When is something proven?

You probably had a whole stupefying section in math class dedicated to answering that very question. If you were like me, that meant nightmares filled with algebra and this elusive three dot triangle. The way I learned it, you started with two seemingly unrelated mathematical statements and had to prove they were equal. Once you had it figured out, you would make a triangle out of three dots to indicate your success. The symbol screams "proven!" but I always struggled to write it with confidence aware of the many mistakes I could have made.

But in the NBA you almost wish it were that simple. A few calculations here, manipulate some statistics, make sure you adjust for pace, and there you have it. Irrefutable proof that the Blazers are for real. Everyone comes to this realization together and we all go about our merry way.

If only it were that easy.

The problem is everyone has their own idea about what it takes to be proven in this league. The Blazers have played 37 games with a top two defense, the key to their status as a contender. Is it proven? Are they? Ask twelve fans and I bet you'd still get a hung jury but the internet intelligentsia is coming around. Look around the NBA, or even at the fan shots for that matter, and it's littered with writer after writer describing their recent epiphany that the Blazers are, ya know, good. It's finally happening. The jury is coming to a decision and we no longer have to scream from the witness stand.

"No really, I saw it. I was there. They were fighting through screens and stunting and everything. You gotta believe me."

That vindication is sweet but it's making this one nagging little question all the more annoying.

A few weeks ago, I checked-in on the Blazers' defense and came to the conclusion it was much improved (duh), probably top ten (double duh), but was primed for a regression out of the top five (wha? This article was so much fun, why the wet blanket?). Most of this was based on the bricky jump shots opponents kept bringing to our games, especially from the three point line. The Trail Blazers are on pace to allow the lowest shooting percentage from three (28.6%) since the Detroit Pistons in 1989, one of the greatest defensive teams of all time in a league that averaged 32% from deep. Portland hasn't been contesting shots at an elite rate so it didn't make sense that opponents were shooting so poorly. It was fun while it lasted but the law of averages meant a couple hot shooting nights were bound to bring the Blazers' defense back down to earth.

Laws shmaws. Over the 17 games since then, opposing teams are actually shooting worse from behind the arc. Thirty-seven games is nothing to sneeze at but it's also not a sample size immune to anomaly. This represents one of the few legitimate legs doubters have left to stand on (as opposed to stuff like "jump shooting teams can't win in the playoffs" or "I don't know if Aldridge can lead a team to the championship"). If there's an explanation for our opponents' poor shooting then it's case closed and we can all enjoy the adoration in peace.

Continuing the allusion to school, we can break up possible explanations into the six key questions: who, what, where, when, why, and how. Shout out to my fourth grade English teacher Mrs. Green. See, I was paying attention.

Who

Who is taking these shots? Perhaps the Blazers are leaving poor shooters open while preventing good shooters from shooting at all? If you only leave 30% shooters open then it wouldn't be surprising if opponents average just 30% on uncontested threes.

This is difficult to test but Nylon Calculus' Expected Points Per Shot (XPPS) provides some inspiration. XPPS is a very advanced (and very cool) stat that looks at the location of shots a defense allows. For a simplified example, suppose a defense allows 30% of opponent's shots at the rim, 30% from three, and 40% from mid-range. Given typical shooting percentages, they calculate how many points you would expect the defense to give up per shot (Nylon Calculus actually uses much more granular and historical data but that's the general idea). In a nutshell, the stat considers a distribution of locations assuming an average player.

We can invert this and look at distribution amongst players while assuming one location. Another simplified example, suppose Kyle Korver (51% from three), Wesley Matthews (40%), and Josh Smith (22%) all played for the same team. If Korver and Matthews take 80% of the threes we would expect the team to shoot a very high percentage from three. If Smith takes 80% of the team's threes we would expect them to shoot a very low percentage as a team. The distribution of three point attempts amongst a team's players greatly affects the team's overall percentage.

If a defense could consistently warp opponents' distribution so their worst shooters took the most shots they would allow an unusually low percentage regardless of how many shots were contested. In order to test this, I started with the distribution of three point attempts the Blazers allowed over the past 20 games. Then I used each player's three point shooting percentage to calculate the shooting percentage we would expect the Blazers to give up.

Unfortunately, this "Expected 3P% Allowed" (35%) was a little higher than what our opponents' average for the season (34.5%) and much higher than the 28.5% they actually shot against Portland.

The Blazers are great at preventing three point attempts overall but they don't change who takes those shots in any appreciable way. Guess we have to look elsewhere for an explanation.

*shout out to CascadianAbroad & jnewhouse for helping me think this through in the comments of my check-in

What

What kind of shots are these? If opponents are taking pull ups or step backs then it would make sense they're shooting poorly, even if they're open. NBAsavant.com tracks the type of shots breaking up three point attempts into jump shots, pull up jump shots, step back jump shots, fadeaway jump shots etc. As a result, we can calculate if the Blazers are consistently forcing teams into more difficult types of shots.

Once again, Portland comes out just about average. They rank 16th in the league with 95% of opponent 3PA being regular ol' jumpers. However, the league is shooting worse on "jump shot threes" compared to threes overall. This doesn't make any sense so there may be a problem with the classification system. Either way, there's no evidence that the Blazers are forcing teams into more difficult types of shots.

Strike two.

Where

Where are these shots coming from? Perhaps the lack of corner threes explains the lower overall percentage. The Blazers are elite at preventing corner threes with less than 22% of three point attempts coming from the corners. That's good for fifth in the league but it still comes up short.

If teams shot league average percentages from the corner and above the break, we would expect Portland to allow opposing teams to shoot 34%, slightly below average. Preventing corner threes is certainly helpful but it doesn't explain how the Blazers are forcing misses at an unprecedented rate. In fact, opposing teams' ineptitude from the corners is one of the toughest things to explain. Opponents are only shooting 31.2% against the Blazers compared to 39.5% league wide. That's a huge discrepancy and there's no apparent reason.

Strike three. Good thing we're not playing baseball.

When

When teams shoot few threes do they shoot worse on the threes they do take? Finding a rhythm is something broadcast crews talk about all the time and it's a widely accepted phenomenon. Perhaps the Blazers force misses simply by not letting teams shoot many shots. The Blazers allow the second fewest three point attempts a game so if this were true it could explain a good portion of their success.

Historically, there doesn't seem to be a relationship. The correlation between three point attempts allowed and three point percentage allowed is very small (~0.2) and it doesn't change much if you look at different timeframes. There's no evidence that allowing fewer attempts also affects how teams shoot. Rhythm may make good sideline chatter but it fails to explain the Blazers' success.

How

How are the Blazers contesting? We know Portland isn't contesting a high percentage of threes but maybe they're contesting the right ones? Portland could be challenging shots in the corner dropping those percentages and only leaving shooters open above the break where percentages are naturally lower.

Nope. The Blazers are actually worse contesting corner threes and the 16.7% opposing field goal percentage is particularly crazy given the league average is 35.6% and the next lowest percentage allowed is 23%.

But what about how strongly they contest? Portland may not contest a particularly high percentage of shots but what if when they do contest they're in the guy's jersey? We can look at this by calculating the percentage of shots the Blazers are within two feet of the shooter.

The Blazers are actually elite ranking in top five. Unfortunately, these are really small percentages. Challenging 5% versus 2% won't have a large affect on the opponents' overall percentage. Certainly not enough to explain the Blazers' historic success.

*Sigh*

Why

Why are opponents shooting so poorly against the Blazers?

I have no idea. That was all I had. I am officially out of ideas.

This makes me nervous because if there's no basketball reason then the only thing left is luck and luck is one unreliable lady. She tends to run out right when you need her most.

The Portland Trail Blazers are absolutely a title contender and people are finally starting to notice. Enjoy it, revel in it even but let's not pretend there aren't any questions. If opponents start shooting like they did last year Portland's defense would fall all the way back down to average. That's not likely to happen since we're already 37 games into the season but that statement is staggering. This is not a small issue and the chance that the Blazers' defense isn't as good as their numbers suggest is the scariest thing currently facing Rip City.

Luckily, the Blazers are not the only Western Conference playoff team with questions still left to answer. From health to depth to free throw shooting, no one's resume is spotless. It's tempting to see the Blazers' record and feel that they're pulling away from the pack. But this question about whether opponents' poor outside shooting will continue is the biggest thing keeping Portland within the morass of the Western playoff picture rather than above it.

I want to write those three little dots next to the Blazers' name so bad but I just can't do it yet.

*All stats are from basketball-reference.com unless otherwise stated.