Okay, this is probably going to be a long blog so you might want to make yourself comfortable.

For some time now I've been wondering about the statistical properties of the Handicap-Adjusted Margin (HAM). Does it, for example, follow a normal distribution with zero mean?

Well firstly we need to deal with the definition of the term HAM, for which there is - at least - two logical definitions.

The first definition, which is the one I usually use, is calculated from the Home Team perspective and is Home Team Score - Away Team Score + Home Team's Handicap (where the Handicap is negative if the Home Team is giving start and positive otherwise). Let's call this Home HAM.

As an example, if the Home Team wins 112 to 80 and was giving 20.5 points start, then Home HAM is 112-80-20.5 = +11.5 points, meaning that the Home Team won by 11.5 points on handicap.

The other approach defines HAM in terms of the Favourite Team and is Favourite Team Score - Underdog Team Score + Favourite Team's Handicap (where the Handicap is always negative as, by definition the Favourite Team is giving start). Let's call this Favourite HAM.

So, if the Favourite Team wins 82 to 75 and was giving 15.5 points start, then Favourite HAM is 82-75-15.5 = -7.5 points, meaning that the Favourite Team lost by 7.5 points on handicap.

Home HAM will be the same as Favourite HAM if the Home Team is Favourite. Otherwise Home HAM and Favourite HAM will have opposite signs.

There is one other definitional detail we need to deal with and that is which handicap to use. Each week a number of betting shops publish line markets and they often differ in the starts and the prices offered for each team. For this blog I'm going to use TAB Sportsbet's handicap markets.

TAB Sportsbet Handicap markets work by offering even money odds (less the vigorish) on both teams, with one team receiving start and the other offering that same start. The only exception to this is when the teams are fairly evenly matched in which case the start is fixed at 6.5 points and the prices varied away from even money as required. So, for example, we might see Essendon +6.5 points against Carlton but priced at $1.70 reflecting the fact that 6.5 points makes Essendon in the bookie's opinion more likely to win on handicap than to lose. Games such as this are problematic for the current analysis because the 'true' handicap is not 6.5 points but is instead something less than 6.5 points. Including these games would bias the analysis - and adjusting the start is too complex - so we'll exclude them.

So, the question now becomes is HAM Home, defined as above and using the TAB Sportsbet handicap and excluding games with 6.5 points start or fewer, normally distributed with zero mean? Similarly, is HAM Favourite so distributed?

We should expect HAM Home and HAM Favourite to have zero means because, if they don't it suggests that the Sportsbet bookie has a bias towards or against Home teams of Favourites. And, as we know, in gambling, bias is often financially exploitable.

There's no particular reason to believe that HAM Home and HAM Favourite should follow a normal distribution, however, apart from the startling ubiquity of that distribution across a range of phenomena.

Consider first the issue of zero means.

The following table provides information about Home HAMs for seasons 2006 to 2008 combined, for season 2009, and for seasons 2006 to 2009. I've isolated this season because, as we'll see, it's been a slightly unusual season for handicap betting.

Each row of this table aggregates the results for different ranges of Home Team handicaps. The first row looks at those games where the Home Team was offering start of 30.5 points or more. In these games, of which there were 53 across seasons 2006 to 2008, the average Home HAM was 1.1 and the standard deviation of the Home HAMs was 39.7. In season 2009 there have been 17 such games for which the average Home HAM has been 14.7 and the standard deviation of the Home HAMs has been 29.1.

The asterisk next to the 14.7 average denotes that this average is statistically significantly different from zero at the 10% level (using a two-tailed test). Looking at other rows you'll see there are a handful more asterisks, most notably two against the 12.5 to 17.5 points row for season 2009 denoting that the average Home HAM of 32.0 is significant at the 5% level (though it is based on only 8 games).

At the foot of the table you can see that the overall average Home HAM across seasons 2006 to 2008 was, as we expected approximately zero. Casting an eye down the column of standard deviations for these same seasons suggests that these are broadly independent of the Home Team handicap, though there is some weak evidence that larger absolute starts are associated with slightly larger standard deviations.

For season 2009, the story's a little different. The overall average is +8.4 points which, the asterisks tell us, is statistically significantly different from zero at the 5% level. The standard deviations are much smaller and, if anything, larger absolute margins seem to be associated with smaller standard deviations.

Combining all the seasons, the aberrations of 2009 are mostly washed out and we find an average Home HAM of just +1.6 points.

Next, consider Favourite HAMs, the data for which appears below:

The first thing to note about this table is the fact that none of the Favourite HAMs are significantly different from zero.

Overall, across seasons 2006 to 2008 the average Favourite HAM is just 0.1 point; in 2009 it's just -3.7 points.

In general there appears to be no systematic relationship between the start given by favourites and the standard deviation of the resulting Favourite HAMs.

Summarising:

• Across seasons 2006 to 2009, Home HAMs and Favourite HAMs average around zero, as we hoped
• With a few notable exceptions, mainly for Home HAMs in 2009, the average is also around zero if we condition on either the handicap given by the Home Team (looking at Home HAMs) or that given by the Favourite Team (looking at Favourite HAMs).

Okay then, are Home HAMs and Favourite HAMs normally distributed?

Here's a histogram of Home HAMs:

And here's a histogram of Favourite HAMs:

There's nothing in either of those that argues strongly for the negative.

More formally, Shapiro-Wilks tests fail to reject the null hypothesis that both distributions are Normal.

Using this fact, I've drawn up a couple of tables that compare the observed frequency of various results with what we'd expect if the generating distributions were Normal.

Here's the one for Home HAMs:

There is a slight over-prediction of negative Home HAMs and a corresponding under-prediction of positive Home HAMs but, overall, the fit is good and the appropriate Chi-Squared test of Goodness of Fit is passed.

And, lastly, here's the one for Home Favourites:

In this case the fit is even better.

We conclude then that it seems reasonable to treat Home HAMs as being normally distributed with zero mean and a standard deviation of 37.7 points and to treat Favourite HAMs as being normally distributed with zero mean and, curiously, the same standard deviation. I should point out for any lurking pedant that I realise neither Home HAMs nor Favourite HAMs can strictly follow a normal distribution since Home HAMs and Favourite HAMs take on only discrete values. The issue really is: practically, how good is the approximation?

This conclusion of normality has important implications for detecting possible imbalances between the line and head-to-head markets for the same game. But, for now, enough.