How do I equate stats when negatives are involved?

I will use hockey stats as an example for what I want to do. I basically want to combine various statistics of a given "hockey player" to get an overall point total that represents him.

For example, a player who has 20 goals, and 30 assists. What I already know how to do is, I would add up all the goals scored in the entire league, and all the assists, and see how many assists there are for every goal, find the ratio between the two stats. Let's say I found out there were two assists for every one goal.

At this point it's easy. I choose a value to give to an assist, say 5 "points," and then because goals are twice as rare as assists, a goal would be 5 points times 2 equals 10 "points." So if assists are worth 5 points, and goals 10, and he has 20 goals and 30 assists, then it's just 20 goals multiplied by 10 points = 200, plus 30 assists multiplied by 5 points equals 150, 200 + 150 = 350. So this example hockey player, between those statistics, has the value of 350 points.

That part is easy. Here is what I don't know how to do. In hockey, and also with the statistics Im dealing with for my project, there are stats that range into the negatives. In hockey, there is a stat called plus-minus. For every goal your team scores while you are on the ice, you get a plus one. For every goal that is scored against you while you are on the ice, you get a negative one, and these compile over the season. Some players get as high as plus 40, some as low as minus 40, but every year it is different. And because approximately half the players end the season with negative plus-minuses, and half end with positives, you can't just add them up like you do with assists. Adding all the negative plus-minuses with the positive plus minuses is just going to land you smack in the middle, somewhere around zero. Say you even ended with 10. If I add up all the goals scored, and it's 3000, for example, and I end up with 10 plus minus... 1 plus-minus point is worth 300 goals? No. Goals are more rare than a plus-minus point, not less rare.

So how do I equate for this? For assists and goals, both strictly positive statistics, I could just add them up to find the ratio. 3000 goals scored this year, 6000 assists? That means 2 assists for every goal, 2:1 ratio. But because half the plus-minuses approximately are negative, I cant add them. So how do I find the ratio? How do I figure out what value to give a plus-minus point in relation to goals?

Thank you for your help!

Do the people trying to help understand what Im describing? Is there anything I should clarify to help get help with this?

Is it completely necessary to group pluses and minuses together? Why cant you say 350 pluses for a team that scored 350 goals, and 200 minuses for all the goals scored against them, for example?

Correct me if I'm wrong, but if a team scores a total of 350 goals in a season, and ends up with plus-minus score of minus 40, then doesn't this mean that there were 390 goals scored against it?

If this is the case, then a plu-minus score is simply an indicator of how successful a particular team has been against it's opponents, and if you want to apply this to several teams, then perhaps you should be looking at a percentage score of plus-minus over total goals scored?

Team plus minus is different than player plus-minus. Team plus minus in hockey would be goal differential, I believe. Player plus minus is this:

Goals scored (for team) while player is on the ice minus goals scored against (team) while player is on the ice.

This means that even on one team, every player can have different plus-minuses, because the same players dont always play together. Players that do play together most of the team tend to have very similar plus-minuses, but rarely are they the same.

So if Im on the ice for 50 goals for in the season, and 70 goals against, I would be a negative 20 for that season in plus-minus.

So no, plus-minus is not an indicator of how successful a particular team has been against its opponents.

What I need ideas for is how to combine the plus-minus stat, a stat with negatives, into one stat or "score" that involves other stats that don't involve negatives.

Take this for instance. A player has 10 goals, 20 assists, and 40 shots on goal in a season. These are all basic hockey stats for this example that have no negatives. If I want to combine those stats into a "player score" for my purposes, I need to figure out how much value to place on each of these categories. Goals are obviously much more important than anything else, and more rare, so I cant just combine goals and shots on goal at a one-to-one ratio.

But, by checking how many goals there are, total, in a season, and how many shots, total, there are, and assists, I can figure out approximately how they equate to each other.

Say there are 4000 shots every season, 2000 assists, and 1000 goals. That would be 4:2:1:, shots, assists, to goals. 4 shots for every 1 goal, 2 assists for every one goal.

So if I assign 10 points to "goals", if assists are twice as common, they get only 5 points. Since shots are four times as common as goals, they only get 1/4 as many points as goals, or 2.5.

Then you look at the player. He has 10 goals, 20 assists, and 40 shots.

10 goals, 10 points each, = 100.
20 assists, 5 points each, = 100.
40 shots, 2.5 points each, = 100.

the player score for this player is 300. I've now combined all three separate stats into one stat to represent the player, the way I wanted to. It doesnt mean this is a fair representation of how good a player he is on the ice, I mean who says shots on goal that dont go in are worth that much at all, but for my purposes I've been able to combine the stats the way I wanted to.

So my question to you is, how do I do the exact same thing I just did there except with plus-minus added in as the fourth stat. Say this same player has a plus-minus of -10, and when I added up the entire league's plus-minuses to try and get the total number so I could equate it to goals like I did with assists, the entire league's plus-minus added up came to 0.

How do I deal with this? -10 plus-minus, 0 total plus-minus for the league. If 1000 goals were scored in the league, I cant make a ratio with zero. Even if I find out the entire league's plus-minus comes out to 10, because of the rare goal that comes shorthanded which is the only instance where pluses can be achieved without the other team getting a minus, so even if it's 10, then what? 1000 goals scored in the entire league, 10 plus-minus in the entire league? 1000 goals equates to 10 plus-minus? 1 plus-minus point is comes around only once every 100 goals? Everyone knows plus-minus points are more common than goals. Generally there are 10 plus-minus points awarded for every goal, 5 plusses for the team that scored, five minuses for the team that got scored on.

So whether it's 0 or 10, it's off. How do I fix this?

My basic problem laid out much more simply.

To find the ratios for goals to assists, or goals to shots on goal, I added up all the goals or assists or shots for the entire league in a given season, and then that was the ratio I used. If 1000 goals were scored all year and 2000 assists, that told me the ratio I should use is 1 goal to 2 assists. The problem Im running into with plus-minus is I cant do this now. I cant add a statistic if it has negatives because the negatives and positives will mostly cancel out.

So how do I find the ratio between goals, an all positive stat, and plus-minus, a stat with positives and negatives? And once I do, how do I incorporate it into the overall score? What I mean is, with goals, I assigned 10 points for goals as a starting point. So if a player has 0 goals, he gets zero points. 1 goal, 10 points. 50 goals, 500 points. But notice even the lowest, 0 goals, just gets 0 points. And the same with assists and shots. So the same needs to be true with plus-minus. What happens when a player is -30? -50? His score should never be lower than zero.

On a slight tangent, how can there be twice as many assists as goals? Surely an assist can't exist without a goal?? I'm pretty clueless about ice hockey, but my understanding of an assist would be that, if anything, they would be MORE rare than goals?

To the question at hand, I'm not sure why you would need to find the ratio between goals and plus/minus? But to my mind, every plus point is equal to a goal. However, that point would be shared between seven(?) players on the ice. So based on 10 points for a goal, you're looking at 1.43 points per plus.

brokenbiscuits,

You're right, it is a tangent, and not relevant. In hockey, two assists are awarded for most goals. The last two people to touch the puck before the goalscorer get assists. The only times there are not two assists given is when the puck is turned over to the other team and the player scores quickly, without any of his teammates having touched the puck before him.

And there are 5 players on the ice for each team so by your logic that would be two points for each plus-minus point if a goal is 10. I see what you're saying, but five minus points are also given for each goal.

So really, each goal equals five plus points, and five minus points. 5 - 5 = 0. So each goal equals zero plus-minus points. Plus-minus equals zero. +40 equals zero. -40 equals zero. Everything equals zero.

Except it doesn't....

That's the problem Im running into.

Why plus-minus? Look at it this way. There are people who are really good at plus-minus just like there are people who are really good at goals. If I can determine 50 goals is 500 points, I would like to also be able to determine what +40 is in points. This would be easy if the range was the same as goals, 0 up. If there were only plusses in this static, I could just do 50 vs 40 and find the ratio (or whatever the total for the league is). But because there's negative, it's a problem.

Anyone have a solution?

I was thinking about, if the worst plus-minus in the league is -50, I was thinking about adding 50 to every player's plus-minus, to basically make them all positive, and then go from there to find the points. Probably isnt the best way or 100% accurate but its the best idea I have.