Thompson; I’d worry more about their loss of Curry’s playmaking. Shaun Livingston is their only other rotation-level point guard, and as good as Andre Iguodala and Draymond Green are at passing they can’t quite fill the same role. The Warriors had to call Quinn Cook out of the G-league to help soak up those point guard minutes, and he actually ended up starting a game — they know better than anyone they can’t just rely on Iggy and Green for those duties.

The return of Kawhi Leonard

The Spurs have been relatively cool this season by their standards, but they’ve been without their superstar player, Kawhi Leonard. Their offense has been average, but they still have one of the best defenses in the league. So how good will they be once he’s back? This is where the Spurs’ brilliance is a double-edged sword. You cannot assume Kawhi will be adding “replacement level” value because their backups are so good. Thus, you have to estimate the value of the minutes he’s replacing and add the difference in value.

First of all, I’ve to guess who’s going to lose the minutes. This is tough because although Rudy Gay was their big offseason acquisition, Kyle Anderson has definitely earned playing time with how he’s been this season. My solution was splitting time between them equally: 14.2 minutes per game. The remaining five minutes I have coming from David Bertans. Based on all their BPM values, including Kawhi’s 2017 numbers, I have the Spurs adding 3.4 points per game of value from his return, which would make them a team that’d win an equivalent of about 58 wins. Due to their depth and quality of options they’re a good team without him, and they won’t be in Houston or Golden State’s tier when he returns. This is no surprise, but with Leonard they’ll be about as good as last season.

Bradley Beal’s 51

Whenever a player tops 50 points, I feel as though I need to take note, but lately these games have been relatively common. (I think we’ll see more than the three we’ve had so far this season.) Beal’s game is the zenith of his growth as a scorer — he’s near the demarcation line of 30 percent usage rate, which is a mark you virtually only see with stars. He’s still a mid-range fiend, and he’s actually taking fewer 3-point shots than last season, partly explaining his dip in efficiency. But that’s how he can pop off a large proportion of Washington’s shots without great size or quickness. And my favorite basket in the below clip, by the way, is the one where he has a defender on his hip, he steps to the side, and hits a bank-shot from around 15 feet.

Bench strength: Pascal

Quietly, the Raptors have been building a case for the best team in the Eastern Conference and have a shot at the best record, despite not having LeBron James or the magic of the Boston Celtics. They’ve improved over last season thanks to a more effective offense primarily driven by significantly more 3-point shots. But they also have a very productive bench again this season despite the loss of their main cogs from last season including PJ Tucker (traded midseason), Cory Joseph, and Patrick Patterson. The development of Pascal Siakim has helped a lot, as the Cameroonian big man has shown a lot of potential in his second year.

Like fellow Cameroonian Joel Embiid, Pascal was discovered and mentored by Luc Richard Mbah a Moute. He’s another young big man, and due to his size — he’s a skinny 6-foot-9 — he’s primarily a power forward, backing up Serge Ibaka. He’s the requisite energy player, going full speed up and down the court like a mad man, which has earned him more playing time. Offensively, he’s attempting to pull an Ibaka this year, as he’s launched a number of outside shots with a ludicrously low conversion rate of 19 percent. But inside of 3 feet he’s at 81 percent — one of the best rates you’ll see out of any player — and he’s made real strides on his passing.

Siakim missed the shot here, but it’s a good example of how he can hold his thumb on the turbo button, so to speak, and out-hustle everyone (however, someone was already waiting there for him at the rim.) He’s great at leaking out and diving at the rim in semi-transition sets. In the half-court, he’s not someone you run plays for; he’ll make hard cuts to the rim or score on putbacks. And his passing is definitely above the average for an energy player now: note that quick and decisive pass to the corner in the play here.

Defensively, he does seem like a plus. He tries to be active, and even though his rebound numbers are small for a player of his type, I believe it’s because he’s happy to cede the easier rebounds for others while he snatches the tougher ones when he needs to. He doesn’t net a ton of blocks, but with his length he is hard to shoot over and he’s a decent rim protector. Overall, he’s an okay defender for a big guy; he has some promise to be even better though. And he’s another in a long line of great bench players the Raptors have had in recent seasons.

Reinventing the Pythagoras Wheel

One of the most popular and widely utilized advanced stats in the NBA community is the Pythagorean win expectation formula. It’s just a way to estimate expected win percentage from how many points a team scores and allows. This is one that’s been used for so long that it’s been taken for granted. It was actually developed by Bill James for baseball, but it was later translated to the NBA game with just a tweak of the exponent by Daryl Morey, Dean Oliver, and later John Hollinger. But it’s only an approximate formula, and I think that the NBA deserves a more precise formula given its importance and how deep our other statistics have gone.

When I was looking to remake the win expectation formula, I first had my eye on probability distributions because it seemed like the most direct and accurate representation of reality. You can see what I mean from the distributions below with the NBA — scores center around some mean and then gradually taper out on both sides. This is true for points scored and allowed, and thus, if you want to know how often a team will score more points than allowed, you just need to figure out that overlapping area. But in a roundabout way, that’s what we’re already doing.

Going back to the Pythagorean, the original form was actually just Runs^2 / (Runs^2 + Runs allowed^2). For those of you rusty with your geometry, that’s why it was called the Pythagoream win formula because it was a copy of the famous triangular formula. However, this is grossly misleading. The exponent of “2” has largely been discarded in favorite of more accurate exponents, like 1.83 and in basketball we use 14 or sometimes 16.5 — that’s certainly not in the famous triangle formula. Why are these non-whole numbers being used? A mathematical derivation of the formula found that it’s actually the result of Weibull distributions, and that makes a lot more sense.

The Weibull function has two important parameters: the scale factor and the shape factor. The scale will stretch the distribution further out along the x-axis, which in our case means points — a higher scale, the higher your points per game. Meanwhile the shape factor will squeeze or squish (technical terms) the distribution itself around the mean. If there’s a greater variety of scores in your games, your shape factor will need to be lower. That exponent referred to earlier is the shape factor, and that’s why it’s not a whole, absolute number and why it differs between sports.

Now that I’ve successfully explained the old wheel, let’s try to make an improvement. What I want isn’t groundbreaking but it should seen necessary after that explanation: a variable exponent factor, the shape, based on the spread of the scores themselves compared to the mean. After some testing, I found the best fit for the shape factor “k” was [ average(PPG + Opp. PPG)/average(St. Dev PPG + St. Dev Opp. PPG)]^1.21. Basically, you divide the average of all the scores a team was involved in (i.e. both points scored and points allowed) by the standard deviation of all those scores, and then take that number and apply the 1.21 exponent. Once you have k, which varies team by team, you use the form of the Pythagorean equation: Win% = Points scored^k/(Points scored^k + Points allowed^k).

Why go through all that trouble of calculating k like that? Because score distributions aren’t constant team-by-team and, especially, from era-to-era. What this variable k does is take the variation of the game score into account, so if you’re a great team with a wide distribution you win fewer games than expected because you have more games at the tail ends where you can be beaten — and vice versa, as a high variance actually helps the worse teams. This makes more sense when you refer to those graphs above. With a wider spread, you’re more likely to overlap your opponent’s scores.

As you can see from the table below, there is an advantage to this method, at least from this initial testing. From what I’ve surmised, the main advantage of my new method is that it can adapt to new eras. I tried it out for the crazy, no-shot-clock era of 1950 to 1954, for example, to see how well it would do — and it was significantly better. Also, in theory, it should do well with partial seasons because you usually see higher standard deviations at first until you have a larger sample; thus, the formula actually regresses to the mean a bit for you.

Table: Root-mean squared-error for win estimates

Seasons Pythag. New method 1950 to 1954 4.09 3.46 1990 to 1999 3.22 3.22 2000 to 2009 2.82 2.83 2010 to 2017 3.01 2.92

(Prorated to 82 games)

I had two primary goals for this new win formula: I wanted something adaptable that wouldn’t break down at the extremes, and I wanted something that would be a real improvement for games now. I just wish it did more to reduce the overall error rate over every season more than it did. It was a dead-heat between the two techniques in the 90’s and 00’s — that’s when people were figuring out which exponent to use, so maybe that’s not a coincidence, but I’d still want better in-season estimates. It’s a good proof of concept, however, and perhaps with more tinkering I can find something even better.

For one, there’s probably a more appropriate statistical distribution for NBA scores that would allow for a better fit. Also, I can adjust for other factors, like some that explain a lot of the end-game “luck” with some common-sense tweaks. And we’ve learned at least one more thing: the Pythagorean formula in sports should be renamed the Weibull formula. The Greeks have gotten enough credit for civilization anyway.