**What are the stats Goals+, Assists+, Shots+, and Shooting %+?**

All four of the “plus” categories are what statisticians would classify as “z-scores,” which intend to compare a given value to the average of a larger sample size. These statistics use all available player data from 2015-20 to create a league average number to compare against.

The “plus” statistics are the quotient found from dividing the difference between a player’s per game average and the five-year league average by the standard deviation of the five-year league average. The number that this formula spits out tells us how many standard deviations above or below the league average a given player’s performance is in a specific category.

For example, a Goals+ value of 2.16 tells us that this player is scoring at a rate of 2.16 standard deviations above average (which is really good). Remember, the standard deviation is a measure of dispersion from the mean. When we use a standard deviation to find a z-score, we are trying to quantify how close to the average a value is. The league average value for each statistic is 0, and any value over 1.00 is generally considered to be quite good.

**What are the invented statistics that have been tracked independently for this project?**

We use five invented statistics that we have created and tracked independently for this project. Those categories are as follows: Unrealized Assists (UA), Penalties Drawn (PD), Pick Assists (PA), First Order Assists (FoA), and Second Order Assists (SoA). Below are the definitions of each:

* Penalties Drawn:* This is fairly self-explanatory, but still important. Some players have a tendency for drawing penalties thanks to aggressively fighting for position, dodging hard at the goal, or some combination of other qualities. We consider the ability to draw penalties a particularly interesting and under-utilized statistic. For penalties drawn, we include all penalties committed out of desperation and those committed after deliberate provocation. Penalties like illegal substitutions and goalie interference are not considered as “drawable” and are therefore excluded.

* Unrealized Assists:* Deliberately facilitative passes that result in quality goal-scoring opportunities that a teammate does not complete. These are passes made with the intention of setting up a teammate. The idea is similar to the concept of batting average on balls in play (BABIP) in baseball: once the pass leaves a facilitating player’s stick, there is nothing he can do to control the result. A quality setup can still go wasted thanks to poor shooting, great goaltending, or some combination of the two. The Unrealized Assist is our effort at quantifying the chances a player creates for his teammates that don’t show up in the box score.

** Pick Assists: **Pick Assists are not a concept unique to this project, but are a statistical category we felt compelled to resurrect. The idea behind tracking pick assists is to try to put a numerical value to a player’s work away from the ball. This can be particularly useful in grading players who do not carry the ball often or are not the primary facilitators in an offensive system. For the purposes of this project, a Pick Assist is considered to be any effort at setting a screen that directly contributes to creating space or a lane for either the primary passer or the scorer to accomplish their goal.

* First Order Assists:* Unlike Pick Assists and Penalties drawn, assists are tracked in the conventional box score. Furthermore, it isn’t difficult to separate out the primary and secondary assists from a box score. The point of the First Order Assist (FoA) is to remove a significant portion of the passive assists that conventional box scores can’t differentiate. First Order Assists are strictly facilitative passes that lead directly to the scoring of a goal. By “facilitative passes” we mean passes thrown to intentionally create a scoring opportunity, not merely to change possession to a different player. These are targeted passes with the expectation that the receiving teammate will likely be shooting. Some of the primary assists that are excluded in FoA are rebounds, flips when getting personnel on and off the floor, and passes that are followed by significant carrying/dodging by the inevitable goal scorer. It is our opinion that generally speaking, a tremendous individual effort by the goal scorer does not warrant the awarding of an assist to a teammate passively swinging the ball in the goal scorer’s direction.

* Second Order Assists:* Similar to First Order Assists, Second Order Assists (SoA) recognize the nuance in secondary assists. While most secondary assists don’t offer any significant value, some do. The goal of SoA is to track the valuable secondary assists that are essential to the scoring of a goal. The general criteria for SoA include cross-floor passes, skip passes, and rapid ball movement that actively moves the defense. Again like First Order Assists, there is a degree of intentionality required for a secondary assist to be considered a Second Order Assist. Second Order Assists are significantly less common than secondary assists in conventional box scores. An average game has between 1 and 5 Second Order Assists combined between two teams.

**What is a Weighted Assist (wA)?**

The weighted assist category attempts to aggregate all four kinds of assists (First Order, Unrealized, Pick Assists, and Second Order) on a weighted basis. Think of it as an adjusted assist total that values some kinds of assists more than others, contrary to the 1-to-1 nature of conventional box score assists. The formula for Weighted Assists is as follows:

**wA** = FoA + .75(UA) + .5(PA) + .5(SoA)

**What is a PTR or PTRe?**

PTR is short for “Possession Termination Ration” and is an attempt at improving upon the idea behind the assist-to-turnover ratio. The PTRe is an experimental version of the PTR that considers non-traditional statistics that you will not find in a standard box score.

In order to find a player’s PTR, we find the following quotient:

**PTR** = (Points + Loose Balls) / (Turnovers + Shots Saved)

The idea of the ratio is to include all of the various ways that possessions (generally) end. The inclusion of loose balls is to try to quantify possessions that are added by a player controlling the ball. Shots saved are counted like turnovers because more often than not a saved shot will be followed by a change in possession.

The PTRe requires a different set of inputs and offers a more interesting, although experimental, look at how a player impacts the flow of possessions. Using three invented statistics that have been defined and tracked independently for this project, we can account for more aspects of a possession than the basic PTR allows. In addition to loose balls, turnovers, and shots saved, we use the following invented statistics to create our ratio:

*Penalties Drawn*

*Weighted Assists*

In order to find a player’s PTRe, we use the following formula:

**PTRe** = (wA + PD + LB) / (TO + Shots Saved)

**What is Understated Production (uPro)?**

This is another value that relies on the inputs of numerous invented statistical categories. The idea behind uPro is to establish a number by which we can quantify things that non-primary forwards can do. When a player doesn’t carry the ball often, he is still able to draw penalties, set quality picks, collect loose balls, and make quick passes to get the ball into the sticks of a team’s primary passers. The formula for Understated Production is as follows:

**uPro** = PA + SoA + PD + 0.25(LB) – TO

**What is a Facilitator Score (fScore)?**

The Facilitator Score (fScore) is an effort at adding all of the various types of assists, subtracting turnovers, and then figuring out how regularly a player’s teammates convert on the quality looks he created. We use most of the same invented categories to find the facilitator score, but we also add a new one into the mix. The “First Order Conversion Rate” is a percentage (or decimal) that indicates how often a player’s teammates convert quality chances into goals. To find the FoCR, we divide a player’s First Order Assists (FoA) by his total First Order Chances, which is the sum of his First Order Assists (FoA) added to his Unrealized Assists (UA). The formula for the facilitator score is as follows:

**fScore** = FoCR(PA + UA + FoA + SoA – TO)

**What is a player’s Experimental Offensive Efficiency (eOE)?**

The eOE is an effort at computing how efficient a player is at all things on the floor that don’t involve goal scoring. Think of it as a blanket score for how efficient a player would be if he never succeeded in scoring a goal ever again. The reason for excluding goal scoring is to try to compare players who are not the primary options offensively against one another. As you can see, players are rewarded for having high PTRe, wA, and UA values. Players are also punished for having a low shooting % relative to the five-year league average and for taking more shots than the five-year league average. The formula for eOE is below:

**eOE** = PTRe[(wA/gm) + (UA/gm) + (SH%+) – (SH+)]

**What is a player’s Experimental Offensive Efficiency Plus (eOE+)?**

The eOE+ is the same concept as the eOE, but adding goal scoring into consideration. eOE+ intends to quantify a player’s efficiency while also rewarding goal scoring. This means that goal scoring is rewarded, as to create a metric to compare the efficiency of primary options against one another. The formula for eOE+ is below:

**eOE+** = PRTe[(Goals+) + (wA/gm) + (UA/gm) + (SH%+) – (SH+)]

#### What is a player’s “Key Sum”?

The Key Sum is a tool we use for determining a player’s eligibility for non-cumulative stats. For example, both eOE and eOE+ are vulnerable to data samples that are too small. To make sure we have a big enough sample to produce a valid, useful eOE or eOE+ score we impose a Key Sum restriction. It’s the same idea behind qualifying for the batting title in baseball. In order to lead MLB in batting average, a player has to have recorded a certain number of at-bats. The Key Sum plays the role of at-bats in our metrics. Generally, we set the minimum Key Sum to 20, but as the season progresses, that number may climb. Below is the formula for the Key Sum:

Key Sum = Shots on Goal (SOG) + Penalties Drawn (PD) + Pick Assists (PA) + Unrealized Assists (UR) + First Order Assists (FoA) + Second Order Assists (SoA)

**What is a player’s Production Score (pScore)?**

The production score of a player is intended to quantify his production without regard to efficiency. For that reason, items like the PTR, Turnovers, Shots Saved, and Shooting Percentage are not considered as inputs. Instead, the pScore is solely intended to demonstrate production. While Unrealized Assists don’t create goal production directly, we think of them as accessories to production, which is why they are included on a one-half weight. The formula for pScore is below:

**pScore** = Goals + wA + .5(UA) + uPro

**What is the difference between a Production Score and a Production Rating?**

The Production Score (pScore) is a raw number, while the Production Rating (pRating) is the z-score of a player’s pScore compared against the league average and standard deviation. The value of the pRating as opposed to the pScore is that the pRating is a contextual value like the various “plus” statistics introduced earlier, while the pScore is purely nominal. The formula for the pRating is below:

**pRating** = (pScore – League Average pScore) / (Standard Deviation of League pScore)

**What is a player’s Most Outstanding Player Score (MOPscore)?**

The Most Outstanding Player Score (MOPscore) is this project’s take on a Pro Football Focus style grade on a 1 to 100 scale. Additionally, the MOPscore virtually guarantees no two players will have the same value, meaning that a hierarchy can be easily created to rank players based on who has been the most outstanding across a number of metrics. In order to find the MOPscore, we take the average percentile ranking of a player across five different metrics: pRating, fScore, uPro, eOE, and eOE+.

**What are Goals Prevented over Average?**

Similar to the idea of Saves Over Average that Ty Merrow has made popular on Twitter, Goals Prevented over Average (GPoA) is a goalie stat that tells us how many more or fewer goals a league average goaltender would allow in the place of a given goalie. Goalie statistics like GPoA use a larger sample size than the position player league averages because there are far fewer goalies in the league each year. To create a satisfactory sample size, we are using all goalie data from 2010-2020. To find the Goals Prevented over Average, we need to use a player’s Goals Allowed and his save percentage compared to the league average save percentage from our sample source. The formula for GPoA is below:

**GPoA** = (GA x Save%) – (GA x League Average Save%)

**What is Goalie WAR (gWAR)?**

Inspired by the baseball statistic of “Wins Above Replacement” this statistic is designed to quantify how many wins a goaltender is directly responsible for creating. To find this number, we have to use our GPoA values to help adjust a team’s win total. To do so, we must compare a team’s real win total against their pythagorean win total, using an exponent of 5.65. This exponent delivers consistently with a margin for error of 1.38%. In winning percentage terms that’s a plus/minus of .0138 points. We can calculate a projected GA number (excluding empty net goals) for a team that will tell us how many goals we could expect a league average goaltender to allow in the same situation. From there we can subtract the win projection of a league average goaltender from that of the goaltender in question. Below are the formulas necessary:

**Pythagorean W%** = (Goals For)^5.65 / (Goals For^5.65 + Goals Against^5.65)

**Pythagorean Win Projection** = 18 x Pythagorean Win%

**gWAR** = [GF^5.65 / (GF^5.65 + (Goals Allowed + GPoA)^5.65)] – Pythagorean Wins