This is the distribution of when scoring plays occur in NFL (American football) games. Each type of scoring is represented by a colored layer whose height represents the instantaneous rate at which points were accumulated over the course of the average game. As such, the area of each layer is the total number of points accumulated, though an inset pie chart presents the breakdown of points by source over entire games more succinctly (overtime scores are included in this figure, but cropped from the time dependence for clarity as overtime is fairly rare in the NFL).
Excepting a brief peak in kickoffs returned for touchdowns, the rate of scoring ramps up over a few minutes at the start of each half and skyrockets at the end of each half, with field goals dramatically peaking in the final seconds. The increased in scoring in the final minutes is more dramatic at the end of the first half than at the end of the game (possibly because many games have effectively already been decided by that point and extremely aggressive play would risk injury for little purpose). Brief increases in scoring are seen immediately after the start of the second and fourth quarters and at the two minute warnings.
As any casual football watcher has probably guessed, safeties contribute vanishingly few points. Two-point conversions finally contribute notably in the fourth quarter.
More points are scored in the first half than the second (mean 11.06 vs. 10.66). However, the mean time to first score (among scoring halves) is slightly longer in the first half (For any team, 8.04 vs. 7:44, compared to just 6:10 in overtime. For a specific team, 12:03 vs. 11:26). Overall, the mean time to first score is 8:13. All Comparisons above are all statistically significant.
Filters can be used to look at a narrower range of seasons or at points scored by or against a specific team. Note that when studying a specific team, the number of games available decreases dramatically, particularly if considering a small number of seasons, which may result in greater volatility. That is to say, this is merely a representation of what has happened historically; be careful in trying to state trends when comparatively few games are available.
Scoring events were extracted from play-by-play data for 11 seasons (2002-2012). After filtering by game, events are binned in 5 second increments. If smoothing is enabled, a low pass filter with a cutoff frequency corresponding to 90 seconds is applied. Halves are smoothed independently, as the halves of an American football game have no continuity.
Aside from swapping sides, nothing changes at the start of the 2nd and 4th quarters, right? Every game is aligned to start a play at the same time, which accounts for the concentration of touchdown plays at exactly 15:00 and 45:00 (turn off smoothing to see this). But even binned over a few minutes, there is a slight increase in scoring at the start of the quarters. Effects of a few minutes rest, perhaps?
I spent way too much time rethinking my binning and filtering approaches on this one and I'm still not completely satisfied with it. Maybe I'll revisit it sometime in the future.