Percentage total involving negative numbers

R0UF

Right, seems an easy one but can't figure it out.

I want to allocate a percentage chance to 6 football teams based on ratings:
If the ratings are:
Team A 70
B 50
C 30
D 25
E 10
F 1

Then it's very straightforward (express each teams points as a percentage of the total points).

But if it's
Team A 50
B 30
C 20
D 10
E -10
F -20

Then I'm hitting a brick wall with the negative numbers.

Anyone any ideas?

click_here!!!

Well, I don't think that calculating a percentage of total points won by a team is a good idea. If a certain team won 50% of the points, what does that actually mean?

A better way might be to give them a rank (first, second, third), or to do a bar chart, which can compare the teams' performances meaningfully and also handle negative numbers.

seamus

R0UF wrote: »

I want to allocate a percentage chance

Of what?

"Chances" can't be negative. Where something has no chance, it is zero. Where something is guaranteed, it's 100%. Everything else lies in between.

The key is to determine a scale. What does -20 mean? -20 of what?

Camarillo Brillo

Just an idea.


Seeing as its only in relation to each other divide all by 10
5
3
2
1
-1
-2
As you can't have a minus probability of something happening
change -2 to 1 as a starting point and work each from there

1 is three times as likely -2
-1 is twice as likely as -2 it becomes 2
1 is three times as likely as -1 it becomes 6
2 is twice as likely as 1 it becomes 12
3 is 1.5 times as likely as 2 it becomes 18
5 is 1.66 times as likely as 3 it becomes 29.9999 or for simplicity 30

so we have
1/69
2/69
6/69
12/69
18/69
30/69


As I said, just an idea.

hivizman

Converting from ratings to probabilities/odds is something that is a common technique for those who bet on horses. Here, handicap ratings as well as commercial rating services provide up to date ratings for racehorses, and these ratings have a straightforward interpretation. The key point is that the ratings are essentially relative - if one horse is rated at 120 and another horse at 115, the interpretation is that the higher-rated horse should carry 5lb more weight than the lower-rated horse to equalise their chances of winning (or, another way of looking at it, if the higher-rated horse carries 5lb more weight than the lower-rated horse, the two horses should, other things being equal, finish in a dead-heat.

Are the football ratings relative? If so, then the first step is to transform the ratings by setting the highest rated team as 0 and all the other teams are then rated at the difference between the highest rating and their original weighting - we simply deduct the rating of each team from the highest-rated team. This would give us:

A 0
B 20
C 30
D 40
E 60
F 70

We then need to transform these adjusted ratings to determine the relative likelihoods that each team will be the winner. This can in practice only be done in a meaningful way if we already have some knowledge of how reliable the ratings are as a prediction of actual outcomes. In horse racing, there is a mass of data that can be used to generate quite reliable parameters, although even here it is necessary to allow for a margin of error in the predicted probabilities.

One technique is to determine relative likelihoods by choosing a parameter X between zero and one, where one means that the ratings have absolutely no predictive value. As the parameter falls, the predicted probabilities show a greater range of values. The relative likelihoods for each team are calculated by raising X to the power represented by the rating. For example, if the parameter X is 0.95, then we get the following relative likelihoods:

A 0.95^0 = 1.0000
B 0.95^20 = 0.3585
C 0.95^30 = 0.2146
D 0.95^40 = 0.1285
E 0.95^60 = 0.0461
F 0.95^70 = 0.0276

The six relative likelihood figures add up to 1.7753, so the relative likelihoods can be converted into percentages by dividing each figure by 1.7753 (numbers below are rounded):

A 56% (about 5 to 4 on)
B 20% (4 to 1 against)
C 12% (about 15 to 2 against)
D 7% (about 13 to 1 against)
E 3% (about 33 to 1 against)
F 2% (about 50 to 1 against)

This can be set out easily in a spreadsheet, but it is important to remember that the odds calculated are sensitive to the choice of parameter X. For example, with a value of 0.98, the odds of team A winning come out at about 9 to 4 against, while with a value of 0.90, the odds of team A winning come out at about 5 to 1 on. If you have an independent prior belief about the probability of team A winning, you can select X to give this probability, and then the probabilities of the other teams winning will take into account the relative ratings in a systematic way.

For example, if you think that team A has an evens chance (50%) of winning, set X to 0.958 and this will give you the following odds:

A Evens
B 15 to 4
C 13 to 2
D 10 to 1
E 25 to 1
F 40 to 1

All these calculations are, of course, only as good as the underlying ratings!