From the FanPosts Joel
Hello AP members and the inevitable crosstraffickers from MHR,
I put together this FanPost for a couple of reasons:
 First, I have seen several posts like this on both sites over the past few weeks which all suffered from similar mathematical errors, so I thought I would gen up my own
 Second, after my attempt to compose an inspirational video failed miserably, I figured I had better stick to the stats
 Third, this is Chiefs vs. Broncos! It's the "Irresistible Force" meets the "Immovable Object!" With a combined record of 171! In a game with obvious playoff implications! Why would we not overanalyze it?
Now the Disclaimer:
This post can tell you almost next to nothing. Continue reading for entertainment value only. What I can assure you is that my math is the best I can make it (which may or may not be "world class" but, hey, I get to write the title, right?) On the spectrum of Lies, Damn Lies, and Statistics, these numbers fall squarely in the "statistics" category. What's the great thing about statistics? When done properly, they tell you how accurate they are. In fact, that's whole point of proper econometrics  determining relevancy. What these numbers tell me is that their predictive power is just a little better than your chances of winning at black jack. Of course, I wouldn't have known that without running through the exercise. Bottom line: all of these numbers are real, yet do little to actually predict anything. But why should that stop us?
So here it is in all its glory: your completely meaningless, yet highly analyzed statistical prediction for the game!
The first thing I had to assess was which stats to look at. People have argued over Defensive and Offensive Yardage vs. Split Stats vs. Points Allowed. Here's what I know, the winner is determined by the score board, thus the ultimate goal is to predict the score. But how can we tell if yards are important or not? We look at correlation:
Yardage
KC’S OPP 
Pass Yards 
Rush Yards 
KCC Yards Allowed 
PTS vs. KCC 
DEN’S OPP 
Pass Yards Against 
Rush Yards Against 
Total Yards 
DEN PTS SCORED 
JAX 
157 
71 
228 
2 
BAL 
462 
65 
527 
49 
DAL 
298 
37 
335 
16 
NYG 
307 
107 
414 
41 
PHI 
201 
264 
465 
16 
OAK 
374 
164 
538 
37 
NYG 
217 
98 
315 
7 
PHI 
337 
141 
478 
52 
TEN 
247 
105 
352 
17 
DAL 
414 
103 
517 
51 
OAK 
216 
121 
337 
7 
JAX 
295 
112 
407 
35 
HOU 
271 
73 
344 
16 
IND 
386 
64 
450 
33 
CLE 
293 
57 
350 
17 
WAS 
354 
107 
461 
45 
BUF 
229 
241 
470 
13 
SDC 
313 
84 
397 
28 
CORR: 
0.77 
0.11 
0.59 

CORR: 
0.48 
0.17 
0.60 

This spreadsheet shows the correlation factor between each yardage stat and points scored on the two sides of the ball we care about: The Chiefs Defense and the Denver Offense. These correlation values actually increased by quite a bit after the DEN vs. SDC game Sunday Night, but they are still fairly weak. A "strong" correlation is 0.8 to 0.99. Moderate correlation is around 0.6 to 0.8. Anything below 0.6 is pretty weak for predicting results (In fact, most true statistical analysis won't report something as statistically significant until PValues reach 95%).
This tells me that I could go through the whole exercise of predicting the yardage of next Sunday's game, but even if we got the yardage perfect, we would be no better off predicting the score from that yardage than we would flipping a coin. But can correlation tell us everything? No, and the 0.6 correlation factor does show some relevance for the stat, so lets look at them side by side. Here are charts of the Chiefs Yards Allowed and Points Allowed sidebyside, and the Broncos Yards Gained and Points Scored sidebyside:
What do we notice? First, we notice that although the individual data points don't correlate very well, the trend lines look very similar. We also see that although the trend lines are both regressing to the norm, that regression is pretty low for both teams. If you look at the "R^2" values that are printed on each graph, they tell you what percentage of the graph is "explained" by the trend line.
For instance, in terms of KCC Points Allowed, the regression to the norm accounts for only 23.21% of the Chiefs trend from week to week. Roughly, it's like saying they are regressing a bit, but 77% of their game is not regressing. What does this mean? It means that again, looking at yardage won't tell us anything additional to what looking at points is going to tell us. It also tells us that we should expect the Chiefs to give up more points than usual, and the Broncos should score less than usual, but not by big margins, both teams are still really good  not exactly ground breaking stuff there, huh? So, enough with yardage.
Win Percent Added
A post at MHR claimed that the Broncos have a better chance of winning the game because, on average, they have spent more time at 100% WPA then the Chiefs. What that tells me is that when the Broncos lock up a game, they do it earlier than the Chiefs. Again, is that news given our contrasting styles of play? Not to me. So instead, I want to look at how much time during each game do the Chiefs have the odds in their favor, and compare that to the Denver numbers.
Therefore, I went to profootballreference.com and looked at the box scores for every Chiefs and Broncos games. I then counted WPA by quarter and broke it into three categories:
 Quarters where KCC/DEN had a "winning" probability, meaning that the WPA graph was above 50% for the entire quarter
 Quarters that were "indoubt," which are any quarters where the WPA line hovers right around the 50% mark or crosses from one team to the other during the quarter
 Quarters in which KCC/DEN were "losing," in that their WPA was never above 50% during the entire quarter
Here's what I found:
# of Qtrs WPA: 

CHIEFS 



BRONCOS 

OPPONENT 
Winning 
In Doubt 
Losing 
OPPONENT 
Winning 
In Doubt 
Losing 
vs. JAC 
4 


BAL 
2 
2 

vs. DAL 
2 
2 

NYG 
2 
2 

vs. PHI 
3 
1 

OAK 
4 


vs. NYG 
4 


PHI 
4 


vs. TEN 
2 
2 

DAL 
1 
3 

vs. OAK 
3 
1 

JAX 
4 


vs. HOU 
3 
1 

IND 
1 
1 
2 
vs. CLE 
4 


WAS 
3 
1 

vs. BUF 
1 
2 
1 
SDC 
4 


TOTAL: 
26 
9 
1 
TOTAL: 
25 
9 
2 
(You can look these numbers up yourself using this link here, and clicking on each game's "boxscore" to find the WPA graph)
Yes, the Chiefs have sustained a winning probability in exactly the same number of quarters as the Broncos, and have only had 1 "losing" quarter of football which came in the Buffalo game. Denver had 2 against Indy, including the most important quarter, the fourth. What does this tell us? It says that even though the Chiefs play in a lot of close games, they have rarely been "out" of a game  just like the Broncos. It also says that because the two teams are so alike, WPA won't help us differentiate them, or affect the score much. So, onto the points.
The Score
Now, several posts have tried to calculate the upcoming scores by looking at the percentage of points that each team scores above or below other teams, averaging those percentages, and then calculating points against each team. There is a problem with using percentages to do this, however. What's the problem? I'm glad you asked.
The problem is that you can't average the percentages because they are different units (now, we could use the Harmonic Means to average unlike numbers, but that would favor the large numbers, which in football are the statistical outliersmeaning that rare games, like our 2 point game against JAX would skew our numbers even more than using arithmetic means, but I digress...). But Bull, you ask, point are points...how are these different units?
"Points against Jacksonville" are not the same thing as "Points against Philly." Why? Mainly because they play different teams  it's the old, "who have you played?" dilemma. If every team in the NFL played every other team exactly once in every season, we wouldn't have that unit problem. Even so, we still couldn't use "percentages above or below mean" to define a team's overall relative strength because those units vary by team also. Seven extra points against JAX may not be the same as 7 extra points against PHI, even if their average scores are the same.
Wait, what? Why not? Look at it this way: Let's imagine that the points scored against JAX in all of their games has been: 20, 0, 20, 0, 15, 5, 15 and 5. Philly has allowed scores of 10, 10, 10, 10, 11, 9, 11 and 9. Both teams have an average of 10 points allowed. Now, the Chiefs score 17 against JAX and 17 against PHI. Which one is more "impressive?" Which one tells us more about the Chiefs?
Well, the extra 7 points against the Eagles is a more "unusual" feat then the ones against the Jags. Luckily, we can capture that "unusualness?" (yes, that's a word... if you can say it, it's a word) and fix our "units" issue with one thing: standard deviations. Stnd Devs give us a normalized error distribution around a determined mean. No matter what group of numbers you have, if you find the standard deviation, around 68% of your values will fall within one Stnd Dev of the average, about 95% of your values will fall within two Stnd Devs, and 99% of your values will fall within three standard deviations  regardless of the units or size of the sample. Presto, our problems are solved, it just takes more work.
To find our two teams' performance using standard deviations, we have to look at the points scored for and against each of our opponents against each of their opponents. We then find each team's mean and standard deviation for points scored and allowed, using all of the games except for the ones against KCC and DEN. If we used our own games in our calculation, it would induce an endogeneity problem, which basically means that we would be using our own performance to measure our own performance. We don't really want to do that. Once we have the Mean and Stnd Dev for each team, we look at the Chiefs' and Broncos' performance against them, calculating how many standard deviations above or below average we scored. We can then average our standard deviations across our 9 games to determine how well our Offenses and Defenses perform as compared to the mean. All of that is here in this table (scroll all the way to the right for the final values):
Opponent: 
JAX 

DAL 

PHI 

NYG 

TEN 

OAK 

HOU 

CLE 

BUF 

BAL 

COLTS 

WASH 

SDC 

Averages 
Stnd Devs 
Week 
PTS SCORED 
PTS ALLOWED 
PTS SCORED 
PTS ALLOWED 
PTS SCORED 
PTS ALLOWED 
PTS SCORED 
PTS ALLOWED 
PTS SCORED 
PTS ALLOWED 
PTS SCORED 
PTS ALLOWED 
PTS SCORED 
PTS ALLOWED 
PTS SCORED 
PTS ALLOWED 
PTS SCORED 
PTS ALLOWED 
PTS SCORED 
PTS ALLOWED 
PTS SCORED 
PTS ALLOWED 
PTS SCORED 
PTS ALLOWED 
PTS SCORED 
PTS ALLOWED 


1 


36 
31 
33 
27 
31 
36 
16 
9 
17 
21 
31 
28 
10 
23 
21 
23 


21 
17 
27 
33 
28 
31 


2 
9 
19 


30 
33 


24 
30 
19 
9 
30 
24 
6 
14 
24 
23 
14 
6 
20 
24 
20 
38 
33 
30 


3 
17 
45 
31 
7 


0 
38 
20 
17 


9 
30 
31 
27 
20 
27 
30 
9 
27 
7 
20 
27 
17 
20 


4 
3 
37 
21 
30 




38 
13 
14 
24 
20 
23 
17 
6 
23 
20 
20 
23 
37 
3 
24 
14 
30 
21 


5 
20 
34 


36 
21 
21 
36 


27 
17 
3 
34 
37 
24 
24 
37 
26 
23 
34 
28 


17 
27 


6 


31 
16 
31 
20 
21 
27 
13 
20 


13 
38 
17 
31 
24 
27 
17 
19 
9 
19 
16 
31 
19 
9 


7 
6 
24 
17 
3 
3 
17 
23 
7 
17 
31 




13 
31 
23 
21 
16 
19 


45 
41 
24 
6 


8 
10 
42 
30 
31 
7 
15 
15 
7 


21 
18 




17 
35 










9 


27 
23 
49 
20 


28 
21 
20 
49 
24 
27 
24 
18 


18 
24 
27 
24 
30 
24 
24 
30 


10 
29 
27 


27 
13 
24 
20 
27 
29 
20 
24 
24 
27 


10 
23 
20 
17 
8 
38 
27 
34 
20 
28 


AVERAGE 
13.43 
32.57 
27.57 
20.14 
27.00 
20.75 
19.29 
24.43 
22.88 
21.25 
19.71 
23.14 
19.25 
28.88 
19.38 
21.75 
20.67 
26.22 
20.13 
17.50 
22.88 
20.00 
26.13 
30.25 
23.56 
22.44 


STD DEV 
8.40 
8.89 
6.04 
10.86 
14.15 
6.10 
9.02 
12.45 
7.59 
7.66 
3.69 
11.58 
9.40 
4.70 
9.91 
8.12 
4.37 
5.69 
5.01 
6.24 
9.89 
10.54 
8.33 
8.00 
5.48 
8.81 


Against CHIEFS 
2 
28 
16 
17 
16 
26 
7 
31 
17 
26 
7 
24 
16 
17 
17 
23 
13 
23 










+/ DEF STND DEV 
1.36 

1.91 

0.78 

1.36 

0.77 

3.44 

0.35 

0.24 

1.75 









1.33 
0.93 
+/ OFF STND DEV 

0.51 

0.29 

0.86 

0.53 

0.62 

0.07 

2.53 

0.15 

0.57 








0.18 
0.95 
Against DEN 
19 
35 
48 
51 
20 
52 
23 
41 


21 
37 






27 
49 
39 
33 
21 
45 
24 
21.75 


+/ DEF STND DEV 
0.66 

3.38 

0.49 

0.41 



0.35 







1.37 

1.63 

0.62 

0.08 

0.75 
1.17 
+/ OFF STND DEV 

0.27 

2.84 

5.12 

1.33 



1.20 







5.04 

1.23 

1.84 

0.08 
2.09 
1.78 
What does this tell us? First, it tells us that Denver's Offense is "better" than our Defense is "good." Our defensive Stnd Dev is 1.33, which essentially means we've performed better than around 80% of others. Denver's Offensive Stnd Dev of 2.09 means they outplay around 95% of others. But, we also see that Denver's defense is "worse" than our offense is "bad." So, which matters more? Well, we don't know that until we know what our own average and standard deviations are. So, repeating the process above for Denver and KC, we get this:

CHIEFS 

DONCOS 

Week 
PTS SCORED 
PTS ALLOWED 
PTS SCORED 
PTS ALLOWED 
1 
28 
2 
49 
27 
2 
17 
16 
41 
23 
3 
26 
16 
37 
21 
4 
31 
7 
52 
20 
5 
26 
17 
51 
48 
6 
24 
7 
35 
19 
7 
17 
16 
33 
39 
8 
23 
17 
45 
21 
9 
23 
13 


10 


28 
20 
AVERAGE 
23.89 
12.33 
41.22 
26.44 
STD DEV 
4.38 
5.25 
8.07 
9.62 
Now, if we multiply our Offense Average Stnd Dev against their Defense numbers, we get a predicted score of 24.67. If we look at their Defensive Average Stnd Dev against our Offense numbers, we get a score of 27.19. These numbers are pretty close, and thus we get a pretty stable prediction of the Chief's score on Sunday: 2527 points.
A Pause in the Action to Look at Sacks
Before we look at the other side of the ball (the one with the intriguing storyline, Manning vs. The Red Crush), I wanted to take a quick look at sacks. One key to this game is going to be getting pressure on Manningbut will it happen? We lead the league in sacks, the Broncos are very good at protecting Manning. Again, we can't compare the two directly  our sacks have come against different offenses than Denver's, and their protection has been against different front 7's then KC. I therefore normalized the sack numbers across the league using total number of sacks allowed and sack percentage.
I took each of the teams we have faced, and normalized them on a scale from 0 to 1, where 0 means you have the best pass protection and 1 means you have the worst, based on allowed sack percentage. I then used that number to compute a "sack ratio," which means that a sack against the worst team equals "1 sack," but a sack against the best team would be worth "3.2 sacks." Why 1 and 3.2? It's arbitrary, but the best team in the league had given up 10 sacks through 9 weeks and the worst had given up 32, so the 1 to 3.2 ratio matched the league extremes. With this number, we can compute the "adjusted sacks" the Chiefs have gotten against each team, and find our average number of adjusted sacks: 7.65. Comparing that to Denver's "3.2 adjusted sacks to sack" ratio, we predict the Chiefs will sack Manning 2.4 times. Here's the work:
BY OPPONENT 
SACK 
FO ASR RANK 
FO ASR 
Normalize 
Ratio 
Adjusted Sacks 
vs. JAC 
6.0 
22 
7.8 
0.4872 
2.1282 
12.7692 
vs. DAL 
3.0 
7 
6.0 
0.2564 
2.6359 
7.9077 
vs. PHI 
6.0 
26 
8.8 
0.6154 
1.8462 
11.0769 
vs. NYG 
3.0 
20 
7.6 
0.4615 
2.1846 
6.5538 
vs. TEN 
3.0 
18 
7.4 
0.4359 
2.2410 
6.7231 
vs. OAK 
9.0 
32 
11.8 
1.0000 
1.0000 
9.0000 
vs. HOU 
5.0 
9 
6.3 
0.2949 
2.5513 
12.7564 
vs. CLE 
1.0 
23 
7.9 
0.5000 
2.1000 
2.1000 
vs. BUF 
0.0 
17 
7.3 
0.4231 
2.2692 
0.0000 
vs. DEN 

2 
4.0 
0.0000 
3.2000 
2.3919 
That's not great, and a bit below our average, but our trend suggests that our pass rush has been slowing down, as you see here:
Why did we take this little detour about sacks? Because our prediction for the Denver score is not as stable as our prediction for the Chief's score was. If we look at it from our Defenses perspective, we give up an average of 12.33 points a game with a Stnd Deviation around 5. Since Denver scores about 2 Stnd Devs above average, this suggests they will score 23.31 points against us.
But, their offense scores an average of 41.22 with a Stnd Dev of 8, and since our defense normally holds teams to 1.33 Stnd Devs below, this predicts a Denver score of 30.49. That 23 to 30 point range is a much larger spread than the 25 to 27 for KC. Thus, math says: Denver: 2330 vs. Kansas City: 2527
I figured up the predicted sacks to tell me which ends I should favor. Although 2.4 sacks is below our average, it is well above Denver's average of 1.4 per game (but not quite the 4 sacks Indy got in Denver's only loss). If it were four sacks, I'd go with the KC number  if it were 1.4 sacks, I'd go with the Denver number. Since it splits the difference  I'll split it, thus my prediction for Denver is 26.90 points. If we compare to our earlier prediction of 2527, we can see it's anyone's game, but to be fair, I took our average as well. This gives us our:
Final Score
Thus, after all this, I mathematically predict:
DENVER  27, KANSAS CITY  26
Now, once again, the certainty of this prediction is so far out of statistical significance that you have a better chance of falling in a manhole tomorrow than you do betting this line, but hey, it was fun. What does my gut tell me? Honestly, I think this season could wind up a lot like 2010, where the score in Denver was a 4929 loss, but the score in Arrowhead was a 106 win. Two teams playing in two games that took on completely different characters. Either way, I think this Sunday is going to be a close one. Either way, we will all know for sure on Sunday.
Hope you enjoyed this.
Go Chiefs!
Edit: Author's PostScript About the "BackUp QB" Argument
In response to a few comments in other threads, I have decided to add a quick blurb about the inevitable critique of these stats that will claim the KC numbers are skewed because we play against backup quarterbacks. While they are right, the Chiefs did play backups, so did many of the other teams that set our baseline averages and standard deviations. Most of the backups have played at least three games, making up a third of our sample size which is more than enough to "unskew" any numberswith one exception: Buffalo. Tuel played only one game, which means the Buffalo numbers are less reliable than all of our others. We can also see that our defense's Stnd Dev against Buffalo was one of our highest at 1.75. So, in fairness, I deleted the Buffalo data as a statistical outlier to see the change in results. What happens? It changes our "final score" from 26.925.93 to 27.1126.16. In other words, it makes no difference whatsoever.
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