Descriptive statistics problem, how do I describe this system?

I am trying to find a good descriptive statistic for a game I'm playing. I would like to compare my kills in a game to a person on the enemy team, and have some sort of normalization for each game. So I'm looking for some f(x,y) where x is my kills and the y is the enemy's kills (I might do the same thing with other stats such as KDA, gold earned, etc.) with the following properties:

1. if x=y, then f(x,y)=0 positive if I do better, negative if they do better

2. if x>y, then f(x,y)>0

3. if x<y, then f(x,y)<0

4. f(x,y)= -f(y,x) statistic is the same size but opposite sign if our kills are reversed

5. some sort of normalization, so if I have a match where I go 2-1, and then a match where I go 20-10, my outputted stat for each game is comparable, something like f(cx,xy)=f(x,y)

6. Is defined for all x>=0, y>=0. Exception: defining f(0,0)=0 is okay

7. The hardest one, an intuitive sense of scale, like, if I do twice as well as my opponent, then the number I get should give that impression somehow.

I might also include other variables in the function if they made sense like the total kills on each team, but I think it would skew how I did versus that one other player.

(Rejected) Candidates:

f(x,y)=x-y gives the first 4 properties, but doesn't normalize.

f(x,y)=(x-y)/(x+y) gives 1-6, but not 7, say that x=2y, then f(x,y)=1/3=33%, which I feel doesn't represent the performance.

f(x,y)=(x-y)/x doesn't give property 4

Everything else that I've thought of has either the same problems as these or worse. Thanks for the help.

Re: Descriptive statistics problem, how do I describe this system?

Hey beebe.

For 5. is it meant to be f(cx,cy) = f(x,y)?

You might want to try f(x,y) = (x-y)/x * H(x-y) where H(u) = 1 if u > 0, 0 if u = 0 and -1 if u < 0.

I would recommend playing around with this H(u) function (also known as the Heaviside function) to get the behaviour you are looking for.

Re: Descriptive statistics problem, how do I describe this system?

Quote:

Originally Posted by

**chiro** Hey beebe.

For 5. is it meant to be f(cx,cy) = f(x,y)?

Yeah, I mistyped that.

Quote:

You might want to try f(x,y) = (x-y)/x * H(x-y) where H(u) = 1 if u > 0, 0 if u = 0 and -1 if u < 0.

I would recommend playing around with this H(u) function (also known as the Heaviside function) to get the behaviour you are looking for.

Good idea, though isn't the function you described the signum function rather than the Heaviside function? I'll play with that. The problem I get with f(x,y)=(x-y)/x is that it doesn't give me any symmetry. For example, f(3,2)=1/3 and f(2,3)= -1/2. It seems like if the player's scores are swapped then the resulting statistic should reflect that by being the additive or multiplicative inverse or something similar.

Thanks for the help.