# A Friendly Wager on the Margin

/You're watching the footy with a mate who leans over and says he reckons the Cats will win by 15 points. How much leeway should you give him to make it a fair even money bet? Surprisingly - to me anyway - the answer is 24 points either way. So, if the Cats were to record any result between a loss by 9 points and a win by 39 points you should pay out. (On the strength of this knowledge though, you can confidently negotiate yourself a far better deal.)

I came up with this result by analysing the line betting results of all games across seasons 2006 to 2010. The statistic I looked at was the absolute value of the handicap-adjusted margin for the favourite for each game. The following rules were used to come up with the handicap for each game:

- If the start given by the favourite was 7.5 points or more, the handicap I used was the start minus half a point. So, a start of +8.5 became a start of 8.
- If the start given by the favourite was 6.5 points or fewer then a further adjustment was necessary because the TAB Sportsbet bookmaker assigns this start for games with a wide range of head-to-head prices. For these games I used the Bookie_3 Margin Predictor formula which converts a head-to-head price to a handicap using -25.7*(ln(Price/(1-Price)))

If, for example, Richmond was giving Melbourne 9.5 start on the line market and they won by 15 points then the absolute handicap-adjusted margin would be the absolute value of 9-15, which is 6 points. In essence, the absolute handicap-adjusted margin is measuring the error in the TAB Sportsbet bookmaker's handicap.

Applying this approach to the 926 games in question yields the following empirical cumulative probability distribution (click on the image for a larger version).

The 50% cumulative probability point is reached at about 24 points - hence my suggestion in the opening paragraph about this being a reasonable range to offer for an even money wager.

Armed with this chart we can also frame markets for other wagers, for example:

- We should offer a price of about $66 for guessing the margin exactly (typically, this pays about $51 on TAB Sportsbet)
- We should offer just over $7 for guessing the margin within one straight goal
- We should offer a bit over $3.80 for guessing the margin within two straight goals

The empirical data presented above is based on all games, regardless of the expected closeness or otherwise of the contest. Surely 24 points either way is far too generous an offer for a game between equal or near-equal favourites; surely these games would be much more likely to produce smaller victory margins.

Actually, that's not the case, at least not to a statistically significant extent. For example, across the 151 games in which the offered start was between 0 and 5 points, 44% of these wound up with an absolute handicap-adjusted margin of 24 points or less.

Similarly, at the other end of the scale, in games where the start on offer is 36 points or more, 50% of these games finish with an absolute handicap-adjusted margin of 24 points or less. The table below gives you all the relevant data:

Regardless of the teams involved then, it's about an even money bet that you can predict the final margin to within 4 straight goals.

(For the especially statistically-minded reader, it's interesting to note that if we assume the handicap-adjusted margin for the favourite is Normally distributed and we use the empirical standard deviation of that margin of about 36.7 points per game, then 50% of the distribution lies between about +/- 24.8 points, which is perilously close to the +/- 24 points we obtained from the actual data.)