Monday, 31 December 2012

Statistical inference and mathematical modelling (part 7)


7.0 Analysis of results

In the final part of the series, it is time to analyse the results of the NFL math model. Here are each's team predictions for season interceptions, together with the outcomes of the 26 individual bets. 
The confidence level of each pick is in the third column. The teams are ranked in order of their unlisted Kelly % - a measure of hypothetical profitability - rather than their win %. This is because the best odds on offer about each proposition varied.

TeamSelectionWin%Result
New Orleans SaintsOVER 12.569.8%WON
New England PatriotsUNDER 22.069.5%WON
Chicago BearsUNDER 20.066.8%LOST
Baltimore RavensOVER 16.567.4%LOST
Philadelphia EaglesUNDER 18.564.7%WON
Dallas CowboysUNDER 18.064.1%WON
Arizona CardinalsOVER 14.063.4%WON
Tennessee TitansOVER 14.062.8%WON
Denver BroncosOVER 13.561.4%WON
Buffalo BillsUNDER 18.562.0%WON
Oakland RaidersOVER 15.560.5%LOST
New York JetsUNDER 17.560.9%WON
Kansas City ChiefsUNDER 18.559.9%WON
Indianapolis ColtsOVER 11.559.2%WON
Jacksonville JaguarsOVER 15.060.1%LOST
New York GiantsUNDER 18.558.6%LOST
Carolina PanthersUNDER 17.559.0%WON
Detroit LionsUNDER 18.058.1%WON
Houston TexansOVER 16.557.0%LOST
Cleveland BrownsOVER 13.558.1%WON
Green Bay PackersUNDER 24.558.9%WON
Miami DolphinsUNDER 16.056.7%WON
Minnesota VikingsOVER 11.557.3%LOST
San Francisco '49ersUNDER 20.555.8%WON
Atlanta FalconsUNDER 19.055.0%LOST
Washington RedskinsOVER 14.554.2%WON


7.1 Summary
  • The 26 Selections went 18-8 overall, a win percentage of 69.2%.
  • The nine best bets went 7-2 (77.8% strike-rate); the next best nine were 6-3 (66.7%) and the remainder 5-3 (62.5%). This is a particularly satisfactory result as far as staking is concerned.
  • The entire portfolio showed a 133.2% return on investment at the listed stakes and odds. Had level staking been adopted (the initial £1000 divided into 26 bundles of £38.46) the ROI would have been 128.3%.
  • There was a total excess of 87.5 interceptions compared with the market's projections, or 3.4 interceptions per team. There is a very large margin and strongly indicates the portfolio was extremely robust. In other words, this is an exploitable market and will remain so - and there are plenty of others like it in sport.

December 31/2012


Thursday, 6 September 2012

Statistical inference and mathematical modelling (Part 6)

6.1 Staking and confidence

In this final part of the series - I can hear distant cheering! - I will be answering the question of how bets should be optimally staked.

In the last part, I produced the results of a mathematical model of an NFL team's expected total interceptions for the 2012 season (xINT). I applied these to pricing from existing betting markets to derive the expected value (EV) I might notionally achieve - if my projections were accurate on average - on each proposition.

The values ranged from a healthy 30.5% profit on betting New Orleans OVER 12.5 to an expected loss of 6.5% on either San Diego UNDER 12.5 or San Diego OVER 12.5. (On the last point, it doesn't matter which bet is struck because my projection coincided with that of the bookmakers.)

It might help if you checked back on the table to get a feel for the range of profit margins I am expecting. If you are a sports bettor with any concept of what to expect from your investment - ie all of you who are taking it seriously - then you might be shocked by how large some of these profits are projected to be. But this is actually just routine.

If you are a punter who makes, say, a 10% profit on investment, the actual profit margin on individual bets (should you be able to know them) would be distributed either side of this 10%. Some of your wagers would have been sucker-bets that would have seen you make a crushing loss over time, while some of your other pokes would have been acts of genius. It is very difficult to know the profit margin on each bet, but much easier to discern the average from your bottom line.

It should be the same when you make projections mathematically or intuitively. Though I have a very strong conviction indeed that the total interceptions market is highly influenced by irrationality - in this case, the degree to which the important variable is a function of randomness or, at least, factors which a team finds extremely difficult to sustain - I don't know for sure that the New Orleans Saints are a lock to intercept 13 or more passes. But the point is that my model has the same degree of confidence that the total will go under as it does about all the other 31 total interception scores of NFL teams.

"Wait a minute", I hope you are thinking, "what about the fact that your projection about the Saints - and corresponding EV on betting the over - is an outlier to the market. Surely that should make you think twice about having the same confidence in the selection as you do with your San Diego projection, which agrees with Vegas?"

6.2 Efficient markets

The answer involves understanding the type of market that you are investing in. In examining the past results of the total interceptions market, I see no evidence that it qualifies as a semi-strong (or, of course strong) market. It is not an exchange market with high volatility, distilling the wisdom of the crowd and expressing the value of all current information known about the teams' capacity to make interceptions. No, it is a novelty market with mispriced commodities; that's why I picked it.

It should be a lot easier to beat (that is different to saying it is easy to beat) than, say, a Betfair market on a Premiership football game at 2.59pm on a Saturday afternoon. (Although, I am given to understand that plenty of you actually beat those well enough, clever chaps...)

The Efficient Market Hypothesis - and the many subsequent academic reactions and contradictions to it - can wait for another blog. But at least you should have a qualitative understanding as a bettor that you have absolutely, literally zero chance of making a 30.5% profit in the long run (aha! this nebulous creature is the subject of another future blog in itself) on all your investments in sports betting. And if many of the projections your math models produce suggest as much, you need to think again. (Or, tell me on the quiet...)

6.3 The Kelly criterion

Underneath is a table of all propositions in the total interceptions market which have an Expected Value (EV) > 0. Also given is the percentage chance of success given by my model. These were used in Part Five to produce the EV calculations, but I did not have room for another column in my table. Also included is the all-important market price, expressed as odds to one.

Team Selection Win % Odds-1
New Orleans Saints OVER 12.5 69.8 % 0.87
New England Patriots UNDER 22.0 69.5 % 0.85
Chicago Bears UNDER 20.0 66.8 % 0.85
Baltimore Ravens OVER 16.5 67.4 % 0.83
Philadelphia Eagles UNDER 18.5 64.7 % 0.87
Dallas Cowboys UNDER 18.0 64.1 % 0.85
Arizona Cardinals OVER 14.0 63.4 % 0.87
Tennessee Titans OVER 14.0 62.8 % 0.87
Denver Broncos OVER 13.5 61.4 % 0.87
Buffalo Bills UNDER 18.5 62.0 % 0.83
Oakland Raiders OVER 15.5 60.5 % 0.87
New York Jets UNDER 17.5 60.9 % 0.85
Kansas City Chiefs UNDER 18.5 59.9 % 0.87
Indianapolis Colts OVER 11.5 59.2 % 0.87
Jacksonville Jaguars OVER 15.0 60.1 % 0.83
New York Giants UNDER 18.5 58.6 % 0.85
Carolina Panthers UNDER 17.5 59.0 % 0.83
Detroit Lions UNDER 18.0 58.1 % 0.85
Houston Texans OVER 16.5 57.0 % 0.87
Cleveland Browns OVER 13.5 58.1 % 0.83
Green Bay Packers UNDER 24.5 58.9 % 0.80
Miami Dolphins UNDER 16.0 56.7 % 0.85
Minnesota Vikings OVER 11.5 57.3 % 0.83
San Francisco '49ers UNDER 20.5 55.8 % 0.85
Atlanta Falcons UNDER 19.0 55.0 % 0.85
Washington Redskins OVER 14.5 54.2 % 0.87

So, those are my selections but, assuming I have £1000 to speculate in the market, the question is how much is optimal to place on each proposition?

In the course of our betting career - which it is easiest to assume for mathematical purposes is infinitely long - we have conflicting ambitions when making a wager. (Actually, in the grand scheme of things we may have many ambitions, but I mean from the standpoint of growing our betting bank.)

On the one hand, we want to maximise our profit. On the other, we want to protect against the ghastly prospect of going busto, referred to technically by one meaning of the term Gambler's Ruin.

Incidentally, as economists among you will point out, we should not think of our money in terms of pounds and pence but, better, in terms of utility - its importance to us or what it can buy. A coin-flip for £25,314 between me and the excellent and very rich Noel Edmonds have the same EV for both of us in terms of cash (Expected Value = zero) but very different prospective outcomes when measured by utility (Expected utility = massively negative for me, zero for him, because of the change in utility of losing for me is now huge as I do not receive an overblown retainer to do Racing UK.)

                                  Edmonds: TFNL hero owns enormous utility buffer

But let's make the assumption - extremely rash, I know - that we all have £1000 on the sideboard to speculate on my total interceptions market with a sustainable impact on utility considerations. How do we do it? Can I ever get to the point?

This is where the Kelly criterion comes in. I expect you might have heard it bandied around, especially if you are in the poker community, given the ubiquitous love among card players of tossing around concepts which they probably then fail to apply to their game. (I'm sure we have all found it a massive relief to rid yourself of the dreaded possibility of polarising your range, no?)

Take a deep breath. If p=the odds-to-one of success at which you are betting (listed in the right-hand column of the table) and x=the calculated expected chance of success, then the optimal fraction of your betting bank, K for Kelly percent, which to stake in order to marry profitability with defence against ruin is given by the formula:

K = [ x (p + 1) - 1 ] / p

So, for instance, if you have a 50% chance of winning, as in a fair coin toss with Noel Edmonds, then x =0.5. If he generously gives you 2-1 about winning - for the sake of one of his tremendous television shows - then p = 2.0, so the Kelly criterion advocates:

K = [ 0.5 * (2 +1) - 1 ] / 2

    = 0.25 or 25% of your bankroll.

Not 67 times your bankroll, as contestants on Deal Or No Deal do regularly when confronted with a parallel situation, often stating the justification "you are only here once" as if there is no other form of legalised gambling involving pure chance open to them outside the studio. (Edmonds is an evil genius; he has to be. Surely the banker has pointed out the inconvenient truth in a production meeting. Indeed, the only here once caveat is a tremendously useful, yet illogical, device for all of mankind, as in: "How can you do this to me? And with the milkman as well, surely you must have realised it was extremely likely you would destroy our marriage?"...."Well, that's true love, but you are only here once...")

Back on point. Here are the Kelly percentages K% for all 26 total interception propositions which are +EV. (If you learn nothing else from this blog, just try saying: "I thought it was super +EV, I mean 'super'," every time another gambler asks you about a losing bet. You'll end up as a paid expert at best or, at the very least, revered by other pseuds. Super is a key word nowadays):

Team Selection Win % Odds-1 K%
New Orleans Saints OVER 12.5 69.8 % 0.87 35.1
New England Patriots UNDER 22.0 69.5 % 0.85 33.6
Chicago Bears UNDER 20.0 66.8 % 0.85 27.7
Baltimore Ravens OVER 16.5 67.4 % 0.83 28.1
Philadelphia Eagles UNDER 18.5 64.7 % 0.87 24.1
Dallas Cowboys UNDER 18.0 64.1 % 0.85 21.9
Arizona Cardinals OVER 14.0 63.4 % 0.87 21.3
Tennessee Titans OVER 14.0 62.8 % 0.87 20.0
Denver Broncos OVER 13.5 61.4 % 0.87 17.0
Buffalo Bills UNDER 18.5 62.0 % 0.83 16.2
Oakland Raiders OVER 15.5 60.5 % 0.87 15.1
New York Jets UNDER 17.5 60.9 % 0.85 14.9
Kansas City Chiefs UNDER 18.5 59.9 % 0.87 13.8
Indianapolis Colts OVER 11.5 59.2 % 0.87 12.3
Jacksonville Jaguars OVER 15.0 60.1 % 0.83 12.0
New York Giants UNDER 18.5 58.6 % 0.85 9.9
Carolina Panthers UNDER 17.5 59.0 % 0.83 9.6
Detroit Lions UNDER 18.0 58.1 % 0.85 8.8
Houston Texans OVER 16.5 57.0 % 0.87 7.6
Cleveland Browns OVER 13.5 58.1 % 0.83 7.6
Green Bay Packers UNDER 24.5 58.9 % 0.80 7.5
Miami Dolphins UNDER 16.0 56.7 % 0.85 5.8
Minnesota Vikings OVER 11.5 57.3 % 0.83 5.9
San Francisco '49ers UNDER 20.5 55.8 % 0.85 3.8
Atlanta Falcons UNDER 19.0 55.0 % 0.85 2.1
Washington Redskins OVER 14.5 54.2 % 0.87 1.6

So, K% gives you the optimal percentage of your money to invest in each proposition in order to grow your bankroll optimally. For the academically minded among you, here are several properties of Kelly staking worth bearing in mind:

1) Kelly staking assumes that all bets are independent and sequential. In this case, total interceptions for each team are indeed independent (or extremely close to it) but they are not sequential. We have to place all our wagers on this market before the season, before the outcome of each bet is known, so K% for all propositions may - and indeed does - add up to more than 100% of our bank. In this case, the sum of all K% is 383.4%.

2) Non-independent propositions, such as the various horses in a race, give rise to a more complicated situation, but one that can also be dealt with optimally. However, that is beyond the scope of this article and subject of another one coming up.

3) In practice, Kelly staking feels rather risk-seeking. Would you really be comfortable betting 35.1% of all your monies available for betting on a single proposition like New Orleans OVER 12.5. It was a cracking bet, in my view, but this percentage sounds - and feels - like too big a risk.

Indeed, there are more than just psychological reasons to be wary of using a full-Kelly approach. As we are talking projections and probabilities, there is the danger of a misapprehension in the calculations. There may be other variables in play which we cannot account for, or a change in preconditions. In the case of total interceptions, for instance, we can't be sure of the effect of replacement referees in the early weeks of the NFL season, or how the ever-increasing rate at which passes are completed will affect the relationships between passes defended and total interceptions in our correlations. In other words, in any dynamic system of variables there is considerable chaos which argues for a riskaverse attitude.

So, the remedy for many investors is to use a scaled percentage of Kelly, such as two-thirds Kelly or even half-Kelly. In effect, this mitigates risk, but it also militates against profitability. But, as relatively riskaverse investors, we have to accept and live with the paradox.

4) Next, though this probably does not affect our calculations here, in semi-strong form (and strong) market situations, the price of a proposition is a variable in and of itself. This is a point I used to belabour tirelessly (and tiresomely for the viewers) on Racing UK - even though one or two of my colleagues still do not accept that horserace betting markets are approaching a semi-strong state.

So, even though all the known information about a horse remains constant, the fact that it is drifting in an exchange market should affect your view of its percentage chance of winning somewhat. This is not suspicion or superstition but can be seen as a result of the obvious paradox, recently (2008) named by the American blackjack genius and mathematician Edward Thorp as Proebsting's Paradox.

Say if you backed a horse with a Kelly percent of your bankroll at 5-1 and it drifted to 10-1. Now, what do you do? Invest a higher percent again because the value has increased substantially? Do that and you will go broke very quickly.

6.31 The importance of bet-sizing

In this case, our profit margin will be the sum of the expected sum of all wagers. Having scaled our bets using Kelly, keeping the ratio between all individual wagers the same as if they were staked sequentially, so that the revised total K% = 100%, we are left with:

Team Selection Win % Odds-1 K%
New Orleans Saints OVER 12.5 69.8 % 0.87 9.2
New England Patriots UNDER 22.0 69.5 % 0.85 8.8
Chicago Bears UNDER 20.0 66.8 % 0.85 7.2
Baltimore Ravens OVER 16.5 67.4 % 0.83 7.3
Philadelphia Eagles UNDER 18.5 64.7 % 0.87 6.3
Dallas Cowboys UNDER 18.0 64.1 % 0.85 5.7
Arizona Cardinals OVER 14.0 63.4 % 0.87 5.6
Tennessee Titans OVER 14.0 62.8 % 0.87 5.2
Denver Broncos OVER 13.5 61.4 % 0.87 4.4
Buffalo Bills UNDER 18.5 62.0 % 0.83 4.2
Oakland Raiders OVER 15.5 60.5 % 0.87 3.9
New York Jets UNDER 17.5 60.9 % 0.85 3.9
Kansas City Chiefs UNDER 18.5 59.9 % 0.87 3.6
Indianapolis Colts OVER 11.5 59.2 % 0.87 3.2
Jacksonville Jaguars OVER 15.0 60.1 % 0.83 3.1
New York Giants UNDER 18.5 58.6 % 0.85 2.6
Carolina Panthers UNDER 17.5 59.0 % 0.83 2.5
Detroit Lions UNDER 18.0 58.1 % 0.85 2.3
Houston Texans OVER 16.5 57.0 % 0.87 2.0
Cleveland Browns OVER 13.5 58.1 % 0.83 2.0
Green Bay Packers UNDER 24.5 58.9 % 0.80 2.0
Miami Dolphins UNDER 16.0 56.7 % 0.85 1.5
Minnesota Vikings OVER 11.5 57.3 % 0.83 1.5
San Francisco '49ers UNDER 20.5 55.8 % 0.85 1.0
Atlanta Falcons UNDER 19.0 55.0 % 0.85 0.5
Washington Redskins OVER 14.5 54.2 % 0.87 0.4

So, if our intended total investment is £1000, our portfolio incorporating all propositions with a positive Expected Value looks like this:

Team Selection Win % Odds-1 £ bet
New Orleans Saints OVER 12.5 69.8 % 0.87 91.52
New England Patriots UNDER 22.0 69.5 % 0.85 87.69
Chicago Bears UNDER 20.0 66.8 % 0.85 72.36
Baltimore Ravens OVER 16.5 67.4 % 0.83 73.36
Philadelphia Eagles UNDER 18.5 64.7 % 0.87 62.93
Dallas Cowboys UNDER 18.0 64.1 % 0.85 57.03
Arizona Cardinals OVER 14.0 63.4 % 0.87 55.64
Tennessee Titans OVER 14.0 62.8 % 0.87 52.28
Denver Broncos OVER 13.5 61.4 % 0.87 44.43
Buffalo Bills UNDER 18.5 62.0 % 0.83 42.30
Oakland Raiders OVER 15.5 60.5 % 0.87 39.38
New York Jets UNDER 17.5 60.9 % 0.85 38.87
Kansas City Chiefs UNDER 18.5 59.9 % 0.87 36.02
Indianapolis Colts OVER 11.5 59.2 % 0.87 32.09
Jacksonville Jaguars OVER 15.0 60.1 % 0.83 31.37
New York Giants UNDER 18.5 58.6 % 0.85 25.81
Carolina Panthers UNDER 17.5 59.0 % 0.83 25.05
Detroit Lions UNDER 18.0 58.1 % 0.85 22.97
Houston Texans OVER 16.5 57.0 % 0.87 19.76
Cleveland Browns OVER 13.5 58.1 % 0.83 19.87
Green Bay Packers UNDER 24.5 58.9 % 0.80 19.63
Miami Dolphins UNDER 16.0 56.7 % 0.85 15.02
Minnesota Vikings OVER 11.5 57.3 % 0.83 15.27
San Francisco '49ers UNDER 20.5 55.8 % 0.85 9.91
Atlanta Falcons UNDER 19.0 55.0 % 0.85 5.37
Washington Redskins OVER 14.5 54.2 % 0.87 4.06

Multiplying the Expected Value of each proposition by the stake and summing for all wagers, the total expected return for the entire portfolio is £1174.92, a profit of 17.49%. I'll report back after the season to see how close to the return is this target sum.

6.4 Introducing Modern Portfolio Theory considerations

One thing about that last sentence might strike you as disappointing. After all that, just 17.49%? But, in the table in Part 5, there were seven teams that yielded a higher Expected Value (EV) in the total interceptions market. So, why not just back them?

You probably know the answer to this intuitively. The consideration is diversifying risk. The more propositions with a positive EV we can add to the portfolio, the more likely that the result will reflect our underlying skill as an investor (which we have to assume or else why bother?) and the less likely it will reflect randomness.

So, there is a see-saw effect. Collect just the tastiest eggs from the henhouse and put them in one basket (I am thinking of my friend Graham Cunningham's superbly funny analogies at this stage) and we have a chance of a right feast but also a chance of no breakfast, but collect every old hairy hen's eggs and put them in loads of baskets and we definitely get breakfast. The problem is, it could be a Little Chef breakfast.

The consideration is more applicable in situations where portfolios in a financial market can be created with assets which are co-dependent or negatively correlated. In this case, we might want to select an equal number of OVERS and UNDERS to hedge against risk in case there is a dramatic change to the level of total interceptions, such as if all NFL teams had installed the no-huddle offense and games had 50% more plays and those plays were associated with a higher risk. It sounds unlikely, but it does happen. Again, this is a wide-ranging topic for another blog.

6.5 Conclusion

I really hope you have enjoyed the series. To be honest, I know it is challenging material but I have tried my very best to explain it in a straightforward fashion. I don't believe in dumbing things down, which is one of many reasons why I have found the media challenging. Perhaps the point is that I don't have the skills to express myself at an intermediate level.

However, I believe that people who are motivated to learn may be inspired by what I have written and start out on a path of their own. That is how it worked - and still works - for me. There is some tremendous material out there on the mathematics of sport and every day you can learn something is a source of joy, if you have a mind like mine.

Thanks again, September '12.

Wednesday, 29 August 2012

Statistical inference and mathematical modelling (Part 5)

5.1 Understanding the importance of distributions

So, I have my projections about the total interceptions market for the 32 NFL teams. How do I know the corresponding market edge?

Let's make an example of the projection I made in the last part for the Tennessee Titans in 2012:

xINT = 16.0
O/U = 14.0

The metric xINT represents the calculated total interceptions which Tennessee are most likely to have in 2012 according to my analysis of the data. But it's very important to have knowledge of the distribution of possible values of total interceptions which this estimate implies.

To establish this we need to examine and evaluate the distribution of past values of total interceptions in past NFL seasons to see how they are distributed (some smoothing of the data is necessary).

(+/-)
distribution %
1
6.8
2
13.3
3
19.2
4
24.2
5
28.5
6
32.1

(NB: It is easy to see that the distribution is much more fat-tailed (wider) than the normal distribution, in which 99% of the sample exists within three standard deviations of the mean. This is a function of regressing to the proposition-specific (ie team) average rather than the league average in calculating xINT.)

What this table describes is an estimate of the percentage of the sample of total interceptions that lies within certain values of the mean.

So, in the case of the Tennessee Titans, our projected mean value (xINT) for total interceptions in 2012 is 16.0. Assuming a symmetrical distribution around this mean (a faulty premise, but convenient for now), the table tells us that 13.3% of the possible outcomes of total interceptions lies between 16.0 and the bookmaker's Over/Under quote of 14.0.

So, if we place a bet on total interceptions for the Tennessee Titans to go over the total of 14, we can expect to win 50% + 13.3% = 63.3% of the time.

The bookmakers in the oddschecker.com table offering the Over/Under quote of 14 total interceptions at the time of writing were Boylesports and Youwin. Both were citing odds of 20/23 associated with the proposition which represents their profit margin or, as it would be referred to in the US, the "juice" or "vig(orish)".

Odds of 20/23 represent roughly 53.5%, so to make a profit on our choice of the Over/Under of 14 total interceptions we need to be right more than 53.5% of the time. In this case, our projection says that the total will go over 63.3% of the time.

5.2 Calculating Expected Value (EV)

The Expected Value (EV) of a proposition is the profit or loss which will accrue as the number of trials of the event tends to infinity. In other words, it is the calculable edge at the given odds which can be expected when the estimate of the event's likelihood is correct.

In the case of betting over the total of 14 interceptions for the Tennessee Titans in 2012, if our estimate of 16 is correct then:

63.3% of the time we would win 20/23 units
36.7% of the time we would lose 1 unit

So, our EV =>    63.3/100 * 20/23 =  0.550
minus                 36.7/100 * -1       = -0.367
equals                                            = +0.183

Assuming our estimate is correct, we can expect to win .183 units for every unit stakes. Our EV on the bet is +18.3% or £18.30 for every £100 staked.

* NB: all decisions - whether involving financial considerations or not - can be evaluated in the same manner. We can think of the units we win and lose as utility in respect to their effect on our life.

5.3 Diminishing Marginal Value (or Utility*)

If you look again at the table which describes the distribution of total interceptions around our notional mean, you should recognise a familiar economic principle. For every extra interception we can project over or under the bookmaker's quoted value, we receive a smaller edge.

One interception is worth 6.8% but two interceptions are only worth 13.3%, or 6.65% each. Three interceptions are worth only 6.4% each and so on.

This is a massively important consideration in all forms of gambling and investment. It will be extremely familiar to you if you have studied, or understand, entry level economics:

As your prediction becomes increasingly outlandish, you receive progressively less value for making it.

The worth of each extra interception is also likely to be reduced the further away from the mean of the overall sample we are making our projection.

This is because, as we depart to values someway from the mean, the actual shape of the distribution is not symmetrical. It can't be.

In the 2007-2011 NFL seasons, for instance, the average number of total interceptions was 15.9. While there is a very small chance that a team could have 20.1 more total interceptions than this (36) there is no chance that a team could have 20.1 less total interceptions (-4.2). For obvious reasons, it is not possible to have fewer than zero interceptions.

It is therefore likely that projecting total interceptions under a quote which is itself less than the mean will result in rapidly diminishing returns. (Projecting total interceptions more than a quote itself higher than the mean is also undesirable, but it is less undesirable.)

5.4 Putting it all together

Using our projections of xINT - expected total interceptions for 2012 - the corresponding EV (Expected Value) for each Over/Under proposition is listed in the table. Note that the calculations do take into account the population distribution as it actually exists (rather, as is likely to do in 2012).

Team xINT  O/U  Diff Odds-1 EV%
New Orleans Saints 15.6 12.5 3.1 0.87 +30.5
New England Patriots 18.8 22.0 3.2 0.85 +28.5
Chicago Bears 17.3 20.0 2.7 0.85 +23.6
Baltimore Ravens 19.4 16.5 2.9 0.83 +23.3
Philadelphia Eagles 16.2 18.5 2.3 0.87 +20.9
Dallas Cowboys 15.8 18.0 2.2 0.85 +18.6
Arizona Cardinals 16.1 14.0 2.1 0.87 +18.6
Tennessee Titans 16.0 14.0 2.0 0.87 +17.4
Denver Broncos 15.3 13.5 1.8 0.87 +14.7
Buffalo Bills 16.6 18.5 1.9 0.83 +13.5
Oakland Raiders 17.2 15.5 1.7 0.87 +13.0
New York Jets 15.8 17.5 1.7 0.85 +12.6
Kansas City Chiefs 16.9 18.5 1.6 0.87 +12.0
Indianapolis Colts 13.1 11.5 1.6 0.87 +10.7
Jacksonville Jaguars 16.6 15.0 1.6 0.83 +10.0
New York Giants 17.1 18.5 1.4 0.85 +8.3
Carolina Panthers 16.1 17.5 1.4 0.83 +8.0
Detroit Lions 16.7 18.0 1.3 0.85 +7.4
Houston Texans 17.7 16.5 1.2 0.87 +6.6
Cleveland Browns 14.8 13.5 1.3 0.83 +6.2
Green Bay Packers 22.7 24.5 1.8 0.80 +6.0
Miami Dolphins 14.9 16.0 1.1 0.85 +4.9
Minnesota Vikings 12.8 11.5 1.3 0.83 +4.8
San Francisco 49ers 19.4 20.5 1.1 0.85 +3.1
Atlanta Falcons 18.1 19.0 0.9 0.85 +1.7
Washington Redskins 15.2 14.5 0.7 0.87 +1.4
Pittsburgh Steelers 15.3 14.5 0.8 0.80 -1.2
Cincinnati Bengals 14.8 14.5 0.3 0.87 -3.7
Tampa Bay Buccs 15.7 15.5 0.2 0.87 -4.3
St. Louis Rams 14.7 15.0 0.3 0.85 -4.8
Seattle Seahawks 17.7 18.0 0.3 0.85 -5.2
San Diego Chargers 16.5 16.5 0.0 0.87 -6.5


The right-hand column enumerates the expected profit from each proposition, assuming xINT represents the team average total interceptions for the 2012 NFL season reasonably accurately.

Note that the table is ordered by Expected Value expressed this time as a percentage. Note that this ranking is not the same as the difference between xINT and the Over/Under quote.

This is a reflection of the distribution. In the case of the Green Bay Packers, for instance, our projection is a healthy 1.8 total interceptions less than the quote. But both these figures are towards the right-hand extreme of t he distribution where relatively few values exist. So, the number of potential outcomes between xINT and the O/U quote (from where we derive our profit) are relatively few.

Now let's use intuition to cross-check some of the findings from the big difference of opinion with the Over/Under we find with New Orleans (a highly profitable Over bet) and New England (a highly profitable Under) to the agreement - resulting in coin-flip expectation minus the bookmaker's juice - about San Diego.

5.5 The acid test of intuition

You might be surprised to find that I believe strongly that the result of a quantitative analysis should have to tally with my intuition. The Figures Never Lie! after all. But, wait a second, to invest wisely there is a need to have confidence, so the figures should be able to "sell" their argument to me. And I should be able to "sell" it to a third-party like you. Not literally, obviously!

So, let's take the opposite case of our two outlying projections, New Orleans and New England. Does it make sense that the bookmaker's Over/Under projections should be a long way wrong?

The total interceptions for New Orleans over the last five seasons are 9, 9, 26, 15, 13 and the Over/Under quote is 12.5. Well, the first thing to notice is that the Saints have indeed gone Over the total (our side) in three of those five campaigns.

In the last two seasons, their total interceptions of 9 is well Under the total, but we know that there is only a weak correlation in this variable from one season to next. As argued in a previous post, there is a lot of randomness at work.

Moreover, the passes defended totals of 99, 77, 109, 106 and 86 suggest that the New Orleans defense has the underlying skills to make significantly more than 12.5 interceptions in 2012. But what about their scheme?

One reason that the Saints have not met expectations in making interceptions is their defensive coordinator for the past three seasons, Greg Williams. Now suspended for his part in the infamous 'Bountygate' scandal (the financial incentivising of Saints players to injure opposing players), Williams runs an extremely aggressive scheme with a multitude of blitzing and man-to-man coverage.

Though the Saints defense has gone out of the frying pan into the fire with his replacement the former St Louis Rams HC Steve Spagnuolo, this alone should produce better results, judged by the latter's history when Defensive Coordinator with the New York Giants in 2007-8.

In other words, given the randomness of interceptions, the underlying skills suggested by passes defended and the change in scheme, betting Over the total of 12.5 interceptions for the New Orleans Saints makes a lot of sense.

What about betting the Under of 22 total interceptions about the New England Patriots? Can we really do this with confidence when the architect of their defense is no less than Bill Belichick? Yes we can!

In the last five seasons, the Patriot defense has exceeded the metric Expected Interceptions - that is enjoyed so-called 'interception luck' - by an average of five interceptions per year.

During this period, their total interceptions were 23, 25, 18, 14 and 19. As with the New Orleans example, it is encouraging that the total would have gone Under  this season's quote of 22 more often than not, but the situation is even more promising than that.

New England's passes defended - which shows a stronger season-to-season correlation that total interceptions - are 84, 102, 86, 71 and 91. In other words, they ranked towards the bottom of the league in a statistic which we do know is related to underlying skills because it tends to persist somewhat from one season to the next. (They also ranked 29th of 32 teams in yards-per-pass against, another strong indicator of defensive skills which survives randomness.)

So, why should a defense which by several measures is one of the worst against the pass have the second-highest (behind Green Bay) Over/Under quote? Well, we know the answer to that: because of their recent history.

5.6 Cognitive biases in investment

It is easy to see why fading the recency bias is such a powerful approach in investment strategy. The average NFL fan might buy the idea that New England's total interceptions of 23 for the 2011 season were an aberration, but not when they also intercepted 25 in 2010.

It's true that in some cases like these, the predictor variables - and indeed intuition - are insufficient to understand why there might have been a sudden change to the likelihood of a variable's output. But we can also observe the statistical significance of an event and understand how randomness profoundly affects the world.

Both our top two plays - Over the total of 12.5 for New Orleans and Under 22 for New England - are the result of an understanding that recent outcomes are subject to considerable flux. With knowledge of the shape of the population of all outcomes, and by using metrics and variables which capture the underlying skills which influence them, we can make often make better projections than the market.

I had intended to get into the nuts-and-bolts of staking and portfolio selection in this post but it has gone on too long already. There will now be a Part 6.

Monday, 27 August 2012

Statistical inference and mathematical modelling (Part 4)

So, now let's apply what we have observed about the variable behaviour of an NFL team's seasonal interception totals to a live betting market.

Over/under quotes on total interceptions of all 32 NFL teams for the 2012 season are available via:

http://www.oddschecker.com/american-football/nfl-specials

** warning: prices referred to in the following copy may be out of date **

To recap, these are the statistical dimensions of interceptions that we established in the previous post:

1) A team's total interceptions for one season is only weakly correlated with its total for the following season;

2) In general, a team's total interceptions is, in effect, a somewhat random draw from its passes defended;

3) In year y, a team's total of passes defended predicts its interception total in year y+1 better than total interceptions.

In particular, result 3) is highly encouraging from the point of view of finding an edge. The Over/Under assessments are, as might be anticipated, highly influenced by each team's recent performance in total interceptions, yet:

There is only a weak correlation between total interceptions in consecutive seasons, r=0.151;
passes defended is a better guide to total interceptions in year y+1, r=0.262

4.1 Introducing linear regression

Linear regression is the bread-and-butter method of sports modelling. It is a mathematical technique of explaining the relationship between two or more quantities, the first described by the dependent variable and the others by explanatory or independent variables.

For now, let us consider the simplest case of two variables a and b linked by the relationship:

b = ma + c

In this case, b = a team's total interceptions for year y+1, while a = a team's total of passes defended for year y. The task is to determine values for m and c which describes a "best-fit line" between the 128 data-points from teams in the NFL seasons 2007-2011.

The answer is b = 0.0924a + 7.47

So, by plugging in a team passes defended for year y as the descriptive variable a, we can predict or project a better estimate of total interceptions for year y+1 than we can by using total interceptions for year y as the descriptive variable instead.

Let's make the calculations for the upcoming 2012 NFL season:

Team Pdef 2011 INT 2011 xINT 2012
Arizona Cardinals 95 10 16.3
Atlanta Falcons 99 19 16.6
Baltimore Ravens 122 15 18.7
Buffalo Bills 87 20 15.5
Carolina Panthers 84 14 15.2
Chicago Bears 85 20 15.3
Cincinnati Bengals 83 10 15.1
Cleveland Browns 78 9 14.7
Dallas Cowboys 72 15 14.1
Denver Broncos 68 9 13.8
Detroit Lions 94 21 16.2
Green Bay Packers 129 31 19.4
Houston Texans 114 17 18.0
Indianapolis Colts 55 8 12.6
Jacksonville Jaguars 79 17 14.8
Kansas City Chiefs 104 20 17.1
Miami Dolphins 80 16 14.9
Minnesota Vikings 58 8 12.8
New England Patriots 84 23 15.2
New Orleans Saints 99 9 16.6
New York Giants 104 20 17.1
New York Jets 90 19 15.8
Oakland Raiders 106 18 17.3
Philadelphia Eagles 78 15 14.7
Pittsburgh Steelers 83 11 15.1
San Diego Chargers 82 17 15.0
San Francisco 49ers 124 23 18.9
Seattle Seahawks 105 22 17.2
St. Louis Rams 77 12 14.6
Tampa Bay Buccs 73 14 14.2
Tennessee Titans 81 11 15.0
Washington Redskins 85 13 15.3

Now, let's now compare our projections of xINT - here, our projected total interceptions for each team in 2012 - with the outlying Over/Under totals among the range of bookmakers offering prices on the market:

Team xINT 2012 O/U quote Difference
New England Patriots 15.2 22.0 -6.8
Green Bay Packers 19.4 24.5 -5.1
Chicago Bears 15.3 20.0 -4.7
New Orleans Saints 16.6 12.5 +4.1
Dallas Cowboys 14.1 18.0 -3.9
Philadelphia Eagles 14.7 18.5 -3.8
Buffalo Bills 15.5 18.5 -3.0
Atlanta Falcons 16.6 19.0 -2.4
Arizona Cardinals 16.3 14.0 2.3
Carolina Panthers 15.2 17.5 -2.3
Baltimore Ravens 18.7 16.5 +2.2
Detroit Lions 16.2 18.0 -1.8
Oakland Raiders 17.3 15.5 +1.8
New York Jets 15.8 17.5 -1.7
San Francisco 49ers 18.9 20.5 -1.6
Houston Texans 18.0 16.5 +1.5
San Diego Chargers 15.0 16.5 -1.5
Kansas City Chiefs 17.1 18.5 -1.4
New York Giants 17.1 18.5 -1.4
Minnesota Vikings 12.8 11.5 1.3
Tampa Bay Buccs 14.2 15.5 -1.3
Cleveland Browns 14.7 13.5 +1.2
Indianapolis Colts 12.6 11.5 +1.1
Miami Dolphins 14.9 16.0 -1.1
Tennessee Titans 15.0 14.0 +1.0
Seattle Seahawks 17.2 18.0 -0.8
Washington Redskins 15.3 14.5 +0.8
Jacksonville Jaguars 14.8 15.5 -0.7
Cincinnati Bengals 15.1 14.5 +0.6
Pittsburgh Steelers 15.1 14.5 +0.6
St. Louis Rams 14.6 15.0 -0.4
Denver Broncos 13.8 13.5 +0.3




average 15.7 16.6 -0.8

The table is sorted by the magnitude of the difference between our estimate of 2012 total interceptions and that of the bookmakers. In other words, the direction of disagreement (the + or - sign) has been removed.

At face value, it appears there are some sizeable differences. So, is it time to bet the under of the Patriots, Packers and Bears and the over of the Saints with everything we can muster?

No. Not yet.

4.2 Establishing and improving forecast accuracy

The correlation between two variables - such as total interceptions and passes defended - amounts to evidence that the two were related (or, to be correct, may have been related) in the past. But it does not mean that one is the cause of the other, or that the same relationship will hold true in the future.

(It is into this bear-trap that the majority of poor forecasts fall and die. I'm not going to talk about statistical significance and other measures of hypothesis testing here because it will slow down my flow. You will just have to trust that I am acutely aware of their importance in forging my conclusions.)

In modelling sports, a highly complex and dynamic interaction of variables is most often the driving force behind outcomes like interceptions or goals. But we need to produce relatively simple models to understand the probability with which they come about, even if we know that the limited array of variables we are using is not the full story.

Having produced our forecasts for the total interceptions of NFL teams in the upcoming 2012 season using just one predictor - passes defended in the 2011 season - we have probably reached a better understanding of what 'causes' interceptions better than the vast majority of American Football fans and - most importantly - bettors.

But, this is not just an exercise in playing with numbers. We are driving towards a sound mathematical conclusion on which we can bet money with the expectation of a positive result. We can improve our forecasts by a good deal yet.

We know that our projection method works in general by observing a correlation between passes defended in year y and total interceptions in year y+1. But let's now look at how the projections have fared in past seasons, importantly, by NFL team:

Team Year Pdef Int INT luck Wt avg
Arizona Cardinals 2007 84 18 3.1 -1.4
Arizona Cardinals 2008 88 13 -2.6
Arizona Cardinals 2009 108 21 1.9
Arizona Cardinals 2010 97 17 -0.2
Arizona Cardinals 2011 95 10 -6.8
Atlanta Falcons 2007 85 16 0.9 1.5
Atlanta Falcons 2008 85 10 -5.1
Atlanta Falcons 2009 78 15 1.2
Atlanta Falcons 2010 91 22 5.9
Atlanta Falcons 2011 99 19 1.4
Baltimore Ravens 2007 106 17 -1.8 -0.1
Baltimore Ravens 2008 125 26 3.8
Baltimore Ravens 2009 100 22 4.3
Baltimore Ravens 2010 99 19 1.4
Baltimore Ravens 2011 122 15 -6.6
Buffalo Bills 2007 98 18 0.6 1.1
Buffalo Bills 2008 83 10 -4.7
Buffalo Bills 2009 109 28 8.7
Buffalo Bills 2010 88 11 -4.6
Buffalo Bills 2011 87 20 4.6
Carolina Panthers 2007 81 14 -0.4 0.8
Carolina Panthers 2008 95 12 -4.8
Carolina Panthers 2009 88 22 6.4
Carolina Panthers 2010 85 17 1.9
Carolina Panthers 2011 84 14 -0.9
Chicago Bears 2007 78 16 2.2 3.0
Chicago Bears 2008 121 22 0.5
Chicago Bears 2009 66 13 1.3
Chicago Bears 2010 94 21 4.3
Chicago Bears 2011 85 20 4.9
Cincinnati Bengals 2007 97 19 1.8 -1.5
Cincinnati Bengals 2008 93 12 -4.5
Cincinnati Bengals 2009 112 19 -0.9
Cincinnati Bengals 2010 83 16 1.3
Cincinnati Bengals 2011 83 10 -4.7
Cleveland Browns 2007 103 17 -1.3 -0.8
Cleveland Browns 2008 90 23 7.0
Cleveland Browns 2009 83 10 -4.7
Cleveland Browns 2010 96 19 2.0
Cleveland Browns 2011 78 9 -4.8
Dallas Cowboys 2007 103 19 0.7 0.1
Dallas Cowboys 2008 68 8 -4.1
Dallas Cowboys 2009 99 11 -6.6
Dallas Cowboys 2010 82 20 5.5
Dallas Cowboys 2011 72 15 2.2
Denver Broncos 2007 80 14 -0.2 -2.8
Denver Broncos 2008 57 6 -4.1
Denver Broncos 2009 97 17 -0.2
Denver Broncos 2010 86 10 -5.2
Denver Broncos 2011 68 9 -3.1
Detroit Lions 2007 76 17 3.5 -0.1
Detroit Lions 2008 48 4 -4.5
Detroit Lions 2009 80 9 -5.2
Detroit Lions 2010 76 14 0.5
Detroit Lions 2011 94 21 4.3
Green Bay Packers 2007 90 19 3.0 5.6
Green Bay Packers 2008 110 22 2.5
Green Bay Packers 2009 126 30 7.7
Green Bay Packers 2010 110 24 4.5
Green Bay Packers 2011 129 31 8.1
Houston Texans 2007 86 11 -4.2 -1.6
Houston Texans 2008 71 12 -0.6
Houston Texans 2009 89 14 -1.8
Houston Texans 2010 69 13 0.8
Houston Texans 2011 114 17 -3.2
Indianapolis Colts 2007 84 22 7.1 0.9
Indianapolis Colts 2008 63 15 3.8
Indianapolis Colts 2009 84 16 1.1
Indianapolis Colts 2010 64 10 -1.3
Indianapolis Colts 2011 55 8 -1.8
Jacksonville Jaguars 2007 83 20 5.3 2.6
Jacksonville Jaguars 2008 69 13 0.8
Jacksonville Jaguars 2009 71 15 2.4
Jacksonville Jaguars 2010 61 13 2.2
Jacksonville Jaguars 2011 79 17 3.0
Kansas City Chiefs 2007 83 14 -0.7 -1.4
Kansas City Chiefs 2008 70 13 0.6
Kansas City Chiefs 2009 96 15 -2.0
Kansas City Chiefs 2010 110 14 -5.5
Kansas City Chiefs 2011 104 20 1.6
Miami Dolphins 2007 70 14 1.6 -0.9
Miami Dolphins 2008 99 18 0.4
Miami Dolphins 2009 90 15 -1.0
Miami Dolphins 2010 93 11 -5.5
Miami Dolphins 2011 80 16 1.8
Minnesota Vikings 2007 88 15 -0.6 -1.0
Minnesota Vikings 2008 78 12 -1.8
Minnesota Vikings 2009 73 11 -1.9
Minnesota Vikings 2010 77 15 1.3
Minnesota Vikings 2011 58 8 -2.3
New England Patriots 2007 91 19 2.9 5.0
New England Patriots 2008 71 14 1.4
New England Patriots 2009 86 18 2.8
New England Patriots 2010 102 25 6.9
New England Patriots 2011 84 23 8.1
New Orleans Saints 2007 86 13 -2.2 -2.9
New Orleans Saints 2008 106 15 -3.8
New Orleans Saints 2009 109 26 6.7
New Orleans Saints 2010 77 9 -4.7
New Orleans Saints 2011 99 9 -8.6
New York Giants 2007 90 15 -1.0 -0.9
New York Giants 2008 92 17 0.7
New York Giants 2009 92 13 -3.3
New York Giants 2010 103 16 -2.3
New York Giants 2011 104 20 1.6
New York Jets 2007 82 15 0.5 -1.0
New York Jets 2008 91 14 -2.1
New York Jets 2009 103 17 -1.3
New York Jets 2010 96 12 -5.0
New York Jets 2011 90 19 3.0
Oakland Raiders 2007 87 18 2.6 -1.2
Oakland Raiders 2008 86 16 0.8
Oakland Raiders 2009 77 8 -5.7
Oakland Raiders 2010 75 12 -1.3
Oakland Raiders 2011 106 18 -0.8
Philadelphia Eagles 2007 75 11 -2.3 1.0
Philadelphia Eagles 2008 107 15 -4.0
Philadelphia Eagles 2009 117 25 4.3
Philadelphia Eagles 2010 113 23 3.0
Philadelphia Eagles 2011 78 15 1.2
Pittsburgh Steelers 2007 88 11 -4.6 -1.4
Pittsburgh Steelers 2008 107 20 1.0
Pittsburgh Steelers 2009 79 12 -2.0
Pittsburgh Steelers 2010 109 21 1.7
Pittsburgh Steelers 2011 83 11 -3.7
San Diego Chargers 2007 119 30 8.9 1.9
San Diego Chargers 2008 90 15 -1.0
San Diego Chargers 2009 80 14 -0.2
San Diego Chargers 2010 83 16 1.3
San Diego Chargers 2011 82 17 2.5
San Francisco 49ers 2007 78 12 -1.8 0.3
San Francisco 49ers 2008 85 12 -3.1
San Francisco 49ers 2009 87 18 2.6
San Francisco 49ers 2010 79 15 1.0
San Francisco 49ers 2011 124 23 1.0
Seattle Seahawks 2007 97 20 2.8 -0.9
Seattle Seahawks 2008 75 9 -4.3
Seattle Seahawks 2009 77 13 -0.7
Seattle Seahawks 2010 97 12 -5.2
Seattle Seahawks 2011 105 22 3.4
St. Louis Rams 2007 89 18 2.2 -1.1
St. Louis Rams 2008 65 12 0.5
St. Louis Rams 2009 59 8 -2.5
St. Louis Rams 2010 91 14 -2.1
St. Louis Rams 2011 77 12 -1.7
Tampa Bay Buccs 2007 84 16 1.1 2.9
Tampa Bay Buccs 2008 95 22 5.2
Tampa Bay Buccas 2009 82 19 4.5
Tampa Bay Buccs 2010 89 19 3.2
Tampa Bay Buccs 2011 73 14 1.1
Tennessee Titans 2007 108 22 2.9 1.1
Tennessee Titans 2008 106 20 1.2
Tennessee Titans 2009 83 20 5.3
Tennessee Titans 2010 89 17 1.2
Tennessee Titans 2011 81 11 -3.4
Washington Redskins 2007 97 14 -3.2 -3.7
Washington Redskins 2008 102 13 -5.1
Washington Redskins 2009 85 11 -4.1
Washington Redskins 2010 103 14 -4.3
Washington Redskins 2011 85 13 -2.1

In several cases, teams like the Green Bay Packers, New England Patriots and Chicago Bears can sustain interception luck - a notably higher percentage of passes defended compared with total interceptions - over several seasons. (I have back-tested this on 10 years of data before 2007 and found a similar effect is at play.)

So, rather than assuming a league-average rate of total interceptions per passes defended it turns out that we can make much better predictions if taking account of each team's historical percentage. The weighted average (in which recent totals count for slightly more) of each team's interception luck is included in the right-hand column.

We can then recalculate by projecting passes defended in year y to total interceptions in year y+1 at a rate nearer that which is typical for the team than the league.

From a technical standpoint, this can be referred to as regressing a rate statistic to a proposition-specific rather than general average.

Again, this is vital technique of sports modelling. Where rate stats are concerned, we need to be careful before assuming that deviations from expectation are not just noise. But, if it is the case there is an underlying signal in this example, from what is its source?

Every NFL team runs a defensive scheme, styled most often by its defensive coordinator (but sometimes, in the case of the New England Patriots, by its defensive-minded head coach).

It turns out that schemes which rely heavily on zone defense (those of the Tampa-2 and Cover 2 family) generally intercept passes at a higher rate than the league average of 17.7% of passes defended, while those schemes which are blitz-heavy or emphasise pressure (typically the 3-4 set-ups of Pittsburgh, Baltimore and Arizona) generally intercept a lower percentage.

The difference, I would guess, is a function of both the number of defenders in coverage on each pass play and, perhaps more importantly, their responsibilities. In zone schemes, defenders face the opposing quarterbacks and thus may get a better read on the ball, whereas man-to-man defenders are focussed on the opposing receivers and may have less time to sight the ball and pick it off, rather than just deflecting it.

4.3 Our final projections

Employing this approach leads to a considerable higher success-rate in projecting total interceptions when back-tested on data of a near-significant sample-size. The correlation between xINT and INT improved from 0.261 to 0.478.

Although the latter figure is still lower than ideal, compensation can be made in our projections for teams which have changed defensive schemes, or are otherwise more likely to face a higher number of pass plays. We must also take into account that there is an upward trend in passes attempted (and hence intercepted) in the NFL.

Team xINT 2012 O/U quote Diff
New England Patriots 18.8 22 -3.2
New Orleans Saints 15.6 12.5 +3.1
Baltimore Ravens 19.4 16.5 +2.9
Chicago Bears 17.3 20 -2.7
Philadelphia Eagles 16.2 18.5 -2.3
Dallas Cowboys 15.8 18 -2.2
Arizona Cardinals 16.1 14 +2.1
Tennessee Titans 16.0 14 +2.0
Buffalo Bills 16.6 18.5 -1.9
Green Bay Packers 22.7 24.5 -1.8
Denver Broncos 15.3 13.5 +1.8
Oakland Raiders 17.2 15.5 +1.7
New York Jets 15.8 17.5 -1.7
Kansas City Chiefs 16.9 18.5 -1.6
Indianapolis Colts 13.1 11.5 +1.6
Carolina Panthers 16.1 17.5 -1.4
New York Giants 17.1 18.5 -1.4
Detroit Lions 16.7 18 -1.3
Minnesota Vikings 12.8 11.5 +1.3
Cleveland Browns 14.8 13.5 +1.3
Houston Texans 17.7 16.5 +1.2
San Francisco 49ers 19.4 20.5 -1.1
Miami Dolphins 14.9 16 -1.1
Jacksonville Jaguars 16.6 15.5 +1.1
Atlanta Falcons 18.1 19 -0.9
Pittsburgh Steelers 15.3 14.5 +0.8
Washington Redskins 15.2 14.5 +0.7
Seattle Seahawks 17.7 18 -0.3
Cincinnati Bengals 14.8 14.5 +0.3
St. Louis Rams 14.7 15 -0.3
Tampa Bay Buccs 15.7 15.5+0.2
San Diego Chargers 16.5 16.5 =


Note that the average of my projections (xINT) = 16.47 while the average of the bookmakers outlying quote = 16.56.

In the next blog, I will be showing how to calculate what these differences mean in terms of percentage chance of cashing an Over/Under bet on these propositions in general.