Introduction
If you watched the Pat Mahomes show in 2018, you know the Kansas City Chiefs had a very explosive offense. As a data nerd, I’d like to know if we can come up with a single number that measures explosiveness. Unfortunately, it’s not straightforward.
A simple solution would be to count plays that go for over 10 yards. If we do that, the Chiefs were third behind New England and the Rams. However, of those plays that went over 10 yards, the Chiefs still had higher combined total yardage. The Chiefs also lead in plays over 15 yards and plays over 20 yards. In fact, they had 17 touchdowns on plays over 10 yards, while New England and the Rams only had 15 touchdowns and 10 respectively. So we know that the Chiefs are explosive, but how do we quantify that? Were they more potent than the Rams and New England?
Note: My data comes from the nflscrapR package. I used only plays outside the redzone with win probabilities between 10%-90%.
Let’s Start With Yardage
Yards are simultaneously a great way to measure football production, and an awful way. Over the sample size of a year, we can rely on total passing yards and total running yards to accurately give us the best quarterbacks and best running backs in a season. However, football seasons by nature are small sample sizes. Every year there will be freak statistical occurrences over stretches of the season. It’s frequent for football analysts to quote stats over the past four games, this month, etc. and those stats lend themselves to be dramatic and unsustainable.
One of the problems with total yardage is that it is heavily skewed by big plays. Adrian Peterson, especially in his prime, demonstrated how yards late in a run can be relatively meaningless. Here he is at 33, still brushing off secondaries:
It didn’t matter where this run started from, it was going to be a TD. Because it began on his own 10 yard line, Adrian got credited with 90 yards. If the same run had occurred on the opponent 45, he would’ve only gotten half the yardage credit for the same awesome run. The Chiefs, it turns out, had a lot of these plays in 2018.
Not All Yards Are The Same
Another way to frame this is to say that some yards are more difficult to gain than others. Once AP gets 10 yards into a run, going full steam into the secondary, the chance that he will go another 3 yards is high. However, if it’s 3rd and 3, the chance that he runs for a full 3 yards against an 8 man box is relatively low. Enough words, here are some visualizations for empirical evidence. Let’s start by plotting the distribution of yards gained of run plays.

Okay, we can extract a ton of important information from this. If you look closely, about 90% of run plays get stopped before they go 10 yards. There’s a better way to show this. A cumulative distribution function will add up all the probabilities from left to right. In other words, we’ll know what percentage of plays go for 0 yards or less, 3 yards or less, etc. For fun, I used Adrian Peterson in his prime (from the beginning of the dataset in 2009–2015). Here’s a cumulative distribution function of his yardage gained.

As you go right along the x-axis, the probability that AP gets tackled increases. If every yard was the same, AP would have the same chance to get tackled from 0–3 yards as he would from 10–13 yards. On the graph on the right, that’s about a 6% chance in both those ranges. Instead, we see the CDF on the left. In that graph, by the time we get to the line of scrimmage at x=0, it’s already at about 40%! That means 40% of AP’s runs were for less than zero yards. By the time we get to 3 yards, it jumps all the way up to 65%. So for roughly 25% of AP’s runs (25% because I’m subtracting 40% from 65%), he was tackled for a gain of 0–3 yards! Alternatively, when he’s speeding at the secondary, he’s only tacked on 3% of his runs between 10 and 13 yards. Put another way, it’s a lot easier for him to gain 3 yards if he’s already gained 10 on the same play. All yards are not created equally.
How is this related to explosiveness?
Well, if a player is explosive, we should see a lot of 20 yard runs, 45 yard runs, and 80 yard runs. Sometimes the only thing that stops a player is the end zone. In football guy/girl terms, they need to be able "get to the second level" and "be good in the open field". In statistical terms, there should be a fat right tail to the distribution.
Let’s see if the 2018 Chiefs have this fat right tail. I’ll map the distribution of only completions below. As much as I would like to include incompletions and sacks, that complicates the distribution and consequentially the math.

Aha! That’s exactly the tail we were looking for. Other teams had a sharper peak because they’re completing a lot more passes from 0–15 yards. Luckily, statisticians smarter than me came up with a way to measure "the sharpness of a peak". That’s kurtosis. But kurtosis isn’t everything. The Chiefs’ distribution is also skewed more to the right. When you combine sufficient right skew and high kurtosis, you end up with a lot of area under that right tail of the distribution. In other words, a bunch of plays that go for 20+ yards. You might also notice that the mean of the distribution of the high powered offenses is to the right of the rest of the NFL. That’s simply yards per completion (YPC). The NFL and the Patriots both averaged about 12 YPC, while the Chiefs and the Rams both averaged 14.5 and 13.7 respectively. Let’s put it all together (I used inverse kurtosis, since kurtosis in this context means less explosiveness). I normalized mean, skew, and inverse kurtosis, and came up with the ugly graph below. I’m kind of embarrassed even to include this graph, but it’s part of a broader point:

This graph takes a second to interpret, and that makes it an unsatisfactory way to compare the explosiveness of these teams. For a potent offense, we want a high yards per completion (blue bar), a high inverse kurtosis (orange bar), and a high skew right (green bar). No grouping has the best in all three, although the Chiefs have the best total balance of the 3.
Reducing to One Metric
Mean, skew, and kurtosis are great if you’re trying to simulate a game. One could even separate out touchdown plays to be more accurate. However, I’m trying to come up with a single metric for explosiveness. As far as I know, there’s no mathematically responsible way to linearly combine kurtosis and skew.
Gini Coefficient
Instead, we’ll borrow a term from economists. We’ll use the Gini coefficient. The Gini coefficient measures inequality in a distribution. Economists most notably use it to measure income inequality. If everyone in a country has the exact same income, the Gini coefficient is equal to 0. If all the country’s wealth is possessed by one person, then the Gini coefficient is 1.
Application to football?
Instead of income inequality, we can measure yardage inequality. In other words, what percentage of a team’s yardage belongs to big plays? If all of a team’s plays went for the same yardage, the Gini coefficient would be zero. However, if one went for 80 yards, and the rest went for zero, it’d be one. In our Chiefs vs. Rams vs. Pats example, we get .436, .380, and .407. The Chiefs have the highest!
The Gini coefficient is independent of average yardage gained. So we also need a term for yards per play, or in the example I’ve been running with, yards per completion. The following is a graph of the teams in my dataset. Unfortunately the "Greatest Show on Turf" is not in the dataset.

So the most explosive passing offenses, in theory, would drift toward the upper right corner. The 2013 Eagles are the only team that seem to be more explosive than the 2018 Chiefs (with relatively similar yards per play). That is the year Nick Foles threw for 27 TDs and 2 interceptions, so it kind of makes sense. Let’s plot the distributions of completions to make sure the Gini coefficient is doing its job. First, I’ll matchup 2018 Chiefs and the 2016 Jaguars. They had similar yards per play, but the Chiefs should have more big plays. The same should be true for the 2016 Dolphins and the 2016 Bucs. The Dolphins had a very high Gini coefficient that year while the Bucs did not.

It seems to have worked! So the two comparisons above involve teams with about the same average yards per completion but very different offenses. The 2016 Dolphins and the 2018 Chiefs are more explosive than their counterparts with similar yards per completion. The 2016 Buccaneers and Jags were still good passing offenses, specializing in "moving the chains", but they were less likely to bust open a big play.
Conclusion
In conclusion, we’ve found a way to measure explosiveness through the Gini coefficient of offensive plays. I think the best way to use this is in combination with Yards Per Completion (YPC). Or even with running plays, in terms of yards per run. Teams that both average a lot of yards and also have a high skew toward big plays are the most exciting teams to watch. Maybe in the future, we can break down how players on a roster might affect this explosiveness 🤔. We could also open Pandora’s box and look at defenses. If you do work with this, please tweet me so I can see it. If you’re interested, this time I remembered to link my code!
Lastly, I want to mention that I’m not the first to try and measure explosiveness. A legend in college football analytics, Bill Connelly, has his own measure of explosiveness – IsoPPP. He’s done a lot of great work with it, and I did my best to replicate it. When I compared Gini to IsoPPP, I have a very interesting result. IsoPPP and Yards Per Completion are highly correlated. However, Gini is correlated to IsoPPP but not Yards Per Completion. Again, more work needed. Stay tuned!
