In data science classification algorithms need to be good at both catching things that are true as well as throwing away things that are false.

One that’s excessively good at one but not the other will under perform its critical mission even if some of its headline accuracy numbers are very good.

For example, if you wanted to design a classification algorithm to predict whether or not someone had an extremely rare case of case of cancer, you could design an algorithm that was nothing more than “y=0”, and it’d be right more than 99% of the time!

However, that algorithm is missing the point in important ways because it’d correctly predict 0% of the true positives, and many people who had the disease would unnecessarily suffer because of it.

The F1-Score

The F1 Score is a metric that’s used to calibrate an algorithm’s proficiency at balancing both its Precision (true positive accuracy) and its Recall (true negative accuracy) in a way that penalizes being deficient in one area or the other.

Its formula is simple but very effective:

2*Precision*Recall/(Precision + Recall)

Since Precision and Recall are both expressed as decimals between 0 and 1, the F1-Score has the benefit of driving a score down to zero if either one is lacking.

For an example, consider two classification algorithms that have equal amounts of Precision and Recall, but are distributed in different quantities.

If one has Precision of 90% and Recall of 10%, its F1-Score will be 2*0.9*0.1/(0.9+0.1) = 0.22. That’s pretty lousy.

However, if another algorithm scores 50% in both its F1-Score will be 0.50. More than twice as good, even though their raw scores are the same.

An F1 Score, Applied to NBA 3 Pt Shooting

Anyone who follows the NBA knows that we’re in the era of the 3 point shot. The Warriors are the best 3 point shooting team of all time, and the 5 most prolific 3 point shooting teams in NBA history have all happened in the last 3 years.

Last year Steph Curry attempted 15.72 3’s per 48 minutes, which is more than 13 teams did 20 years ago.

And the ubiquity of the 3 point shot is becoming more pervasive. For example, treys from behind the arc were once the sole domain of backcourt players, but the last 10 years has seen its use crawl up the height curve.

This effect is most easily seen by looking at players that enter the draft. In the past, being over 6’8 meant that your job was to stick close to the basket, grab rebounds, and have a competent jump hook when you had the ball in the post. However, today’s young players have had the benefit of observing the rise of the 3-point shot during their youth, and were able to get the repetition and coaching necessary to perfect it, regardless of height. (In fact, you could argue that shooting is more valuable the taller you are because it draws defenders out of the paint, creating larger driving lanes for penetrating guards).

In the 2016 draft, more than half of the frontcourt players drafted had the ability to hit the 3 point shot. The draft from ten years before there was only one. (Andrea Bargnani).

The 3-point shot now comes at defenses from every position on the court.

Volume vs. Accuracy

The sea change in the 3 point shot is due to shooting volume, not shooting accuracy. For example, here’s a histogram that plots volume and accuracy over the last 20 years.

As you can see, volume has clearly gone up a notch, with the highest volume team in 2005-2006 the average in 2015-16:

Coincidentally, the team with the highest 3-ball volume in 05-06 were the Steve Nash Suns, who’s offense was considered an anomaly at the time, mostly due to their over-reliance on the outside shot. They’d be completely average by today’s standards.

Accuracy, meanwhile, hasn’t really changed at all during that same time:

Presently, there isn’t a useful way to measure a player’s ability to combine both volume and accuracy in one number, even though it’s your ability to perform both that makes you valuable.

There is a reason Steph Curry is a generational shooter while Steve Novak is not, even though they both shoot similar percentages.

With that in mind, the purpose of the rest of this post is to elaborate on something we’ll call the Shot Score: a metric that uses the F1-Score to calibrate a player’s ability to shoot with both accuracy and volume.

What the stat does is standardize a player’s volume and accuracy by dividing it by the league leader in both categories, and then uses an F1-Score to generate a number between 0 and 1.

Additional analysis has been done to rank players against people who regularly take jumpshots as well as their own positions in order to make useful comparisons for each player.

We’ll call this rebranded F1-Score the Shot Score to make it more semantic.

The Shot Score – The Big Winners

Here’s the head of the data used to make the shotscore, with an overview of what each of these numbers mean.

Player, Year, Team, Height – Self explanatory, with the caveat that height was converted to inches to make calculations easier.

Att – 3PA/40.

Acc – 3 point accuracy.

sAtt – Standardized attempts. This is each players 3PA divided by the league leader in that category (Steph Curry).

sAcc – Standardized accuracy. Works the same way as sAtt.

F-Score – The F-Score derived from each players sAtt and sAcc.

Z – How many standard deviations each player’s Shot Score is above or below the league as a whole.

zShot – How many standard deviations each player’s Shot Score is above or below the league’s jumpshooters. (Ie, any player that attempted more than 1 3PA/40 minutes).

zPos – How many standard deviations each player’s Shot Score is above or below the league average at their position.

What Moves the Numbers

Before we begin to dig deeper it’s worth observing the distribution of sAtt vs. sAcc, which can be seen here:

The take home point from this picture is that league wide accuracy bunches together towards the top, whereas league wide volume bunches together towards the bottom. Ie, there’s more separation to be had in a player’s volume than their accuracy.

If you glance again at the league leaders, you can see this effect in plain sight. Kyle Korver and Doug McDermott are nowhere to be found, but gunners like JR Smith and Isaiah Canaan comfortably sit atop the league.

Volume it turns out, is an elite skill.

How Dominant is Steph Curry?

A picture tells a thousand words. Here’s the distribution of each player’s Shot Score. Can you guess which one belongs to Steph Curry?

He’s the one furthest to the right.

Steph Curry is 1 standard deviation above his nearest competitor when compared to the league at large, its jump-shooters and his position.

The Effect of Volume – Surprisingly Large

It’s very easy to compare how accurate shooters are because you can just use their percentages. It’s less intuitive to determine how you can compare a player’s accuracy in light of how many shots they take.

For example, take two players who both carry the perception of being good at shooting: Jose Calderon and Damian Lilliard.

They’re both scouted as players you have to guard behind the arc, and shoot very respectable percentages at 0.414 and 0.375. At first glance you might consider Calderon to be the better shooter since he has a higher 3-point percentage.

However, how much separation is there between them when you account for how many shots they take to arrive at their volume?

Per 40 minutes Jose Calderon shoots 4.0, and Damian Lilliard shoots more than twice as much at 9.1.

The difference in volume might mean that Lillard’s a chucker, but we’ll give him the benefit of the doubt since he’s an all-star and the best player on a playoff team…….he must be doing something right, or else Terry Stotts wouldn’t play him so much.

If we assume both take the majority of their shots in the best interest of the offense, then the difference in volume could be interpreted as Lilliard having twice the ability to put pressure on opposing defenses, and therefore twice as hard to guard.

And maybe…..he ought to be considered as twice as good of a shooter since he can double Calderon’s volume while maintaining similar accuracy.

When you blend their volume and accuracy together this is how the two compare to one another:

Here you can see the difference between the two is very large. Lillard comes out 1.64 standard deviations above Calderon.

Calderon and Lillard are both good shot makers, but only Lillard excels as a shot-taker, making his accuracy more valuable.

The effect of volume is most acute at the PG position, where an inability to take a high volume of shots will relegate you to the middle of the pack at best. Steph Curry all by himself has a big impact on the volume curve with his insane shooting volume, but most of the other league’s best gunners do it while playing PG as well.

Isaiah Canaan and Trey Burke’s agents ought to send me a thank you note, because I might be the only guy who’s tried to credibly position them as top-10 at their position in anything.

Here are the 10 highest shot scores for PG’s in the league:

Meanwhile the bottom of the list is populated by a few players who are noted for their accuracy, but get pulled down because of their inability to shoot with sufficient volume, such as Jose Calderon and Darren Collison:

The difference between shot taking and shot making evens the score between many other players who have very different reputations as shooters.

For example, when you account for both volume and accuracy, this is how JR Smith and Kyle Korver rank against their position:

Jr Smith comes out ahead by a modest but substantial amount, even though Korver carries the reputation as the league’s most pristine shooter.

One might scoff at this number, but it’s worth noting what the best lineups on the best teams usually comprise of.

Consider last year’s Finals matchup. Both teams had vaunted ‘Death Lineups’ with efficiency differentials well above +10, which is usually the marker for a championship contender.

They looked like this:

Cavs:

Kyrie Irving, Iman Shumpert, JR Smith, Lebron James, Kevin Love.

Warriors:

Steph Curry, Klay Thompson, Andre Iguodala, Harrison Barnes, Draymond Green

The one quality both lineups have in common is each position has a versatile player that can score from different spots on the court. One dimensional players like Matthew Dellavedova and Andrew Bogut were on the bench, despite being more effective at their respective specialties.

One trick ponies increasingly find themselves sandboxed during the most intense competition, and Kyle Korver himself has been a useful anecdote to make this point.

Despite his legendary shooting and ability to deftly move off-ball, his WS/48 is 50% lower in the playoffs than the regular season.

Meanwhile, someone else was on the court for the Cavs in crunchtime against the Warriors……….

Unsung Heroes

When you account for volume some of the league’s most noted marksmen aren’t as impressive as they seem.

However, the Shot Score shines a favorable light on players who can manage to shoot a good percentage while having to endure the burden of maintaining very high volume.

Klay and Steph are unsurprisingly atop the league in this regard, but this article warrants a closer look at three players who soldier on with little recognition despite combining volume and accuracy in elite quantities.

They are Mirza Teletovic, Robert Covington, and Isaiah Canaan.

To visualize, here’s how each of these players compare to their positions.

Mirza dominates NBA frontcourt players, with a shot score that’s 2.41 standard deviations above the league at his position:

Robert Covington does very good as well, with a shot score that’s 2.11 standard deviations higher than the average wing player:

Isaiah Canaan’s 9.9 3’s per minute was second only to Steph Curry among PG’s, and his volume was a respectable 76% of the league’s best, giving him a shot score that was 1.76 standard deviations above his position:

This measure clearly has some drawbacks, because volume becomes a bad thing if said player takes the shots in a way that subtracts from team’s overall efficiency. But clearly accuracy is most useful when it’s calibrated with volume, and this measure might provide us with a useful way of observing that.