# Learn a Stat: individual Defensive Rating

Welcome back to Hack a Stat! After told about individual Offensive Rating, in this post we discover the individual Defensive Rating. For this stat, there are important considerations to take into account.

ITALIAN VERSION

### Introduction

Individual Defensive Rating is a fascinating and insidious statistic. It gives reliable information, but due to its calculations, some data are partially lost: that’s why it is defined insidious.

### Definition and starting data

The individual Defensive Rating provides points allowed to the opponent team by a single player for every 100 possessions played. The lower the value, the better the player’s defensive contribution. It is, therefore, necessary to extrapolate from the box score all the player’s contributions that influence the opponent’s attack.

These contributions are three:

• Defensive rebounds [DR];
• Steals [ST];
• Blocks [BL];

However, consider only these contributions is reductive: they are all important actions in a team’s defensive economy, but there are many other situations that determine a good defense.
In fact, Dean Oliver in his book “Basketball on paper” explains that these other elements should also be accounted for:

• Forced turnovers by the player: all opponent’s turnovers (but not steals) should be counted. For example, it can be forced traveling;
• Missed contested shots: all missed shots contested by the player should be counted;
• Free throw granted by the player: all free throws granted by the player through fouls should be counted;

These additional three data are not available in box scores. So how to get them? They have to be found indirectly and distributed to all the players based on the minutes played. How accurate is this distribution? It is acceptable: in the absence of more precise information, it is the most effective method.

In order to find these data, the following values has to be counted:

• Opponent offensive rebounds [OppOR];
• Team defensive rebounds [TeDR];
• Opponent field goals attempted [OppFGA], made [OppFGM] and field goal percentage [OppFG%];
• Team blocks [TeBL];
• Player’s minutes played [MP];
• Team minutes played [TeMP];
• Opponent turnovers [OppTO];
• Team steals [TeST];
• Personal fouls [PF];
• Team fouls [TePF];
• Opponent free throws made [OppFTM], attempted [OppFTA] and free throw percentage [OppFT%];
• Team Defensive Rating [TeDef Rtg];
• Opponent points [OppPts];
• Opponent possessions [OppPoss];
• Opponent scoring possessions [OppScPoss];

Let’s move on the calculation.

### Formulas and calculation

The individual Defensive Rating is based on the calculation of the intermediate data defined by Dean Oliver as Stop. Stop is the number of opponent’s interrupted actions. This term is defined by two elements, each dependent on the various contributions. The first, Stop1, depends on the terms found on the box score; the second, Stop2, instead is dependent on “hidden” terms and so will require a longer calculation.

$\boldsymbol{Stop_{1}=ST+BL\cdot&space;FMwt\cdot&space;(1-1,07\cdot&space;OppOR\%)+DR\cdot&space;(1-FMwt)}$

Steals are simply added together, while blocks and defensive rebounds are instead multiplied by coefficients that take into account the rebounding fight. To understand the reason, take a look at the formula.

$\boldsymbol{OppOR\%=\frac{OppOR}{OppOR+TeDR}}$

OppOR% is simply the percentage of offensive rebounds grabbed by the opponent. Taking into account the player’s rebound opportunities instead of the number of grabbed defensive rebound is better. With the same number of rebounds grabbed, a defender may have lost 1 or 100 rebounds, then turned into offensive for the opponents. In addition, a rebound not grabbed by a player can also be taken by his teammates; this eventuality will also be taken into account with the other coefficient:

$\boldsymbol{FMwt=\frac{OppFG\%\cdot&space;(1-OppOR\%)}{OppFG\%\cdot&space;(1-OppOR\%)+OppOR\%\cdot&space;(1-OppFG\%)}}$

This second coefficient takes into account the opponent’s shooting percentage. Furthermore, thanks to the difference (1 – OppOR%) the team defensive rebound percentage is calculated, just to take into account the teammates’ defensive rebounds.

Why do the blocks also have to be multiplied by the coefficients that take into account the rebounds? The reason is simple. Following a block, these situations can occur:

1. Ball recovered by an opponent player: this is counted as an offensive rebound;
2. Ball recovered by a defender: this is counted as a defensive rebound;
3. Ball goes out of bounds: this is counted as an opponent offensive rebound and possession continues;

Statistically, the blocks and the rebounds are closely related. In addition, the term inside the parenthesis that multiplies the player’s blocks with the FMwt wants to give greater weight to a block that turns into a defensive rebound rather than an offensive one.

So, we identify the first part of the opponent’s interrupted actions by the player through steals, defensive rebounds, and blocks with the term Stop1.

Let’s move on to the second term, Stop2. For convenience we will divide the formula into three sub formulas:

$\boldsymbol{Stop_{2FG}=\frac{(oppFGA-OppFGM-TeBL)\cdot&space;FMwt\cdot&space;(1-1,07\cdot&space;OppOR\%)}{TeMP}\cdot&space;MP}$

$\boldsymbol{Stop_{2TO}=\frac{OppTO-TeST}{TeMP}\cdot&space;MP}$

$\boldsymbol{Stop_{2FT}=\frac{PF}{TePF}\cdot&space;0,4\cdot&space;OppFTA\cdot&space;(1-OppFT\%)^{2}}$

$\boldsymbol{Stop_{2}=Stop_{2FG}+Stop_{2TO}+Stop_{2FT}}$

That’s how those terms not shown in the box score are found: each sub-formula is related to the three data defined in “Definition and starting data”. The first defines the contested shots, the second the forced turnovers, while the third identifies the free throws granted. As mentioned above, the first two terms are multiplied by the player’s minutes/team minutes ratio, in order to distribute the contributions to the various players.

This, however, raises a question: the minutes played do not certify a player’s defensive ability. For example, a good defender and a bad defender may play the same amount of minutes, but will not give the same effort on the defensive side. Unfortunately, the limit of the Defensive Rating is this one. The assumption behind the Defensive Rating is that there are five players on the field with the same defensive ability and effort.

The sum of the two stops gives us the value of the actions interrupted by a single player:

$\boldsymbol{Stop=Stop_{1}+Stop_{2}}$

The next step is to determine the Stop percentage. In fact, you don’t want the total amount of stopped actions but rather the number compared to the total number of opposing actions. This term is defined as Stop% and is calculated with the following formula:

$\boldsymbol{Stop\%=\frac{Stop}{OppPoss\cdot&space;\frac{Mp}{OppMP}}}$

The opponent’s actions interrupted by the player are divided by the total opponent’s possessions multiplied by the percentage of minutes played. Also, in this case, this value (MP / OppMP) is used to take into account the player’s presence on the field to identify the number of possessions in which he participated.

Before arriving at the final calculation we still have to find the opponent’s possessions ended successfully:

$\boldsymbol{OppScPoss=OppFGM+\left&space;[&space;1-(1-OppFT\%)^{2}&space;\right&space;]0,4\cdot&space;FTA&space;}$

After doing that, we can calculate the individual Defensive Rating:

$\boldsymbol{DefRtg=TeDefRtg+0,2\cdot&space;\left&space;[&space;100\cdot&space;\frac{OppPts}{OppScPoss}\cdot&space;(1-Stop\%)-TeDefRtg&space;\right&space;]}$

As you can see, the individual Defensive Rating is not a simple division like the individual Offensive Rating. The most remarkable presence is for sure the team Defensive Rating: the inclusion of this data in the calculation of the individual Defensive Rating helps to set the average team’s defensive level as the base value and then adding the part related to players’ contribution. In other words, the team Defensive Rating helps to find all the contributions not available from the box score which are a fundamental element of the defense. For example, a good defense that doesn’t allow the opponent to make a good cut.

Furthermore, a player’s defensive contribution can be assumed positive when he is tangible (steal a ball, force a turnover), but still remains positive even when it does not allow the opponent to receive the ball. In the end, the first part of the formula tries to consider all the intangibles efforts.

The tangible (Stop1) and partially tangible (Stop2) contributions are counted with the second part of the formula.

### How to read and analyze

Let’s take as an example the best team in Defensive Rating of the 2016/2017 championship, that is Brescia. At the time, he allowed just 94 points per 100 possessions. We choose to analyze the Def Rtg of Moore, Hunt, and Landry, three players in three different roles. Here are the players’, team’s, and opponent’s averages in the first eight rounds.

The first step is to calculate the two coefficients FMwt and OppOR% and then Stop1.

Hunt is the best of the three in terms of Stop1: defensive rebounds and blocks affect his Stop1. Immediately behind we find Moore who, thanks to almost two steals per game, manages to keep up with Hunt. We have noticed that steal is added up without applying any coefficient to it, so among the three (steals, rebounds, and blocks), it is the one that has the greatest impact on the result.

Stop2, on the other hand, is very similar for the three players: this is because the minutes played by the three players are quite similar; minutes played are the only player’s data that influence Stop2. Moore obtains the lowest value among the three because he plays less: even if, by hypothesis, the guard forced twice the turnovers compared to Landry, both would have similar data. Hunt will likely contest more shots than his teammates, but this can never be seen from the available game stats.

Let’s add the values to get the Stop and then calculate the Stop%:

Thanks to his high Stop1, Moore manages to be a better defender than Hunt and Landry. Landry, on the other hand, suffers from the low Stop1 value and obtains the lowest Stop% among the three.

We are ready to calculate the individual Defensive Rating:

The higher the Stop%, the lower the Defensive Rating. Moore (the player with the highest Stop%) is so the most effective defender.

As can be seen in the next table, the individual values oscillate around the team value.

The various individual Defensive Ratings are always around the team value, 94. If we take the current worst defense (Sassari with about 111 points allowed per 100 possessions) we note that the closeness is also visible here:

This is because the main factor in the final formula is the team Defensive Rating. Stop% of players correct the value according to tangible contributions, but you can never deviate much from the team value.

This Learn a Stat ends here. See you soon, your friendly neighborhood Cappe!