I wonder how much of the week 17 dropoff is caused by teams resting up before the playoffs, since that would disproportionately hit the best teams, which are usually the teams most likely to score points. Also, any way we could see a breakdown of special teams touchdowns by week? I think there are more kick/punt returns and blocked kicks in week 1-2 than other weeks.

#1: Wouldn't we expect both the offense and defense to be affected equally by that though? Is there any reason to think coaches would be quicker to rest offensive players than defensive?

Perhaps so. I would think QB and RB would be the first positions likely to be benched for resting/avoid-injury purposes.

Great! By mid-season the cowboys should score 50 points!

Could a part of the explanation be that the offenses need more time to gel together after the preseason, but are more dramatically affected by injuries as they pile up during the season? It could make for an offensive saddle point in the middle of the season.

Additionally, the NFL seems to avoid playing outdoor cold weather games, which is why the Broncos get a lot of early home games and a lot of late away games. I wouldn't think that cold weather would have a strong affect on gameday, though it might have an effect during the practice week.

The downward drift looks like injury to me, specifically to qb's since there is no equally important counterpart on defense--except Bob Sanders.

It looks from the graph like most of the effect is from dome teams, actually. Any chance of graphing the trendline for non-dome teams?

Are the Colts and Rams enough to skew the dome data quite a bit in favor of offense on their own?

Another myth slain by Aaron, and this one didn't require use of any "advanced metrics." Why don't Prisco or Clayton every write articles solving such simple questions?

#8 I'm sure that they'll write an 800 word essay about it in PFP 2008, so I'm glad that I got the cliff notes version first.

Maybe the drop has to do with the defenses getting ahead of the offense again as they have more film to study.

It seems clear to me there is less scoring the first 2 weeks of the year. After that it really sort of looks like random noise to be honest. If you fit a line to weeks 3-16 it seems like it would be nearly flat at around 21 points.

I'm with you Brian, I don't see any downward trend in the data. Remove week 17 from the indoor data and you see the same trend, slightly down weeks 1 and 2 flat weeks 3-16.

Agreed with points 11 & 12. The downward trend does not appear to have statisitcal significance. You've got a slightly slow start, some slowdown at week 17, and a lot of noise.

#3: Great! By mid-season the cowboys should score 50 points!

and Altanta should be getting to that lofty scoring plateau by then as well! Oh, you meant in a single game...

Aaron,
We're all asking:

Is there another thing in football that starts slightly behind, peaks at midseason, and trails off very slightly at the end.

(there's probably be a joke in there but I didn't find it.)

15: Theo...
It's week 14 to 16

Regarding the downward trend, note that Aaron did say "very slightly."

If you remove Week 1, 2, and 17, you'll get a slightly negative trend. I don't think it'd be insignificant statistically (it's being anchored by about 5 points out of 13 - that's not that statistically likely to happen from chance), but it definitely would be weak.

Is there any reason to believe that the "trendline" should be a second order polynomial? Why not third order, or fifth. Why not use a linear trendline? The overall scoring includes the dome scoring in it so of course it is influenced and probably looks something like the dome line at second order curve fit. This is a silly chart and there is no reason to look at those curves and see anything.

Re 18 Milo.
I see "week 1: don't go for Dome Team; week 5,6,7: go for Dome Team."

re: 18

Wouldn't a higher order polynomial be overfitting the data? Besides the main point of the plot is to help answer Mike's questions, the second of which specifically said "Are more points scored early or later in season and why?"

Note the range is only 3 points per game, or about one shootout per year.

A second order polynomial has to either go only up, go only down, go up and then come down, or go down and then come up. It's important to remember the limitations of these curve fits.

To me the data looks awfully scattered. Is there necessarily any trend at all, or is the week-by-week deviation from the average possibly due to random chance? I.e. are these trends statistically significant? Do we have enough data to say? I'm not a statistician...help please?

I'm too lazy to do the tests, but from eyeballing, if we start from a null hypothesis stating that week and score are unrelated:

Alt. hypothesis that scoring goes up during the season: not enough evidence to reject null

Alt. hypothesis that first two weeks of the season are atypical: good evidence for rejecting null (except it's a post-hoc test)

Alt. hypothesis that after the first two weeks, scoring trends either up or down: not enough evidence to reject null

(In case any of my stathead friends read this, yes, I'm being very loose with hypothesis statements here.)

Well since the games are played under the same rules each week, why wouldn't a least squares (linear) fit work the best. See the link where it shows scores going up in domes but relatively flat line for all games. If the dome scores were removed, the outside scores might even be flat or downward trending. This would indicate scores in general go up (see domes) but the trend is mitigated outdoors by weather.

What do consistent rules have to do with linearity?

Honestly, I don't see the sense in fitting any degree polynomial here. If you must join the dots, use a moving average or something.

Oh yeah, this might not need to be said, but the other reason to be careful with comparisons between the average and the dome average is that certain teams (in particular: Indy) are overrepresented in the dome stats.

What do consistent rules have to do with linearity?

Since the game's the same each week, the zeroth-order approximation should be flat, and the first-order change you would expect would be either linear or exponential with time (depending on whether the effect is additive or multiplicative).

If you must join the dots, use a moving average or something.

There's no real difference. A moving average requires that you define a timescale over which the fluctuations are minimal. A polynomial requires that you presuppose the number of nodes in the data.

Just draw a line by eye, fit it to a polynomial of order "number of direction changes + 1". If you don't really care, that's a decent simple trendline.

A moving average gives a much wider range of possible shapes, e.g. one that rises for a couple of weeks, then is horizontal. You can approximate this with a polynomial, but changing the number of points you smooth over in a MA doesn't affect the shape nearly as much as changing the degree of your polynomial.

Although now that I think about, the endpoint effects die off to quickly for the MA to be much use.

(This is the second FO thread I've commented on. In the other, I was defending linearity...)

I'll leave the math/stats analysis to those of you who know what you're talking about.

Some possible explanations:

- Close games tend to be lower-scoring. With every team at 0-0 and integrating new players, it might be feasible to imagine there are more early-season close games. The season-long average for games decided by 3 or fewer points is 1 in 4, and there were more than that this weekend past, all with low scores.

- Peak scoring occurs during the epoch of the bye weeks. Is there any correlation between increased scoring and two weeks of preparation?

Alas, I don't have time or energy to look up the data to (dis)prove these hypotheses.

For those who care, the uncertainty in each weekly average is around 0.7 points (I worked it out for week 1 and week 9 overs 2001-2006 seasons: 0.71 for week 1, 0.72 for week 9). Weekly average scores have a mean of 20.95 and an RMS of 0.76. So they appear to be normally distributed about the mean with an RMS corresponding to each measurement's uncertainty - exactly what you'd expect really.

Can there still be an effect where the lowest average scores cluster together at certain points? Possibly. But look at the distribution of scores above and below the mean on a week by week basis (+ is above mean, - is below mean):

--+++-++-+-++-+--

There's a fairly crude (but robust, and surprisingly sensitive) test using that sign data, but it is pretty obvious at a glance that it isn't going to show anything significant.

Eyeballing it, the only think that really stands out is that the three of the four most extreme values (furthest from the mean) are in the first 5 weeks. But that is a) not terribly significant on its own, and b) a post-hoc idea from looking at the data, not from a test devised before looking at the data.

So there is no significant evidence, to my mind, of anything other than random distribution of average scores around the mean, and no evidence of variation in time. No evidence of more points being scored early [i]or[/i] late.

I have to ask, had Aaron not put that chart up there and used just straight statistics in a chart or something, would people be more willing to accept the theory?

2: Yes, both offense and defense would be effected by resting starters, but the net result is less scoring. It's kind of a bad offense looks like good defense effect, nicely demonstrated by SF/ARI.
28: I think you're getting cause and effect mixed up. Low scoring games tend to be close, but close games don't necessarily tend to be low scoring. When a game isn't close, both teams will abandon efficient point-scoring strategy in exchange for desperation on the trailing team side and clock-killing on the leading team side. When a game is close, both teams are trying to score efficiently.

30: I'd say no, because the data as presented doesn't show a significant trend. How it's presented doesn't change the actual data. Aaron states scoring tails off, but the overall data shows the tail off is far lower the variance in the data. The data suggests that scoring week to week generally changes by less than a 1 point. Clearly there’s a heck of a lot more data than 2001-2006 and that data is clearly need to find a less than 1 point trend through the noise in the current presented data.

It seems to me natural to expect defense to have a slight edge at the end of the season. Offense is proactive, defense reactive. Offenses use a huge range of personnel sets with regularity, defenses as few as four (base, nickel, dime and goal line), and I would speculate that an offensive playbook would contain more different plays than a defensive one would. Offense, I think, is simply more complex than defense, and therefore it should not be surprising if players take longer to become comfortable in an offense than in a defense.

Bill Walsh reportedly (per The Blind Side) maintained that offense was primarily about scheme, defense about talent, which I think is a function of the same phenomenon I'm trying to describe here.

the isn't the first graph i've seen recently. did aaron get a nifty new graph program he wants to show off?