The Tim Tebow show rolled into Foxborough on Tuesday, just a day after the news of his imminent signing broke. The NFL’s most polarizing figure inked a bargain-basement two-year deal with the Patriots and took the field as Bill Belichick’s third-string quarterback, eager to continue his NFL career under the tutelage of his biggest proponent in the pro game, offensive coordinator Josh McDaniels.

Critics and supporters alike agree that Tebow’s throwing is substandard for a starting NFL quarterback, but barring a Tom Brady injury, there’s zero chance that he’ll start in New England. So are his statistics subpar for a backup? A quick look at his passer rating shows that Tebow ranks 46th among 77 active quarterbacks with NFL playing experience (no minimums) at a middling 75.3. That puts him among the poor-starter-yet-solid-understudy class like Dan Orlovsky (76.0), Chad Henne (74.9) and David Carr (74.9). If you trust the rating system as a QB quality indicator, you can do a lot worse among the league’s signal-callers, a list that includes his 2012 Jets teammate Mark Sanchez (71.9), former Patriots backup Brian Hoyer (72.2), Mike Kafka, the man cut to clear a roster space for Tebow, (47.7) and current New England No. 2 man Ryan Mallett (5.2, lowest among all active QBs).

FULL ENTRY

By Andrew Mooney, Boston.com Correspondent

The most recent report from our own Greg Bedard says that “the odds are strong [Wes] Welker will be back” with the Patriots in 2013, the latest in conflicting news regarding the 31-year old wideout.

I don’t think there’s any way to get a good read on what the New England front office is thinking outside of inside information—past history suggests they could go either way, offering Welker a new deal or parting ways with him for good. The Patriots have shown themselves to be decidedly unsentimental when making personnel decisions, especially with older players, even before age has started to take its inevitable toll. However, there is nothing the Pats value more than solid, no-nonsense production, and Welker has certainly brought that since coming over from Miami in 2007.

I decided to examine just how much Welker has meant to the Patriots offense using statistics pulled from Advanced NFL Stats. Though it doesn’t separate the contributions of a receiver from those of his quarterback, one of the best stats for evaluating a wideout’s production is Expected Points Added (EPA), which translates a team’s movements down the field into the points it can historically expect to score from that position, given down and distance. For a more thorough explanation of EPA, click here.

In the chart below, I’ve graphed Expected Points Added for wide receivers in the NFL from 2007-2012. Alongside Welker’s own production, I’ve plotted the league average, calculated from wide receivers that played at least half the regular season, and the NFL’s top performer for each season.

In every season he has played in New England, Welker has been above average, and in most of them he’s been excellent, regularly producing at least twice the amount of expected points as the average of his peers. Just two seasons ago, after completing a 122-catch, 1,569-yard season, he led the NFL in this category. The only years in which Welker did not lead the Patriots in EPA were 2010, his least productive season, and 2007, when he finished second in the league behind Randy Moss, who set an NFL-record for touchdown receptions.

It’s clear that Welker has been a high-usage player for Tom Brady and the New England offense; his target rate, a measure of the percentage of a team’s passes thrown a receiver’s way, has been about 40 percent higher than the league average from 2007-2012. But Welker has also made the most of his targets, catching them at a high rate. Once again, I’ve plotted below how Welker stacks up with his competition in this category.

Again, Welker ranks well above average in each season, often at the top of the league—though it’s worth noting that his catch rate has started to fall off slightly in recent years. If that’s an indication that he can’t get the same separation from defenders that he once did, you can be sure Bill Belichick has taken notice.

Whatever the fate of the Welker contract negotiations, there’s no denying the integral role he has played in the Patriots’ record-setting offense. It is difficult to know whether he can sustain his incredible rate of production for many more years, but team management will know the answer to that question better than anybody. For now, let’s appreciate the place that Welker—all 5’9” of him—has carved out in a league of giants.

By Andrew Mooney, Boston.com Correspondent

For interested fan or gambler, the NFL playoffs can be infinitely maddening to comprehend. Those Cincinnati Bengals that had won seven of their last eight? Well, they’re out, held to just -6 passing yards in the first half by the team that allowed 68 points to the Jaguars and Lions in consecutive games. And let’s not forget the 2008 Super Bowl, in which probably the best team in history lost to the worst team ever to win a championship. If you’re skeptical of the influence game-to-game variance holds in the NFL, the playoffs have a way of making you a believer.

This post from Football Perspective lends further credence to the idea that playoff results have been getting even more random in recent years, edging closer to 50-50. We’ve long known parity is a very real phenomenon in the NFL, but is there a way to cut through all the randomness and noise to determine what makes teams win in the playoffs?

To start with, I considered upsets. In college basketball, an underdog stands the best chance of shocking a more highly rated team in the NCAA tournament if it employs a high variance style of play—pressing, shooting lots of three-pointers, etc. A strategy like this is termed “high variance” because it can lead to extreme outcomes; a hot shooting night and a sloppy opponent could lay the groundwork for a Cinderella story, but if the threes aren’t falling and the favorite takes care of the ball, the underdog is in for a blowout.

I adapted this theory to the NFL, hypothesizing that a team with a greater amount of variation in its regular season play stood a better chance of getting hot enough to pull off a few upsets. For every playoff team of the last ten years, I took the variance of a team’s point differentials (margin of defeat or victory) in its 16 regular season games. Next, I ran a few regressions to determine whether variance was a significant predictor of playoff wins or upset wins, which I defined as any victory by a team seeded two or more places lower than its opponent. In each test, I controlled for the teams’ point differentials per game from their respective regular seasons, as a proxy for team quality.

My results indicated no such relationship. The variance of a team’s week-to-week performance in the regular season had no measurable effect on the number of wins or upsets it pulled off, even when I split the data up into thirds and quarters, ordered by mean point differential. In other words, even for the worst teams to make the playoffs, regular season variance played no role in determining a potential Cinderella.

The difference I perceive in how variance affects college basketball and football upsets is that, generally speaking, variance in college basketball is more a product of a team’s style of play, which isn’t necessarily true for football. Few NFL offenses and defenses stray far from the mainstream in terms of concepts and schemes, which can’t be said for college basketball. In the NFL, game-to-game variance might be the result of a number of factors that aren’t necessarily predictive: turnovers, luck, etc.

I did find one variable that was slightly predictive of playoff success, however: team quality, as measured by mean point differential. Teams in the top third of mean point differential averaged about half a playoff win more than teams in the bottom third. This is about as hammer-to-the-head obvious as it comes—better teams do better in the playoffs. There are the NFL playoffs for you: the only things we can know for sure are the ones staring us right in the face.

By Andrew Mooney, Boston.com Correspondent

It appears that, for another week, a Patriots opponent will not be without bulletin board material as it prepares for its trip to Gillette Stadium. This time, however, the responsibility lies not with an employee of this site, but with the oddsmakers in Las Vegas.

Depending on which online sportsbook you consult, New England currently stands as 9- or 9.5-point favorites over the Baltimore Ravens, a team that it lost to earlier in the season in a Week 3 thriller. No doubt Ray Lewis and crew will feel a little slighted by such a spread, which suggests they have about a 20 percent chance of victory, but the Patriots should find some encouragement in the history of large postseason lines.

Pro Football Reference has compiled point spreads for NFL games since 1978 and counts exactly 71 games in which a team has been favored by nine or more points in the playoffs. Of these 71 favorites, 40 ended up covering the point spread, comfortably advancing to the next round, and another 17 still won their playoff game, despite not covering. That means that the historical rate at which these heavy favorites won—57 out of 71, or 80 percent—exactly mirrors the theoretical win probability implied by the spread in Sunday’s game.

Yet I don’t anticipate the Patriots will be particularly overconfident when they take the field this weekend; they own two of those fourteen losses, both during the Belichick era. They were 12.5-point favorites over the Giants in Super Bowl XLII and 9.5-point favorites over the Jets two seasons ago, but each of their New York opponents produced a performance that belied the talent discrepancy on paper. New England enters much the same situation against the Ravens on Sunday, and as this year’s playoffs have proved, things don’t generally stick to the script.

By Andrew Mooney, Boston.com Correspondent

Last week’s blowout win over the Indianapolis Colts was characteristic of this season’s Patriots: another game played, another opportunity to change one’s opinion of the team. Two weeks ago, a lackluster defensive performance against the Bills suggested an inevitable first-round playoff exit, though just one week earlier the Patriots had convinced two countries of their dominance with a blowout victory over the Rams in London. The picture wasn’t any clearer earlier in the season, with a convincing win against the AFC contender Broncos followed up by a fourth quarter collapse against the Seahawks.

From game to game, the “eye test” doesn’t seem a reliable way to evaluate this team; you just can’t be certain which Patriots are going to show up. For that reason, it makes more sense to look at the Patriots’ full body of work over the course of the season to get a sense of what their prospects are of playing into February.

The most distinctive feature of the team is quite clearly the offense, which has put up points at a historic rate. This year’s Patriots are one of nine teams in history to score 350 points through 10 games, and only the fifth since 1950. They have twice scored over 50 points in a game, something only one team since 1969—the 2007 Patriots—has accomplished. They even have a formidable running game, something that hasn’t been seen in Foxborough for some years.

The ever-present worry is, as always, on the other side of the ball. The defense has continued following Bill Belichick’s formula: give up truckloads of yards, limit teams in the red zone, and force turnovers. Though creating turnovers is generally not thought of in the statistical community as a skill—that is, replicable from year-to-year—the Patriots have finished in the top ten in turnover margin in ten of their 13 seasons with Belichick at the helm. But they need every takeaway they can get, as their 30th-ranked passing defense has been quite kind to the opposing quarterbacks they’ve played—which, by the way, is not a very impressive list. The Patriots have played only two top-ten passing attacks in the Colts and Broncos.

This all goes to say that the team is essentially a carbon-copy (if not slightly upgraded) of the group that came within a few plays of winning a Super Bowl last year. I could basically re-run this article a year later, and the conclusions would be about the same.

It’s hard to win without an elite unit on one side of the ball—which the Patriots have—but beyond that, the playoffs can be a crapshoot. They are currently second in the league in Football Outsiders’ team efficiency ratings and are listed as having the highest odds of any team to win the Super Bowl at 18.5 percent…but that still means the Patriots would fall short of their ultimate goal four out of five times.

To me, this betrays the danger of using a binary definition of success—championship or no championship—for a season. The Patriots have been the NFL’s most successful franchise for a decade, but all anyone wants to talk about as the playoffs approach is that they haven’t won a Super Bowl since 2004, as if that represents some historic drought. The breaks find a way to even out over the long run; they had the benefit of some good luck in winning three titles in four years and suffered some bad luck in losing their last two Super Bowl appearances. But once again, the Patriots are among the league’s best teams and are as well-positioned as anyone to make a run at the title, and that’s all a fan can really ask for.

By Andrew Mooney, Boston.com Correspondent

A year and a half into the NFL’s amended kickoff rules, it’s safe to say they’ve achieved their intended purpose. In 2010, 16.4 percent of kickoffs resulted in touchbacks; a year later, that number rose to 43.5 percent. The teeth were taken out of what was thought to be one of the most dangerous plays in the game by simply reducing the number of times it occurs.

The kickoff rule change—specifically, kicking off from the 35-yard line—interests me not as a safety regulation, but as a method to examine teams’ decision-making. Have NFL coaches properly accounted for it in the way they instruct their special teams units?

My gut reaction says no. On a number of occasions this season, I’ve watched returners take the ball out of the end zone from eight or nine yards deep only to be stuffed at the 15, needlessly hampering their offenses with poor field position. It seems like teams should be settling for even more touchbacks than they currently are.

Of course, gut reactions are worthless for evaluations of this sort, so it’s time to go to the numbers. Using play-by-play data for the first four weeks of the 2012 season, I picked out every instance in which a team kicked the ball deep (i.e. not a squib or onside kick) and noted the receiving team’s resulting field position following the return.

It turns out that NFL special teams units are not being so irrational after all. On all long kickoffs (defined as any kick fielded by a returner inside his own ten-yard line), the average starting field position of the receiving team is at their own 23.5-yard line. Admittedly, this doesn’t offer a comprehensive picture of the question, since I’m most interested in just kicks fielded deep in the end zone, so I broke the data down further. Even on kicks fielded five yards deep in the end zone or further back, the average kick return yields better starting field position (a team’s own 22.5-yard line) than simply taking the touchback and heading to the 20.

But the decision to encourage kick returners’ apparent greediness can be good or bad depending on the game situation. The variance in field position for kicks returned from five yards deep in the end zone or deeper is much higher than for kicks returned from further up, meaning that the set of outcomes is much more spread out. In other words, with a kick fielded deep, a team is relatively more likely to have a starting field position that is quite a bit different from the average. That means that if a team is behind late in the game, it’s probably worth while to take the risk of bringing a kick out of the end zone for the possibility of a big return and great field position. But if a team is protecting a late lead, it may want to simply take the touchback, instead of running the risk of starting its drive at its own 10-yard line and sacrificing ten yards of precious field position.

I have yet to examine the data for previous seasons to determine what the existing trend is here, but the verdict seems to be that, even with the new kickoff rules, a returner is usually better off bringing the ball out than settling for a kneeldown in the end zone—that is, unless he happens to be fielding a kick from Neil Rackers.

By Andrew Mooney, Boston.com Correspondent

Yes, the Patriots lost last week. Yes, they blew a substantial fourth quarter lead. But before the "Brady's a choker!" train gets any momentum, I'd like to insert a word of reason.

To put it in the simplest terms I can muster, the Patriots are really good at football. They have been for over a decade, and they're even good in the fourth quarter, believe it or not. But to see just how good, we'll need to look beyond the dark memories of Week 6, soul-rending though it may have been.

More specifically, we'll examine the period from 2000 to the present, when Bill Belichick took the Pats' head coaching position, and the organization began to take on his personality. Certainly, if these occasional late-game lapses are something to fret about, we can pin them squarely on him and his personnel.

The bad news: since 2000, the Patriots have lost 11 games in which they were leading at the start of the fourth quarter. The good news: only two teams in the NFL have lost fewer of these games, the Falcons (eight) and the Ravens (nine). More good news: over this time span, the Patriots have played more games in which they started the fourth quarter with a lead than any other team in the league.

Far from being chokers, the Patriots have been among the best closers in the league over the last decade. They have had more chances to blow leads than anyone else, but they've actually done it at one of the lowest rates in the NFL. In fact, only the Ravens have been more efficient at preserving leads, by a few percentage points.

When the question is framed this way, the easy counterargument is to point out that, in many of those games, the Patriots were blowing teams out, and it's not very difficult to hold a 21-point fourth quarter lead. But when I limit the results to one-score leads entering the fourth quarter (8 points or less), the verdict is essentially the same. The Pats drop a few places to fifth, but the rate at which they preserve leads is still among the leaders in the NFL.

What about comebacks? By virtue of being an elite team for so long, the Patriots have had fewer chances at them than any other team; on only 45 occasions in 11-plus seasons, the Pats have been trailing entering the fourth quarter. Yet the Patriots have successfully come from behind for the win in 17 of those games, giving them the second-best comeback rate in the league, trailing only Indianapolis.

Admittedly, this "trend" is getting more attention because the Pats have been less steely is preserving leads in recent years, with seven of their 11 losses coming in the past four seasons and four in the 2009 season alone. However, conducting a quick binomial test shows that this is not a statistically significant difference; in other words, this is much more likely the product of random variation rather than a change in some fundamental quality of the team.

Instead of bashing Brady and company for giving one away in Seattle, let's marvel at the fact that it happens so seldom. The Brady/Belichick era has been one of the most decorated and successful periods in league history. It may be difficult to remember that when Russell Wilson snatches away victory from them in the fourth quarter, but that's why careers and trends need to be evaluated based on the big picture, not isolated, out of character incidents.

By Andrew Mooney, Boston.com Correspondent

A road game in the NFL can be a daunting proposition. Facing a sufficiently amped home crowd, a team has to worry about things they normally wouldn't at home: snap counts, hearing play calls, and hey, what is that biker-looking gentleman doing with a blown-up picture of my family? Every NFL fan base would like to believe its savage passion is the most intense in the league, but there can't be 32 best home field advantages. To that end, I set about finding which team's home field gives them the largest boost.

First, I don't think it makes sense to attribute the effects of a team's home field advantage solely to the fans; you can't separate the noise they make from the environment in which they make it, the stadium. It's possible that fans in Seattle truly go bananas for the Seahawks, but the stadium's architecture (or even its PA system) likely contributes at least some part to pumping up the volume at CenturyLink Field. Similarly, Boston fans exhibit plenty of passion and noise at Celtics, Bruins, and Red Sox games. Why should they be so notoriously quiet for the Patriots, unless Gillette Stadium itself (or, I suppose, the commute there) muted them?

For that reason, I centered my analysis on which team's stadium—the combined effect of the building and the fans—produces the greatest home field advantage. Instead of using home and away win-loss records, I looked up teams' point differentials at home and on the road from the time they moved into their current stadium, using the ultra-handy Play Index at Pro Football Reference. (There is considerable evidence that point differentials provide a more reliable measure of team quality than do simple wins and losses.)

Next, I calculated the difference between each team's point differential at home and on the road, then divided that by the number of seasons the team has occupied its stadium. This gave me the average number of additional points per season provided by a team's home field advantage. The Chargers have played in what is now Qualcomm Stadium since 1967, but assuming the home field advantage effect it affords has been constant over that time, its per-season average should be as valid as that of Philadelphia's Lincoln Financial Field, which began hosting NFL play in 2003. Finally, in order to ensure that I had a sufficient sample of games, I omitted teams who moved into a new stadium within the last five seasons, which included the Colts, the Jets, the Giants, and the Cowboys. I also counted the major overhaul to Soldier Field in 2003 as a “new” stadium, given the way it drastically altered the building's structure. My results are presented below.

This analysis bears out the conventional wisdom, at least at the extremes: Seattle enjoys the greatest home field advantage in the NFL, and Gillette Stadium ranks at the bottom of the league. Recently, the Ravens have borne out that M&T Bank Stadium is one of the hardest places to play in the country; they currently boast the NFL's longest home winning streak at 13 games.

It's surprising to see the Superdome so low, with its at least anecdotal status as one of the loudest stadiums in the NFL. The top ten features only three enclosed stadiums, and not necessarily the ones you expect—Ford Field, the Edward Jones Dome, and the Metrodome. Again, team quality over this time should have nothing to do with the final outcome, since we're comparing the same teams' home and away results.

So what's the deal with Gillette? As Chris Gasper noted a couple of years ago, the stadium itself is largely to blame: its open-ended architecture doesn't hold sound well, its location forces everyone to rush for the exits to avoid traffic once the result is assured, and it's one of the priciest stadiums in the NFL, especially when you factor in transportation costs to get there. Mr. Kraft may be happy with the money the place rakes in, but if he wanted to give an assist to his product on the field, he might consider phasing out the wine-and-cheese folks and drawing a slightly more populist crowd.

By Andrew Mooney, Boston.com Correspondent

Here’s a look inside the final sequence of Sunday’s wild 20-18 loss to the Cardinals. For my numbers, I’m making use of the Win Probability Calculator made available by advancednflstats.com. Specifically, I’ll be investigating the Patriots’ decision to settle for a 42-yard field goal at the end of regulation rather than attempting to improve their field position.

Game situation: 1st-and-15 at the Cardinals’ 23-yard line, 00:46 remaining in the 4th quarter

At this point, things are looking awfully fortunate for the Patriots. They just recovered a miracle fumble, and though a couple of Gronk malfunctions (a holding penalty that nullified a touchdown and a false start) may have cost them a sure victory, they still hold a 76 percent chance of winning the game. Their expected points value here is +3.56, meaning that teams score touchdowns more often than field goals in this scenario. However, the Pats elect to take the conservative route, with Tom Brady centering the ball in the middle of the field, sustaining a one-yard loss on the play. The clock winds down to seven seconds before Brady spikes the ball to stop it.

Game situation: 3rd-and-16 at the Cardinals’ 24-yard line, 00:07 remaining in the 4th quarter

The game will be decided by a Steven Gostkowski field goal attempt. According to a separate bit of research from Advanced NFL Stats, kickers convert from this distance just under 80 percent of the time. The Win Probability Calculator concurs with this figure, giving the Patriots a 78 percent chance of victory—still comfortably in their favor and slightly higher than before, but not significantly so. Gostkowski misses the ensuing field goal try, and the Patriots lose a shocker.

Now let’s imagine a scenario in which the Pats continue their drive toward the end zone. Forty-six seconds is enough time within which to run two plays, and taking the Patriots’ average yards per play for the game (5.0), let’s assume they gained an additional ten yards before spiking the ball with seven seconds left.

Hypothetical game situation: 4th-and-5 at the Cardinals’ 13-yard line, 00:07 remaining in the 4th quarter

This is chip shot distance. From this range, kickers successfully make field goals about 90 percent of the time, again coinciding with the Win Probability Calculator’s figures, which give the Patriots an 86 percent chance of winning—a better outcome than the scenario in which they ultimately placed themselves.

The main takeaway here is not that the Patriots sacrificed exactly 0.08 points of Win Probability by playing it safe, but rather that a 42-yard field goal is far from a sure thing. Why settle for it when you have an incredibly competent quarterback—one with a miniscule 2.1 percent career interception rate, I might add—who’s more than capable of pushing the result closer to “sure thing”? A slide into the middle of the field is hardly the optimal use of Brady’s talents.

By Andrew Mooney, Boston.com Correspondent

For the last decade, the beginning of a new Patriots season has brought with it the realistic prospect of a Super Bowl in a few months’ time. Much of the reason for this—and one of the only things remaining from those teams ten years ago—is, of course, Tom Brady.

You wouldn’t know it by his performance, but Tom Brady is 35 years old, and that fact has led to a slightly different air surrounding this year’s Patriots. The most successful franchise of the new millennium finds itself face to face with mortality. Life after Brady is no longer some vaguely unpleasant event to be dealt with in the distant future, but a reality that needs to confronted sooner rather than later.

For the present, however, that sense of hopeful anticipation remains. The Patriots are heavily favored to win the division and look like the smart pick to once again emerge from the AFC playoffs; Tom Brady is under center, and all is right with the world. But for how many more seasons will that be true? As the grey hairs proliferate, when will Brady’s performance slip?

To get some idea of the answer to this question, it makes sense to look at Brady’s peers, of which, admittedly, there aren’t very many. I took a list of career comparables for Brady from pro-football-reference.com—players who had careers of “similar quality and shape” through their first ten seasons—and examined how the ends of their respective football careers played out, hoping to find a hint of when we might expect age to finally catch up with number 12. The list is composed of nine (mostly) legendary quarterbacks: Roger Staubach, Joe Montana, Boomer Esiason, Dan Marino, Dan Fouts, Brett Favre, Donovan McNabb (?), Trent Green (uh...), and Mark Brunell (eesh). I’ll dub those last three “worst case scenario.”

To measure each quarterback’s aging patterns, I calculated the percent change for each season from the previous year in a few key passing metrics: yards per passing attempt, adjusted yards per attempt (factors in touchdowns and interceptions), and quarterback rating (an imperfect statistic, but one that should reveal any significant underlying trends). If the quarterbacks suffered any sort of dropoff after ten years in the league, a trend of negative percent changes should be evident. Finally, I took note of playing time, as a low number of starts or games played wouldn’t be automatically obvious from looking at percentages.

As these quarterbacks entered their 11th, 12th, and 13th seasons, I found that, on average, their play didn’t suffer any noticeable decline. In fact, I didn’t find any pattern to suggest their performance differed at all; the fluctuation graphed below (up one year, down the next, then back up) looks more like the natural year-to-year variation in players’ statistics than any definite trend.

The reason I didn’t include any more seasons in this analysis is the same reason that the Patriots are likely to lose Brady, when he finally chooses to hang up the cleats: the number of players in my sample kept dropping as quarterbacks simply chose to retire. I can imagine that 14 years of NFL football isn’t exactly kind to the body, and it seems that, though the on-field product these quarterbacks delivered stayed at a reasonably high level, they felt the effort necessary to produce it wasn’t worth it anymore. Obviously, that wasn’t true for all nine of these players; I don’t think you’d hear anyone reminisce fondly about the golden twilight of McNabb’s or Green’s careers, but I also don’t think they’re the best comparables for Brady. I foresee the end of his career playing out like that of Roger Staubach or Brett Favre: capable from a talent standpoint of being a starting quarterback for another year, but physically unable or unwilling to bear the punishment of it.

The number of comparables used in this analysis is far too low to draw any firm conclusions, but it suggests that Brady will be just fine for at least the next few years. After that, however, anything is possible. He might decide that enough is enough and go out in fine form, or he could try to hang on past his expiration date, like Montana. Either way, there’s still time for Pats’ fans to appreciate what they have before it’s too late; as the 2011 Colts showed, there comes a time when the bill for prolonged dominance finally comes due.

By Andrew Mooney, Boston.com Correspondent

The most telling statistic from Sunday’s devastating loss to the New York Giants was the time of possession for each team: the Giants held the ball for 37:05, more than 14 minutes longer than the Patriots (22:55). Excluding their kneel-down to end the first half, the Giants’ drives averaged nine plays and 4:37 each, none of which netted fewer than two first downs.

Though early on, the Giants (average starting field position: NYG 25) were unable to convert their time-consuming drives into many points, they dominated the field position battle, severely disadvantaging Tom Brady and the Patriots offense (average starting field position: NE 16). Consider that the Patriots’ best field position to start a drive was their own 29-yard line, and it becomes clearer why they could only manage 17 points; it’s incredibly difficult to drive 80 and 90 yards on a consistent basis.

When the Giants’ drives stalled, punter Sam Weatherford — whose performance merited serious consideration for Super Bowl MVP — pinned the Patriots deep in their own territory with lethally precise kicks, three of which landed inside the 10-yard line. His first punt, a lob wedge that spun back to the New England six-yard line, contributed directly to the game’s first points, the safety forced by the Giants’ pass rush on the ensuing play.

The Giants exposed the true weakness of the Patriots’ “bend-but-don’t-break” defense; when they don’t force turnovers (which play an enormous role in swinging field position), the team becomes tremendously beatable, especially against teams with more than competent offenses. Ordinarily, Brady is able to counter the offensive output of the opposition, putting together his own long drives and effectively neutralizing the opponent’s field position advantage — the Patriots were second in the league in opponent starting field position this year — which makes the three possessions in which the Pats failed to register a first down all the more critical.

Yet, for all that, the Patriots were still in excellent position to win the game, their win probability as high as 80 percent with 4:40 left. But they failed to capitalize on two pivotal opportunities to extend their slim lead on their first two possessions of the fourth quarter: Brady’s underthrown bomb that was intercepted by Chase Blackburn at the Giants’ eight-yard line and Wes Welker’s drop at the Giants’ 20-yard line. Had either play been converted, it’s likely that the Lombardi Trophy would have returned to New England, and the Tom Brady-Joe Montana comparisons would have continued for the next seven months.

As it was, the Giants did convert their big chance, emerging from the shadow of their own end zone with a 38-yard heave from Eli Manning to Mario Manningham. In the words of Indianapolis native Kurt Vonnegut, “So it goes.”

By Andrew Mooney, Boston.com Correspondent

If you’ve paid attention to the interminable television babble that is this week’s Super Bowl coverage, you’ve probably been forced to confront the following tired cliché at least once: “Defense wins championships.” All season, we’ve heard that the Patriots’ shaky defense will ultimately prevent them from claiming the NFL’s top prize, or perhaps even coming close.

Well, here we are. Both the Patriots and the Green Bay Packers (combined record: 28-4) proved that, in the regular season, defense is hardly a necessity when it’s backed up by an elite offense.

But does this change when the stakes get raised? Perhaps, when the best teams face off in the most important contests, the ability to grind out a defensive victory is particularly valuable. Specifically, does defense win championship games? Using measures of defensive efficiency (DVOA) from Football Outsiders, I analyzed playoff and Super Bowl results from the last 20 years (1992-2011) to determine whether the Patriots are, in fact, doomed this coming Sunday.

I found little reason to fret. In the playoffs, teams with a ranking in defensive efficiency superior to their opponent won games about 53 percent of the time — almost exactly what we’d expect from random chance, given equal teams. The results were similar when limited to just the Super Bowl; just over half (11 out of 20) of the championship-winning teams possessed a better defense than the runners-up, again yielding an inconclusive result.

Further discrediting the adage is the fact that, in this sample of Super Bowls, teams were almost as likely to win while allowing the opposition to outgain them in total yardage. Eight times in the last 20 years, the Super Bowl champions allowed more yards than the runners-up in the big game, and in a few instances, it wasn’t close. On average, those eight teams were outgained by 78.25 yards, the most extreme example of which came in Super Bowl XXXVI; the St. Louis Rams accounted for 427 yards, yet were defeated by some no-name second-year quarterback whose offense only managed 267 yards.

So not only are the defenses’ established identities marginalized, but their in-game performance doesn’t even determine the outcome of the game. Except, that is, for one crucial component: turnovers. As I wrote a couple weeks ago, the occurrence of turnovers is generally unpredictable, but they play a large role in both scoring offense and scoring defense. Only once in the last 20 Super Bowls has a team won the championship despite losing the turnover battle: the Pittsburgh Steelers in Super Bowl XL, who defeated the Seahawks in one of the most contentiously officiated Super Bowls in recent memory.

Now, this analysis is not particularly rigorous; it does not consider, for instance, the quality of the teams’ offenses or the size of the disparity in defensive efficiency in each game. However, if defense truly were the decisive factor in winning a title, we’d expect to see more compelling results than the ones shown above. There’s just far too much variability in a one-game sample to make any one aspect of the game predictive of the final outcome (see Super Bowl XLII).

Even so, the Patriots are entering uncharted territory. Should they emerge victorious on Sunday, their 30th-ranked defense would be the worst to win a Super Bowl over the period I examined, displacing the 2006 Indianapolis Colts. If it hasn’t already been dispelled, the image of cornerback(!) Julian Edelman raising the Lombardi Trophy would retire this cliché for good.

By Andrew Mooney, Boston.com Correspondent

A few significant matchups ahead of Sunday’s AFC Championship game:

In last week’s divisional round, the Patriots’ passing attack was keyed — as it has been many times this year — by its pair of rampaging tight ends, Rob Gronkowski and Aaron Hernandez. The two were generally unstoppable all night, combining for 14 receptions, 200 receiving yards, 61 rushing yards, and four touchdowns.

Enter the Ravens, whose defense against opposing tight ends has been downright stingy. During the regular season, Baltimore allowed a mere 3.9 catches and 41.6 yards to tight ends on a per-game basis, ranking sixth and second in the league, respectively. This wasn’t a product of weak competition, either; the Ravens faced seven of the top 20 tight ends (as measured by receiving yards), and didn’t allow more than four receptions or 42 receiving yards to any of them.

But they haven’t faced two elite tight ends at the same time, and Gronkowski (first among tight ends in receiving yards and touchdowns) and Hernandez (fourth in both categories) are just that. Only the Jets, who boast arguably the league’s strongest secondary, have been able to hold the pair under 100 combined yards, and the two still teamed up for 9 catches and 87 yards. The Ravens might be able to neutralize one of them, but not both.

Advantage: Patriots

Another crucial battle will occur at the line of scrimmage when the Patriots have the ball. This may be where the game is decided; in their four losses, the Ravens have accumulated only five sacks, as compared to 43 sacks in their 12 regular season wins. And when given time to throw, a big day from Tom Brady is almost inevitable, no matter who’s in coverage.

But that’s easier said than done. The Ravens’ pass rush, led by Terrell Suggs (14.0 sacks in 2011), amassed 48 sacks during the regular season, third-best in the NFL. They’re well aware of how surgical Brady can be with the football, so expect a complex package of blitzes and coverage designed to disrupt Brady’s rhythm. Though the Patriots have been slightly above average (11th) in preventing sacks, I can’t foresee the offensive line keeping Brady’s jersey completely clean.

Advantage: Ravens (slightly)

The Patriots have faced only one other elite defensive unit like the Ravens (top five in both scoring and yards allowed) this year, and the results weren’t pretty. The Pittsburgh Steelers held the Pats’ offense to 17 points and just 213 total yards, both season lows.

This offensive futility actually had as much to do with the Patriots’ defense as the Steelers’ D. Pittsburgh possessed the ball for nearly 40 minutes — converting ten of 16 third downs — and ran 28 more plays than the Patriots. When they did get the ball, the Patriots couldn’t sustain any offensive pressure, going 3-for-10 on third downs.

Which brings me to my final matchup: the battle of the much-maligned Patriots’ defense and Ravens’ offense. We’ve heard all about the Pats’ struggles to, in the eloquent words of Bart Scott, “stop a nosebleed,” but Joe Flacco is, well, Joe Flacco. Flacco submitted a Tebow-like performance last week against the Texans, completing 14 of 27 passes for 176 yards, taking five sacks, and converting only four of 16 third downs.

If the Ravens follow the Steelers’ blueprint — possess the ball for large chunks of time, force Brady into some three-and-outs, get to the quarterback, and avoid turnovers — they’ll be representing the AFC in the Super Bowl. I just can’t see Flacco playing the type of game the Ravens require to win the game. As they proved last week, the Patriots are capable of stifling a below-average quarterback, even with the threat of a dangerous running game. Hello, Indy.

Advantage: Patriots

By Andrew Mooney, Boston.com Correspondent

The Patriots’ defense has extended the “bend-but-don’t-break” concept to its limit. Their mighty struggles to prevent opposing offenses from cruising up and down the field are well-documented, yet somehow they retain an average ranking (14th) in the one defensive category that truly counts: scoring defense. The bend-but-don’t-break defense is one that’s been ascribed to Bill Belichick at various times over the years, but it’s not clear that it’s a sustainable strategy; after all, how could a defense that’s bad on 80 percent of the field perform consistently better in the last one-fifth?

Over at Smart Football, analysis conducted by Chase Stuart may have uncovered the reasons for this discrepancy. Interestingly, the bend-but-don't-break defense might be best explained by another familiar cliché: "the best defense is a good offense."

Because of a great offense and a good punting unit, the Patriots defense is rarely placed in a bad situation. New England rarely turns the ball over (third fewest in the league) and gains so many yards (2nd most) that they’re not giving the opponent the ball in a position to score. In fact, New England’s opponents have the 2nd worst average starting drive position of any team in the league (#1 is San Francisco, a team that seems to have been teleported straight from the ’70s) — the 24-yard-line.

The offense’s influence on the D’s effectiveness is not limited to the ground they chew up, but also, just as valuable, the time. Opposing offenses don’t score as much as might be expected against the Pats in part because, with less clock with which to work, they simply get the ball fewer times.

Because New England goes on many long drives on offense and allows long drives on defense, New England’s defense has faced the 6th fewest drives against this year (and the 4th fewest drives on offense). The Patriots have allowed 38 yards per drive (most in the league by over two yards) and 1.91 points per drive, 23rd best. Points per drive allowed excludes non-offensive touchdowns, so a 23rd-place ranking in points per drive allowed is a better measures of New England’s defense than their 14th-place ranking in points allowed.

I covered recently that a defense’s ability to force turnovers is essentially random, but this also implies that we would expect more to occur as the number of plays increases (larger sample size). And as the Patriots’ defense has faced the fourth-most passing attempts this year — due largely to Tom Brady’s ability to turn games into shootouts, while limiting his interceptions — it’s no surprise the unit ranks 2nd in the league in interceptions forced.

New England’s often playing with a lead, which forces their opponents into riskier tactics, which explains why the Pats are [3rd] in the league in turnovers forced despite not having much individual talent on defense. And, of course, every turnover forced is a drive that does not require any more defense, and the Patriots are 3rd in turnovers forced per drive.

But what about the most important aspect of the bend-but-don’t-break defense: the “don’t break,” or red zone defense? Is the Patriots’ ability to stop teams before reaching the end zone a product of purposeful defensive scheming or the difficult starting positions in which they place opposing offenses?

As is usually the case, the truth lies somewhere in the middle. The Patriots’ red zone efficiency on defense has remained remarkably consistent over the last five years; their percentage of touchdowns allowed per red zone trip has held relatively firm around 56 percent.

The regularity of this number leads me to believe their red zone efficiency is indeed the result of a repeatable defensive strategy. Play your safeties deep, and you won’t get burned deep, but you’ll allow a ton of yards — until the red zone, that is, when those safeties get forced up by the goal line and involve themselves in the play.

The only problem is that the Pats’ red zone defense is not particularly good, consistently well below the league average. In reality, the Patriots are no better at “not breaking” than the majority of NFL teams. If the defense can’t take all the credit, then, it’s fair to acknowledge the Pats’ offense for helping to keep the opposition off the scoreboard. Bad field position, limited possessions, and a victory in the turnover battle can all make a defense appear much stiffer than it really is.

Luckily, the Patriots will not be exposed by an elite offense this postseason unless they reach the Super Bowl; opposing quarterbacks in the AFC playoffs include two guys still learning the position (Tim Tebow and T.J. Yates) and the mighty Joe Flacco. But looming NFC foes look much more frightening. It may be at the hands of Drew Brees or Aaron Rodgers that the Patriots’ bending limbo act on defense finally collapses upon itself.

By Andrew Mooney and Ben Blatt, Boston.com Correspondents

Bill Belichick and the postgame podium are notorious for being a lethally boring tandem. Week after week, the Patriots' coach stymies reporters with concise non-answers, vague summaries, and robotic praise of players, giving them nothing of substance to write about his team.

Belichick's public persona may be as inscrutable as his game plans, but that doesn't mean we can't try to break it down, as he would do to an opposing offense. To get a more in-depth look at the man, we analyzed Belichick's speaking patterns from the transcripts of his postgame press conferences this season and, for purposes of comparison, gave the same treatment to the more media-friendly Tom Brady, Belichick's quarterback.

We started by looking at how long each of them is willing to stand in front of the podium. There's actually not much variation between the two; Belichick answers questions with an average of 72 words, whereas Brady's answers average 60 words. However, the difference in the two personalities becomes apparent when we split these averages up by wins and losses. Brady's answer length barely changes; after a win, his average response length is 59 words, as compared to 62 words following a loss. Belichick is quite a bit less eager to talk to the media after a defeat. Though he averages 82-word answers following a win, he cuts his responses to an average of 25 words after a loss, less than one-third of his post-victory average.

We also examined the word frequencies for each speaker. Of course, the most commonly used words were ordinary ones like 'a', 'the', and 'it'. It is more interesting to look at which words had the largest difference in frequency between the two (i.e. which words had the greatest ratio difference for the two speakers). Below are the top ten words used most commonly by Belichick and Brady in proportion to one another. (Note: words used less than three times by any one of the speakers were not counted.)

Unsurprisingly, the results for Belichick include a list of profoundly uninteresting words, matching his style at the podium. And Brady, though his list includes a few more syllables, isn't conveying much more meaning; "We gave them another chance" tells us just as much (or little) as "We didn't take advantage of our opportunities."

We also wondered if it was possible to determine the speaker at each press conference just by the style of their speech, examining the frequency of the various words used. Statistician Frederick Mostellar famously used this method to ascertain if Alexander Hamilton, James Madison, or John Jay was the author of the articles of the anonymously published Federalist Papers. At HSAC, we also used it last year to determine the authorship of sports articles, based on individual writing styles -- for those interested, an explanation of the math involved here can also be found in this article. Using these same probabilistic methods, we randomly selected four post-game press conferences (Brady's and Belichick's post-game press conferences from last season's victory over the Packers and loss to the Browns) and tested the question: can a set of equations detect the true speaker?

The answer is a solid affirmative. The model works by producing a probability that a given set of words belongs to either Brady or Belichick, given the speaking patterns analyzed above. In all four cases, the generated probabilities were lopsided (more than 99.9 percent in favor of one speaker), and in all four cases the model was correct in identifying the true speaker.

All in all, this underscores just how little can actually be learned from sports interviews, given the short, recycled nature of their content. Personally, we appreciate Belichick's approach. The Hoodie has no intention of giving comprehensive answers to anyone's questions, but at least he doesn't pretend to do so by dragging out the same timeworn clichés heard all over the sporting world. His concern is winning football games, and the less other people know about how he does it, the better.

By Andrew Mooney, Boston.com Correspondent

You’d have to examine this year’s GOP field to find a frontrunner inspiring this much doubt. For the second consecutive year, the Patriots have earned the AFC’s No. 1 seed, and the overwhelming fear is that, for the third consecutive year, the Pats will bow out in their first postseason contest.

The cause of this agonizing is unquestionably the Patriots’ defense, which just finished a campaign in which it allowed more yards (6,577) than any team ever to make the playoffs. If, for some reason, Tom Brady and the offense should falter in concocting one of the blistering comebacks we’ve seen the past few Sundays, the Pats may be spending their February weekends on the golf course rather than the gridiron.

Burdened with this historically porous defense, the Patriots’ Super Bowl hopes may hinge on a fickle, yet vital aspect of the game: the turnover battle. If you’re searching for a statistical trend underlying the Pats’ current eight-game winning streak, look no further: the Patriots have turned the ball over just three times on offense, while the defense has forced 20 turnovers over that same span, for a margin of plus-17, the best mark in the NFL over this period.

Recent history certainly testifies to the importance of the turnover battle. In two of the last three weeks, the Patriots’ early deficits were quickly reversed with the help of timely takeaways. Tim Tebow’s Broncos watched a 9-point lead evaporate into an 8-point deficit in the span of seven minutes after two fumbles in their own territory. Against the Dolphins, a third quarter fumble by quarterback Matt Moore positioned the Pats for their first touchdown and turned a game that had threatened to become ugly in the first half into a one-score game.

Just how crucial are turnovers to a game’s outcome? The chart below shows teams’ winning percentages in games with a given turnover margin, derived from a ten-year sample of games from 2002 to 2011. Assuming equally matched teams, forcing just one turnover more than the opposition boosts a team’s chances of victory 19 percent (50 to 69 percent). With a turnover margin of plus-2, a team’s chances of victory rise to 84 percent, and at plus-3 or above, they are virtually secured at 93 percent.

This would seem to favor the Patriots, whose plus-17 turnover margin this year ranks third in the NFL. However, analysis done by Football Outsiders suggests that, for as critical as turnovers are to a game’s result, they seem to be more attributable to random chance than to players’ skill.

A bit of clarification: if the ability to force turnovers were a skill, we’d expect to see it show up on a consistent basis, over multiple seasons. But the year-to-year correlations for turnovers, especially on defense, are actually quite weak, meaning the numbers from one year have very little ability to predict those of the next year. (For those interested, the exact figures can be found here.) Offensive turnovers are slightly more predictable, due largely to a quarterback’s ability to limit his interceptions, but on defense, takeaways essentially occur at random. In other words, the Patriots haven’t been “good” at forcing turnovers; they’ve been lucky.

The only real measure of control over turnovers lies with quarterbacks in avoiding interceptions, and fortunately for the Patriots, they have Tom Brady, whose 2.2 percent career interception rate is the third-lowest in NFL history. Even if the Pats’ luck runs out on defense, Brady gives them a better chance of maintaining a favorable turnover margin.

But it’s important to remember that much of the early-season panic surrounding this team — if it’s possible to panic after a 5-3 start — was a result of Brady’s uncharacteristically poor protection of the football. Brady’s four interceptions were almost solely to blame for the Patriots’ Week 3 loss to what has proved to be a considerably inferior Bills team, and the fact that two of them came on batted balls — which could just as easily have fallen harmlessly to the turf — underscores the role luck can play in swinging games.

Likewise, in the one-game, do-or-die format of the NFL playoffs, the best teams don’t always win. Flukes happen; sometimes Brady throws four picks. As cruel as it sounds, chance could easily determine the outcome of the Patriots’ playoff run. If the bounces continue to favor the Pats’ defense, they’ll be able to curb the scoring of the opposition (if not their total yards) and not overly disadvantage Brady and the offense. But an ill-timed deflection of a Brady pass could portend yet another early exit.

By Andrew Mooney, Boston.com Correspondent

As 2011 comes to a close, so too does the “Year of the Quarterback,” ESPN's year-long investigation into the position and all its intricacies. Unlike other manufactured, hype-driven initiatives, this one actually turned out to be quite prescient; it’s been a record-setting year for NFL quarterbacks, who have been heaving the pigskin at an unprecedented rate. Most notably, not one, but two quarterbacks — Tom Brady and Drew Brees — are challenging one of football’s longest-standing records: Dan Marino’s single-season passing mark of 5,084 yards, set in 1984.

Whether it’s Brees (4,780 yards, with two games remaining), Brady (4,897 yards), or both, the record will almost surely fall sometime in Week 17. The two have indeed been outstanding this season, but their assault on the record books is somewhat overshadowed by the excellence of their peers. Six quarterbacks — Brees, Brady, Aaron Rodgers, Matthew Stafford, Eli Manning, and Philip Rivers — have already broken, or are poised to break, the 4,500 yard threshold, a feat accomplished just 22 times in history, and no more than three times in the same year.

So unless this year’s passing explosion was an anomaly (and I doubt it was), Marino’s mark will soon be no more than a curiosity of history. But as with baseball, it’s impossible to truly understand the significance of such a record without examining the context in which it took place. When Babe Ruth hit 60 home runs in 1927, only one other player in baseball hit more than 30. When Roger Maris broke that record in 1961, 12 others joined him with 30 or more homers. In some cases, relative performance can tell us some things that absolute numbers cannot.

Likewise, what distinguishes Marino’s record is the era in which it was set.

The mid-80’s is not considered a particularly high-flying era in the NFL, and the numbers back that up. Teams passed for, on average, 228.2 yards/game in Marino’s record-breaking season; in 2011, that number is up to 243.3.

Marino did not have the benefit of a rulebook that considered coughing on an open receiver a flag-worthy offense. In 1984, defensive backs could still maul and chuck receivers, particularly near the line of scrimmage. On a related note, Marino was one of only three players to throw for 4,500 yards in a season over the span of his career (Drew Bledsoe and Warren Moon were the other two). If he managed 5,000 yards in 1984, what might he accomplish in 2011?

To adjust for the difference in offensive environments, I’ll rate how the performances of these three quarterbacks — Brees, Brady, and Marino — compare to the league average during their respective (potentially) record-setting seasons. In other words, the question no longer concerns exactly how many yards they amassed, but how much better each quarterback performed than his peers. To better account for injuries, platoons, or other circumstances that caused quarterbacks to play less than a full season, I’ll use team passing yards to calculate the average in each year.

With this approach, Marino’s ’84 season looks even more prolific. His 5,084 passing yards were 2.42 standard deviations (a measure of the extent to which points in a data set differ from the mean) greater than the league average total. Relative to their peers, Brees’ (projected through Week 16) and Brady’s numbers do not look quite as spectacular; their passing yards are, respectively, 2.36 and 2.04 standard deviations above the ’11 league average.

Jargon aside, what this means is that, for his time, Dan Marino in 1984 put up more impressive numbers than those soon to pass him in 2011. Had a quarterback this season exceeded his peers to the degree that Marino did in ‘84 (2.42 standard deviations above the mean), he would have thrown for just over 5,150 yards — with one game still left to play.

This is not a perfect way to compare eras, as it assumes that the average talent level of NFL quarterbacks has remained generally static over the last 25 years, but it highlights the fact that Marino’s record deserves some recognition before it fades into the obscurity of second (or, more likely, third) place. Simply transporting Marino 25 seasons into the future wouldn’t automatically make him a better player, of course, but it’s safe to assume that, in the modern-day NFL, Marino’s ’84 season would look a lot more untouchable.

By Andrew Mooney, Boston.com Correspondent

Enough exclamation points will be printed in the coming days regarding Tim Tebow's latest escape act, so I thought I'd let the numbers do the talking. Below is the wildly fluctuating win probability graph from advancednflstats.com, covering the entirety of the Broncos' ridiculous 13-10 overtime victory.

The biggest single-play swings (+/- refers to Denver's chances of victory):

+0.38 WP -- Matt Prater's game-tying 59-yard field goal at the end of regulation. I actually think this drastically understates Prater's chances of making the field goal; a win probability of 0.01 at the start of the play seems far too low. Conversion rates from this distance hover between 35 and 40 percent, and may be even higher for a kicker with a leg like Prater's. Regardless, it was still arguably the game's largest turning point and most unlikely play -- until the next one on this list.

+0.35 WP -- Marion Barber's fumble in overtime, derailing the Bears' potential game-winning drive. By the way, Barber had lost exactly five fumbles in 1,298 career touches entering this week, a rate of 0.39 percent.

+0.29 WP -- Matt Prater's 51-yard, game-winning field goal in overtime, extinguishing whatever doubt remained in the football world that this sport makes sense.

-0.23 WP -- Tebow's scramble for no gain on 3rd and 10, forcing the Broncos into Prater's desperation 59-yard attempt with 14 seconds left in regulation.

+0.18 WP -- Tebow's 19-yard pass to Matt Willis with 30 seconds left in regulation that advanced the Broncos from their own 40-yard line to the Chicago 41, putting Denver within Prater's range.

For all his late heroics, remember that, at one point, Tebow was 3-for-16 on his passing attempts; that is, until the Bears decided to extend Tebowmania another week by playing an extreme prevent defense, with two deep safeties about 30 yards from the line of scrimmage. Tebow took what they gave him, completed 18 of his next 24 passes, and set the stage for a crazy spectacle next weekend in Denver.

As great as I think the disparity in overall team quality is between the Patriots and the Broncos, the head-to-head matchup may be much closer than expected. Denver's problem has been scoring, especially in the first half; New England's problem has been stopping anyone from scoring. If the offense sputters early, with the Broncos' fierce, young pass rush in Tom Brady's face, the Pats may find themselves face-to-face with the prospect of more late-game Tebow magic -- and, as we've seen, that's when you can throw team quality and win probabilities right out the window.

By Andrew Mooney and Chris Bruce, Boston.com Correspondents

Week 15 looms large among the games remaining on the Patriots’ schedule –– not because of its playoff implications or a meeting with a heated rival, but because New England fans will get an up-close look at the NFL’s biggest sideshow: the Tim Tebow-led Denver Broncos.

Tebowmania has inspired everything from “Jesus” jerseys to a ubiquitous signature pose. What it hasn’t inspired in anyone who has watched Tebow play is confidence that he can complete a pass 10 yards downfield.

As a traditional NFL quarterback, Tim Tebow looks lost. Take his first two starts this year: unable to make the necessary reads in time, Tebow lingered in the pocket interminably, drawing 14 sacks in 80 dropbacks. His 44.8 completion percentage ranks last in the NFL for quarterbacks with more than 100 passing attempts. His own coach, as justification for overhauling the Broncos’ offensive schemes, admitted “he’d be screwed” in a regular offense.

But Tim Tebow is not a traditional NFL quarterback, so it doesn’t make sense to evaluate him like one. Pure passing statistics won’t tell the whole story. To better assess the respective contributions of Tebow and the quarterback he replaced, Kyle Orton, we’ll use Expected Points Added (EPA), a statistic that records the result of each play –– given down, distance, and yard-line –– and converts it to a point value. (A more thorough explanation can be found here, at advancednflstats.com).

Without applying context to the numbers, the insertion of Tebow has actually hurt the Broncos –– his EPA this season (-13.4) is 13 points lower than that of Orton (-0.3). This includes his contributions on the ground; Tebow’s inadequacies in the passing game have been so pronounced that his overall statistics are still worse than Orton’s, despite also being the team’s second-leading rusher.

Yet inexplicably, the Broncos are 4-1 with Tebow at the helm. In fact, correcting for strength of schedule with Jeff Sagarin’s team ratings, the Broncos would have been expected to win, on average, only 1.9 of Tebow’s five starts. By what sort of heavenly magic has Tebow been able to conjure up those additional 2.1 wins?

Tebow has saved his best for the biggest moments. By examining his Win Probability Added (WPA), which measures the effect of each play on the odds of victory of a player’s team (much the same as EPA, except that it also factors in score and time remaining), we can see that his value has been greater than what his raw numbers indicate. Much of Tebow’s production –– his game-winning drives against the Dolphins and Jets, for example –– have occurred with the game’s result largely in doubt, when individual plays have a larger impact on a team’s chances of victory. As a consequence, his WPA, though still negative, is much higher than his EPA (-0.06) and also exceeds Orton’s WPA (-0.45). So while, on the whole, Tebow has contributed fewer expected points than Orton, the points he has generated have come at the most important times, making up for his otherwise inferior performance.

How long can Tebow’s remarkable stretch continue? As illustrated in the graph below, which plots the EPA and WPA for each season played by all NFL quarterbacks currently in the league, there is a strongly linear relationship between the two metrics. Over the course of a season, players like Tebow, who possess significant disparities between their EPA and WPA, tend to regress toward their expected means along this line, implying that Tebow’s current streak is unsustainable for any extended period of time.

It’s possible that the Broncos keep up their winning ways under Tebow –– he’ll just need to play a lot better. If he continues to rely on a series of coordinated miracles to bail him out, his record won’t look so spectacular in the long run.

Thus far, Tebow has certainly exhibited a knack for excelling in crunch time, but to overlook his flaws and simply anoint him as a “winner” is a drastic oversimplification. Excepting divine intervention, nothing related to Tim Tebow bears responsibility for the crucial plays that allowed for his late-game heroics on multiple occasions: the onside kick recovery in Week 7 against the Dolphins, or Andre Goodman’s pick-six against the Jets.

NFL defenses are simply too savvy to be bested by a gimmicky, one-dimensional offense like the read option for long (anyone seen the Wildcat lately?). If Tebow hopes to remain the Broncos' quarterback of the future, he’d better start watching some old Elway game film. Pretty soon, his guardian angel is going to need a vacation.

By Andrew Mooney, Boston.com Correspondent

Pack away your foam fingers, head south for the winter, and tell Pat Patriot he’s free to spend his weekends however he pleases. According to a CSNNE interview with Kerry Byrne from Cold Hard Football Facts, it is “statistically impossible” for the Patriots to win a Super Bowl if their defense continues to play at their current substandard level. Good season, boys – we gave it our best.

Before we proceed, a word on hyperbole: a statistical impossibility implies that the probability of an event’s occurrence is zero – as in, under no circumstances could this event occur on planet Earth. The probability of the Rams playing the Dolphins in this year’s Super Bowl is not zero. The probability of John Lackey pitching a perfect game with his left arm is not zero.

All quibbling aside, the Patriots’ defense has certainly been a cause for concern, ranking last in the NFL in yards per play and yards per game, though, they do maintain a solidly average 16th in scoring defense. Is Byrne on the right track? How unlikely is it for a team with a defense as porous as the Patriots’ to make a deep run in the playoffs?

Here’s a look at all the Super Bowl champions since 1992, plotted according to their season ranks in offensive and defensive efficiency – measures which record the results of every play during a season, adjust for strength of schedule, and compare the figures to those of a league average team – taken from footballoutsiders.com. Though the sample size here is too small to reveal any robust statistical conclusions, the plot can yield a few practically significant insights.

It’s difficult to win a championship without an elite unit on at least one side of the ball. All of these teams except two possessed either a top-5 offense or a top-5 defense. Yet it is possible to win a Super Bowl when one of these facets is well below average. The 2006 Colts won it all despite a 25th-ranked defense, while the 2000 Ravens took home their rings on the strength of Trent Dilfer and a mighty 23rd-ranked offense. However, only three teams managed the feat with a defense ranking in the lower half of the league.

In Byrne’s defense, none of these teams’ defenses ranked as low as New England’s, currently 28th in defensive efficiency. But remember that the Pats are only six games into the season. The extreme degree of their poor performance may simply be an aberration that will diminish as the season wears on, regressing toward more moderate results.

The Pats meet the criterion of one exceptional unit; their offense is tops in the NFL in terms of efficiency. Even if the defense doesn’t improve significantly, the ’06 Colts provide a good parallel – their defense rated about one standard deviation below the league mean efficiency that season, approximately the same position the Patriots find themselves in now. And in each case, the offense usually had something to say about the final result.

So while a poor defense hampers a team’s chances at a Super Bowl, it does not render them nil. It’s certainly statistically possible that the Pats win it all, even with a defense playing this poorly. In fact, if you asked the folks offshore, or in Vegas, they’d tell you it’s not much of a long shot – the Pats are 4/1 favorites to win the Super Bowl on sportsbook.com.

There’s just no quantifiable measure of the likelihood of winning a championship, so categorical predictions don’t carry much weight. In the playoffs, anything can happen. It would have been much more justified to talk about the impossibility of the 2001 Patriots winning a title after Tom Brady stepped into the huddle for the first time – they weren’t among the best in the league on offense or defense. Let’s just not talk about the other outlier on the plot: the ’07 New York Giants.

By Andrew Mooney, Boston.com Correspondent

Sunday was vintage Tom Brady. Two minute drill, down by three, the game on the line. And for the 32d time, he delivered a fourth quarter comeback victory.

Yet, had the previous two Cowboys possessions played out differently, Brady may never have had that chance. As Chad Finn pointed out on Monday, the choices of Dallas' coaching staff down the stretch "proved ripe for second guessing." Did the Cowboys’ conservative decision-making swing the game in the Pats’ favor?

With about 5:35 left in Sunday’s fourth quarter, the Cowboys faced the first of two critical decisions. Staring at 4th and goal on the New England 8 yard line, the safe choice was to kick the field goal, though it would mean giving the ball back to Tom Brady, up by only three. A Cowboys touchdown in this situation certainly would have put severe pressure on the Pats, who had managed only one touchdown the entire game.

One way to decide the optimal course is to compare the Expected Points (EP) of each decision, here measured according to the system compiled by advancednflstats.com. Every yard line on the field has an Expected Point value: the amount of points, on average, a team can expect to gain, possessing a 1st and 10 at that spot. For example, having a first down at midfield is worth two EP; in other words, averaging all the points scored on drives with a first down at the 50, a team can expect to score two points for each such trip.

On 4th and goal from the eight, the Cowboys’ probability of scoring a touchdown was about 30 percent, and the probability of converting a field goal was approximately 96 percent. Failing to convert either, and giving the Patriots the ball around their own 10, would yield an EP of -0.2; the Patriots, from that field position, can expect to score 0.2 points on average. We can find the total EP value of each decision by multiplying the probability of each outcome by its EP and adding them together.

(Note: under the Expected Point system, a touchdown (and extra point) is not actually worth seven points. The scoring team must also kick off to the opposition, giving them, in turn, about 0.4 Expected Points, which is the value of a possession beginning at their own 20 yard line.)

Aggressive Strategy
(Probability of TD)(EP value of TD) + (Probability of no TD)(EP value of no TD)

.30 * 6.6 + (1-.30) * (-0.2) = 1.84 EP

Conservative Strategy
(Probability of FG)(EP value of FG) + (Probability of no FG)(EP value of no FG)

.96 * 2.6 + (1-.96) * (-0.2) = 2.49 EP

Since, in a tie game, it is in the team’s best interest to maximize Expected Points, Cowboys head coach Jason Garrett made the right decision to kick the field goal.

The results seemed to vindicate Garrett, as the Cowboys’ defense again stifled Brady and the Pats, forcing a 3-and-out and a Zoltan Mesko punt. Less than four minutes remained on the clock, and Garrett once more had to make a game-changing choice: would he open up the offense and try to ice the game with a first down via the pass, or would he resign himself to three simple running plays and a punt, forcing the Pats to use their remaining timeouts?

This one is a little trickier to calculate, requiring a few rough estimates. Considering the outcome of this choice likely would determine the result of the game, it's better to use Win Probability (WP) here – the percent chance a team has of victory, given a certain game situation.

For a series beginning with a 1st and 10, a team has a 66 percent chance of earning another first down, assuming they use their regular offense, and not the “play it safe” approach of three runs. A first down for the Cowboys at this stage in the game forces the Patriots to use all their remaining timeouts and, in the best case scenario, the Pats get the ball back deep in their own territory with less than a minute remaining. For simplicity’s sake, let’s put the Cowboys' Win Probability in this case at 90 percent.

In running the ball three times, the Cowboys would essentially give up the chance of gaining a first down, especially against a defense that’s expecting the run. But the probability of a first down is not zero; there’s always a chance a running back could break a tackle and move the chains. Let’s put the chance of a first down here at 15 percent.

Now we need to figure out the Cowboys' Win Probability if they fail to get the first down – the outcome that was actually observed. According to this graph provided by advancednflstats.com, the Cowboys' chances of victory upon punting the ball back to the Patriots with just over 2:30 on the clock stood at 82 percent.

However, this estimate is based on a league-average opposing offense and quarterback, and Brady is anything but average. In reality, the Cowboys' Win Probability was much lower – let’s say, 65 percent. Had the Cowboys attempted to pass for the first down and thrown an incompletion, their Win Probability would be lower still – perhaps another five percent – as the Patriots would have held an additional timeout.

Now that we have all the appropriate probabilities, or at least reasonable proxies for them, we can calculate the relative merit of each strategy.

Aggressive Strategy
(Probability of first down)(WP) + (Probability of no first down)(WP)

.66 * .90 + (1-.66) * .60 = .80 WP

Conservative Strategy
(Probability of first down)(WP) + (Probability of no first down)(WP)

.15 * .90 + (1-.15) * .65 = .69 WP

Again, this second series of calculations is based heavily on estimates. Tweaking some of the percentages might give you a narrower margin, but, for practical purposes, the conclusion remains the same: the Cowboys made a mistake in playing it safe the second time. Their strategy failed to maximize their odds of winning, and they ultimately paid for it with a heart-breaking road defeat.

There's a growing body of evidence that NFL coaches are far too conservative in the manner in which they manage games: they shouldn't clam up in the fourth quarter, and they should attempt more 4th down conversions, for example. Yet many coaches fear the sort of public backlash Bill Belichick received after his infamous decision to go for it on 4th and 2 against the Colts in 2009, even though the statistics supported his call.

If you asked Belichick about that choice, he'd probably tell you he'd do it again, if he had the chance. Sure, sometimes it comes back to bite you, but you'll come out ahead in the long run if you play the percentages. In the future, Jason Garrett would do well to learn from his example.

By Andrew Mooney, Boston.com Correspondent

No matter what else might be occupying your time this fall, be sure to plant yourself firmly in front of a television on Sunday afternoons, popcorn at the ready. The Patriots' offense is putting on a show.

Nine-hundred-and-forty yards. That’s what Tom Brady has produced through the air in the first two weeks, shattering the NFL record of 854 set by Cam Newton only a few hours earlier. He’s been just as efficient as he has been prolific, ranking second in the league in completion percentage (71.6 percent) while leading the NFL in passing attempts. Not to mention that he’s accomplished these feats against solid defenses; the Chargers and Dolphins both ranked in the top half of the NFL in scoring defense and yards allowed in 2010.

He’s done it with a couple of old stalwarts at wide receiver. Wes Welker and Deion Branch have each totaled over 200 yards receiving, benefiting from a quasi-telepathic connection with their quarterback.

He’s done it with the liberal use of a pair of rhinoceros at tight end. Through two weeks, Aaron Hernandez and Rob Gronkowski have accounted for 337 yards and five touchdowns, including all three of Brady’s passing TDs against Miami San Diego. Brady has targeted the two on 31 of his 88 passing attempts; only Tony Romo of the Cowboys has thrown to his tight ends at a greater rate (26 out of 69 attempts). With Hernandez expected to miss 1-2 weeks with an MCL strain, Gronkowski could become an even bigger threat.

Brady has done it behind a quick-striking no-huddle attack that’s left defenses gassed, unable to substitute the appropriate personnel. The Patriots have used the no-huddle at least once on 55 percent of their drives (12 out of 22), producing six touchdowns and three field goals.

He’s done it against every type of look defenses can throw at him. According to data compiled by ESPN Stats and Information, the Dolphins blitzed Brady on 42 percent of the plays following the injury to Dan Koppen in Week 1; against those blitzes, Brady completed 71.4 percent of his passes and threw three touchdowns. Sunday, the Chargers took the opposite approach. On twelve of the Patriots’ pass plays, they rushed only three linemen, leaving eight men in coverage. Brady responded by completing 10 of 11 passes for 120 yards and a touchdown.

Though there’s almost no chance Brady will sustain his current 470 yards/game pace, there’s also no reason to believe his workload will lighten any time soon. “Establishing the run” is probably not a phrase found anywhere in the Patriots’ game plans; the offense will continue to do the heavy lifting through the air.

And as long as the Patriots’ defense continues to make opposing quarterbacks look like…well, Tom Brady, you can bet we’ll be seeing plenty of shootouts. Sure, they made the plays when it mattered Sunday -- Vince Wilfork’s interception and the goal-line stand come to mind -- but a defense can’t consistently depend on timely turnovers to bail them out, especially when they allow opposing offenses to move the ball up and down the field at will.

So plan accordingly, folks. The next screening of the Tom Brady Show will be held in Buffalo, Sunday at 1 p.m. The Bills’ D allowed 35 points last week to the great Jason Campbell and the Oakland Raiders. Sounds like a blockbuster to me.

By Andrew Mooney, Boston.com Correspondent

There was one key point lost in the general hand-wringing that followed the Patriots’ lackluster showing this preseason: it was the preseason. These are the games during which you fall asleep in the second quarter, not pull your hair out over a blown coverage. Granted, there’s nothing encouraging to be taken from a beating like the one received at the hands of the Detroit Lions, but there isn’t much to lament, either. Preseason results have almost no correlation to regular season success.

To support this assertion, I performed a regression analysis, which shows the extent that one or more variables (in this case, various measures of preseason success) predict another dependent variable (regular season win total). If preseason performance reflects anything meaningful about the true quality of a team, then this performance should correlate with how the team ultimately fares in the regular season, measured in wins.

First, I collected data from the past five preseasons and tested the significance of a number of variables measuring both total offense and total defense—yards/game, points/game, yards/play, penalties, penalty yards, total points, turnover margin, and third down percentage. I also included scoring margin and, to better gauge the effect of teams’ starters on preseason games, first quarter and first half scoring margin.

For the statistically minded, my regression results can be found here. The numbers aside, the conclusion is simple: just one of the preseason variables showed any statistical significance in predicting regular season wins, and only one other was even close.

The significant variable? Offensive penalty yards—and its counterpart, offensive penalties, is close behind it. This is in line with a finding by Football Outsiders: teams in the regular season that commit more penalties on offense usually lose more games, though, oddly, the same doesn’t apply for defense. It’s interesting that offensive discipline is the only element remaining consistent from preseason to the regular season. Perhaps tendencies to hold and jump at the line can’t be broken when the games get serious.

The Pats rank eighth in offensive penalty yards this preseason, and they’re tied with the Broncos and Texans for ninth in offensive penalties. That’s somewhat disconcerting, given that they’ve been in the lower third of the league in those categories the last five seasons. However, this could be more a function of their abbreviated training camp than an accurate predictor of what’s to come. The offense may just have used the preseason as a time to shake off the rust of an extended offseason. Oh, and the team ninth in penalty yards, and seventh in penalties? The New York Jets.

You may wonder why I didn’t include wins in my regression; after all, might not preseason wins correlate with regular season wins? For my purposes, there aren’t enough outcomes possible -- the totals only range from 0-4 -- to draw any solid conclusions.

But a 2004 study conducted by twominutewarning.com was able to provide some insight on the subject. The group found that preseason wins matter most for teams with poor to average records the year before. Three or four wins in this case can herald a successful regular season. For the Pats, their findings were largely insignificant; good teams with at least two preseason wins were just as likely to stay successful as they were to decline.

So the chorus of panicked voices that have arisen over the past few weeks -- should we re-sign Randy Moss? Can we block anybody? Our two defensive tackles weigh what? -- can be largely ignored. It’s useless to read anything into games like these in which the two teams have drastically different levels of urgency in their preparation and execution. Come Monday night, Tom Brady and crew will finally be playing a meaningful game. You can bet they’ll treat it like one.

By Andrew Mooney, Boston.com Correspondent

Mark it down: Chad Ochocinco will have a comeback season in 2011. No, it’s not because of a voodoo brainwashing spell cast by Bill Belichick, nor will it be due to the homey wisdom of a hospitable fan, but rather the result of one seemingly obvious change: he’s now catching passes from Tom Brady.

I don’t know if you’ve noticed, but Tom Brady is pretty good at throwing footballs, a quality from which his receivers are happy to benefit. Let’s look at Brady’s past with other new additions at wideout to see what we might expect from his partnership with Ochocinco.

Since Brady became the starter in 2001, the Patriots have picked up seven starter-caliber wide receivers from other teams -- players who, at some point in their New England tenure, were among the top three in the receiving corps in single-season yards. Ochocinco figures to be the eighth to fit this classification.

Here’s how these signings performed the year before their arrival in New England and in the following season, as a Patriot.

Most of these receivers dealt with middling to below-average quarterback play prior to their tenure with the Pats, catching passes from the likes of Andrew Walter, Joey Harrington, and a graying Matt Hasselbeck. Ochocinco ran routes in 2010 for Carson Palmer. Though once considered a top-5 NFL quarterback, the increasingly disgruntled Palmer, who spent much of the last two years trying to force his way out of Cincinnati, can’t be considered too much of an upgrade on those listed above.

Not surprisingly, the new signings, as a group, improved considerably -- their receiving yards rose by 51 percent and their yards/game by 42 percent after teaming up with Brady.

What does this mean for Ochocinco? None of the Pats’ previous signings is a perfect analogue to him. At 33, he ranks as the oldest of this group, but he’s also the most productive coming into New England; his 831 yards and 59.4 yards/game last season are first and second, respectively, on this list. When applied to Ochocinco’s 2010 stats, the average increases above project to over 1,200 receiving yards in 2011, a level achieved only by Stanley Morgan, Randy Moss, and Wes Welker in Patriots’ history.

The best parallel available here is Moss. Upon joining the Patriots, both were declining superstars with a history of disciplinary issues whose best years were thought to be behind them. Moss, 30 years old when he joined the Patriots, rediscovered his status as an elite deep threat and put up numbers in 2007 that Ochocinco will probably never approach this year, given the presence of a Pro Bowl incumbent in Welker, with whom he’ll have to split catches, and the reality that he may indeed have lost a step. He’s not quite the home-run threat who rips off 70-plus-yard catches anymore, like he did in each of the ’05, ’06, and ’07 seasons.

But if Ochocinco’s still got any of his superstar form left in the tank, Brady will bring it out of him. According to Football Outsiders’ advanced metrics, he’s been the top quarterback in the NFL in each of his last three healthy seasons, and there’s no reason to think he’ll slow down now. He’s made the likes of Danny Woodhead and Julian Edelman into productive players, and he got 1,000 yards out of the tight end position last year from two rookies. We’ve seen what he can do with a legitimate superstar in Moss; if he can recreate even a fraction of that magic with Ochocinco, the Pats’ passing offense will look as destructive as Rex Ryan in a buffet line.