Posts tagged ‘football’
Guess the Graph
The bar graph below was created because of a recent discussion with my wife. The title and axis labels have been removed. Can you identify the data set used to create the graph? I’ll give you some hints:
- The data set contains 32 elements.
- It’s based on a real-world phenomenon from this year.
- The middle five categories account for 81% of the data.
- The special points marked by A, B, and C won’t help you identify the data set, but they will be discussed below.
Got a guess?
No clue? Okay, one more hint:
- The vertical axis represents “Teams.”
Still not sure? Final hints:
- Point A represents the lowly J-E-T-S, who are currently winless.
- The region outlined by B shows that 26 teams have from 3 to 7 wins.
- Point C on the graph represents my Pittsburgh Steelers, whose record is a perfect 10‑0. (It’s my hope that I’ll still be able to gloat on Friday morning, after the Steelers host the Ravens on Thanksgiving night.)
This graph was generated while discussing the current standings in the NFL with my wife, who speculated that there seemed to be a lot of really good teams and a lot of really bad teams this year. The horizontal axis represents the number of wins. As it turns out, the distribution above is somewhat typical at this point in the season. At the end of most seasons, about 2/3 of the teams finish a 16-game season with 5 to 10 wins. It may be a little unusual that there are 8 teams with 7 wins, but it’s not statistically cray-cray.
If you’ve read this far, then you may enjoy these other math-related football trivia questions:
- Describe two ways in which an NFL game can end with a score of 2‑0.
- What’s the greatest score that cannot be attained by scoring only touchdowns (7 points) and field goals (3 points)?
- Express the ratio of width:length of a football field. For length, include the end zones.
- What are the only positions allowed to wear single-digit uniform numbers?
- During a typical broadcast of an NFL game, approximately what percent of the time is spent actually playing football (as opposed to commercials, half time, or just milling around between snaps)?
Happy Drinksgiving! And, go Stillers!
—
Answers
- A game can end 2‑0 if one team scores a safety and the other team doesn’t score at all. It can also end 2‑0 if one team forfeits before either team has scored, by league rule. (In high school and college, a forfeit is officially recorded as a 1‑0 loss.)
- 11 points. Any point total above that is (theoretically) possible. Below that, it’s not possible to score 1, 2, 4, 5, or 8 points.
- A field is 53 1/3 yards wide and 120 yards long. In feet, that’s 160:360, which can be reduced to 4:9.
- Quarterbacks and kickers.
- According to several analyses, 11 minutes of a three-hour broadcast is spent actually playing. That’s about 3%. Sheesh.
When a Half Is More Than a Half (and When It Ain’t)
Tonight, the dreadful Philadelphia Eagles defeated the pathetic New York Giants 22‑21 in a match-up of horrendous one-win teams. But not all one-win teams are created equal: in late September, the Eagles played the Bengals to a 23‑23 tie in a game that might have featured the all-time worst ending ever. As a result, the Eagles entered tonight’s game with a horrid 1‑4‑1 record, but not to be outdone, the Giants entered the game with a slightly more putrid 1‑5 record.
In football, a tie counts as a half-win (and a half-loss). But half-wins are sometimes worth more than half a win, sometimes they’re worth less than half a win, and sometimes they’re worth exactly half a win. Let me ‘splain.
After their win tonight, the Eagles record is 2‑4‑1. For the time being, that puts them atop the lowly NFC East:
Eagles | 2-4-1 |
Cowboys | 2-4 |
Washington | 1-5 |
Giants | 1-6 |
Philadelphia has played 7 games and won 2 1/2 of them. That is, they’ve won
of their games. That puts them ahead of Dallas, who has won
of their games. So, the Eagles are currently in first place by 1/42 of a game.
But let’s say the Eagles had entered tonight with a 3‑2‑1 record and the Cowboys were 4‑2. After tonight’s win, the Eagles would be 4-2-1, and they would’ve won
of their games. The Cowboys, on the other hand, would have won
of their games, and the Cowboys would be leading the division by 1/42 of a game.
So that half-win tie? It’s worth more to the Eagles because they’re terrible. Were they at least mediocre, that tie wouldn’t be as valuable.
On the other hand, if the Eagles had entered tonight with a 2‑3‑1 record and the Cowboys were 3‑3, then the Eagles would have been 3-3-1 after tonight’s win, and they would’ve won
of their games. Similarly, the Cowboys would have won
of their games, and the teams would’ve been tied for first in the pitiful, talentless, miserable NFC East.
(Yes, I’m being hard on the NFC East, but it isn’t unwarranted. The average power ranking of the four teams is 28, when the lowest possible is 30.5. The four starting quarterbacks have thrown nearly as many interceptions as touchdowns (24 TDs, 22 Ints), and the four teams’ top running backs have more fumbles than touchdowns (11 TDs, 12 Fum). Seriously, this division may be all-time bad.)
All that said, it’s highly unlikely that the season will end with the Eagles having played more games than the Cowboys. Then again, with COVID‑19, who knows what might happen?
It’s often been said that football is “a game of inches.” But given the importance of half-wins, isn’t it time we started saying that football is “a game of fractions”?
Which is Closest?
Not too long ago, I published a blog post about end-to-end comparisons, those silly feats of computational gymnastics that try to reduce an overwhelming statistic to something more tangible. Something like this:
If each piece of candy corn sold in a year by Brach’s — the top manufacturer of the waxy confection — were laid end to end, they would circle the Earth 4.25 times.
In writing that post, I inadvertently formulated a statistic that rather surprised me:
If all the players on an NFL team were laid end to end, they’d stretch from the back of one end zone to the opposite goal line.
That the players would almost line the entire field struck me as an amazing coincidence. And it got me to thinking — might this be true for other sports?
Not one to let sleeping dogs — or professional athletes — lie, I decided to investigate. Based on that research, here’s a simple, one-question quiz for you.
Which of the following comparisons is the most accurate?
- If all of the players on an NHL (hockey) roster were laid end to end, they would reach from one end of the rink to the other.
- If all of the players on an NBA (basketball) roster were laid end to end, they would reach from one end of the court to the other.
- If all of the players on an NFL (football) roster were laid end to end, they would reach from one end line to the other.
- If all of the players on an MLB (baseball) roster were laid end to end, they would reach from home plate to second base.
- If all of the players on an MLS (soccer) roster were laid end to end, they would reach from one end to the other.
As you begin to think about that question, some notes:
- Every professional baseball stadium has different measurements. Fenway Park (Boston) is a mere 310′ from home plate to the right field wall, whereas Comerica Park (Chicago) extends 420′ from home plate to straightaway center. Consequently, the distance from second to home is used in the fourth answer choice, because it’s the same for every field.
- To my surprise, MLS stadiums are not uniform in length and width. Who knew? The length of the field must be at least 100 meters, at most 110 meters, and anywhere in between is fine. Assume an average length of 105 meters for the fifth answer choice.
Before you read much further, let me say how much fun I’ve had discussing this question around the dinner table and at the local pub. In spite of hard facts, there is resolute disagreement about player height, roster size, and field dimensions. And the shocking (or should I say predictable?) results raise an eyebrow every time. I only mention that to persuade you to think about the question, alone or with some friends, before continuing.
Okay, you’ve cogitated? Then let’s roll.
In researching the answer to the question, I was struck by how close the total length of all players on the roster is to the length of the field, court, or rink. Coincidence? Of course, a larger field requires more players, so perhaps this is the evolution of roster size that one would expect.
To answer the question, you need to know the height of an average player, the number of players on a roster, and the dimensions of professional venues. All of that data can be found in a matter of minutes with an online search, but I’ll save you the trouble.
League | Average Height (in.) | Players on a Roster | Combined Height, Laid End to End (ft.) | Dimensions |
NHL | 73 | 23 | 140 | 200 feet (from end to end) |
NFL | 74 | 53 | 327 | 120 yards (360 feet, from end to end) |
NBA | 79 | 14 | 92 | 94 feet (from end to end) |
MLB | 73 | 23 | 140 | 127 feet (from home to second) |
MLS | 71 | 28 | 166 | 105 meters (345 feet, from end to end) |
As it turns out, the MLS comparison is the least accurate. The combined heights of soccer players is only 48% of the length of their field. The NHL comparison is a little better, with players’ heights extending 70% of the length of the field. But the NFL and MLB are both very close, with the players’ heights equalling 91% of the field length and 110% of the distance from home to second, respectively. Astoundingly, if the players on an NBA team were laid end to end, they’d come just 22 inches short of covering the entire court, accounting for a miraculous 98% of the length!
So there you have it. D, final answer.
One last thought about this. I play ultimate frisbee, a sport with a field that measures 120 yards (360 feet). For tournaments, our rosters are capped at 29 players, and I suspect my amateur teammates are, on average, shorter than most professional athletes. If we assume a height of 5’10” for a typical frisbee player, then the combined height is 172 feet. That puts us in the realm of soccer, with our combined length covering just 48% of the field.
If, like me, you play a sport that isn’t one of the Big 5 in the U.S., I’d love to hear about your sport’s field and roster size, and how it ranks with the comparisons above.
Probability, the Playoffs, and the Indianapolis Colts
With one week left in the NFL season, Dan Graziano had this to say about the Indianpolis Colts’ chances of making the playoffs:
Indianapolis can still win a third straight AFC South title. Really, it can. All it needs is to win and then have the Texans, Bengals, Chargers, Jets, Saints, Chiefs, Patriots and Browns all lose. The league will throw in the partridge in a pear tree.
If the Colts win and the Houston Texans lose, both would be 8-8, and the first four tiebreakers for deciding which team makes the playoffs – record against one another, record against divisional opponents, record against common opponents, and record within the conference – would not be enough to decide who makes the cut.
It then comes down to strength of victory and strength of schedule. And for things to play out in the Colts’ favor, a lot of things have to go their way.
Fox Sports referred to this as long shot, comparing it to a recent win by a horse who was 200-to-1:
Sure, going 9 for 9 here looks dim, but long shots come in every once in a while.
Are the Colts’ odds as good as that horse’s? Seems not.
Using a simplistic model, assume that each of the nine necessary outcomes are equally likely. That alone would put the Colts’ odds at 511-to-1. (Since 2^{9} = 512.)
But it’s not that simple. The following chart from 538.com gives the probability of each team winning their game this weekend:
The good news is that the Colts have an 81% chance of winning their game against Tennessee. The bad news is that it seems unlikely that any of the Texans, Bengals, Chiefs, or Patriots will lose, let alone all four of them. So putting all those numbers together, the Colts’ chances of making the playoffs are:
0.81 × 0.20 × 0.22 × 0.84 × 0.54 × 0.69 × 0.15 × 0.18 × 0.77 = 0.0002 = 0.02%
or, more precisely, about 4,311-to-1. That’s more than a long shot; that’s an extended-to-an-unfathomable-distance shot.
The Colts are in the unenviable position of Lloyd Christmas in Dumb and Dumber, “So, you’re telling me there’s a chance…”
P (NFL ∪ Math) > 0
John Urschel is an offensive lineman for the Baltimore Ravens and admits, “I love hitting people.” As it turns out, he loves hitting the books, too. He earned a masters degree in mathematics from Penn State, and he recently published a paper with the impressive title A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians in the Journal for Computational Mathematics.
Note that Urschel was the lead author, even though his three co-authors were an associate math professor from Tufts and two math professors from Penn State.
I have to wonder if the paper was fairly refereed. I mean, honestly, who in the math community is gonna tell a 6’3″, 308‑pound football player that he made an error?
A la Paul Erdös, Urschel doesn’t need much to be happy. In an essay published March 18, he wrote:
I drive a used hatchback Nissan Versa and live on less than $25k a year. It’s not because I’m frugal or trying to save for some big purchase, it’s because the things I love the most in this world (reading math, doing research, playing chess) are very, very inexpensive.
I was thinking about how Urschel has superior talent in two fields, when I saw this comment on an article on Deadspin:
Here’s the thing.
There are 1,596 players in the National Football League at any given time (32 teams with 53 players each). Throw in a few more who serve on practice squads and occasionally get a chance when someone else gets hurt, so maybe that number climbs to 2,000. Still, the chance of making it to the NFL is unbelievably remote. Recruit 757 claims that only 0.008% of all high school athletes get drafted by the NFL.
And if you can believe Wolfram Alpha, there are 2,770 mathematicians in the United States, or approximately 1/47,165 of the U.S. workforce.
Point is, the probability of becoming either a professional football player or a mathematician is ridiculously small. Becoming both is smaller still. Though John Urschel proved it’s greater than 0. The saving grace is that he seems like a down-to-earth guy who realizes how lucky he is.
To read a math article written by John Urschel, check out 1 in 600 Billion.
6 Degrees of Bad Math Jokes
I once read an article that said, “To a greater or lesser degree, everything tastes like chicken.” Well, that’s true, but it’s also true that everything tastes like broccoli, to a greater or lesser degree. Carrots, to a greater degree; mint chocolate chip ice cream, to a lesser degree.
To a greater or lesser degree, some of the following jokes are funny.
What did the thermometer say to the graduated cylinder?
A scientist dropped a thermometer and a candle from the roof of a building. He observed that both objects reached the ground at the same time. Conclusion: A thermometer falls at the speed of light.
A doctor walks into a meeting, and a nurse asks why he has a rectal thermometer behind his ear. “Damn,” says the doctor, “some asshole has my pen!”
The star college football player was taking a math exam. The coach desperately needed him for the big game on Saturday, so the professor agreed to an oral exam.
“All right,” said the professor. “How many degrees are in a circle?”
“That depends,” said the boy. “How big is the circle?”
If you’re cold and there’s a right triangle nearby, stand in the corner opposite the hypotenuse. It’s always 90° over there.
The number you have dialed is imaginary. Please rotate your phone 90°, and try again.
The MJ4MF Bowl
It’s college bowl season, and there is an impressive line-up of games, from the Famous Idaho Potato Bowl, to the Little Caesars Pizza Bowl, to the Buffalo Wild Wings Bowl, to the GoDaddy.com Bowl, to the…
Oh, for Pete’s sake.
There are no fewer than eight college football bowls that have completely abandoned any pretense of respecting tradition. The name of the bowl is isomorphic with the name of the sponsoring company. Sure, some bowls give a nod to tradition by appending the name of the sponsor to the historical name, such as the Allstate Sugar Bowl or the Discover Orange Bowl. But even in those cases, the sponsor is listed before the bowl itself.
What can you do? My daddy always told me, “If you can’t beat ’em, join ’em!”
Following his sage advice, I’d like to announce the 2013 Math Jokes 4 Mathy Folks Bowl, which already has a spiffy logo…
But the MJ4MF Bowl will be different than the others. There have to be rules. My rules.
First, the game must be played on January 3, 2013, which can be written as 1/3/13. (Nice, huh?)
Second, both teams would have to be willing to modify their nicknames — only temporarily, of course — to make them more mathy. For example,
- Arizona State Sum Devils
- East Carolina πrates
- Navy Midpoint Men
- North Texas Median Green
- Penn State Nittany Lines
- Standford Cardinality
- Tulane Sine Wave
- UCLA de Bruijn Sequences
- Western Kentucky Hilltopologists
Third, and most importantly, the yard lines on the field would need to be renumbered. Currently, they are numbered as follows:
|0 1|0 2|0 3|0 4|0 5|0 4|0 3|0 2|0 1|0 0|
That’s just dumb. For the MJ4MF Bowl, the yard lines will be numbered like this:
-5|0 -4|0 -3|0 -2|0 -1|0 0 1|0 2|0 3|0 4|0 5|0
Honestly, doesn’t that make more sense? The middle of the field would be the 0-yard line, which seems appropriate; and, now when you hear, “The Lions have the ball on the 10-yard line,” you won’t have to wonder, “Which 10-yard line?”
Finally, teams will not have to meet the onerous NCAA bowl eligibility requirements to participate in the MJ4MF Bowl. Why does a team need six wins to be bowl eligible, anyway? That just means they’ll demand a big pay-out, and unless a rich, eccentric math geek buys a million copies of Math Jokes 4 Mathy Folks in the next week, well, that’s just not gonna happen.
Two exciting teams are currently sought to play in the inaugural MJ4MF Bowl. Notre Dame and Alabama are required to play for the national championship, and the likes of Georgia, Kansas State, and Nebraska have already agreed to other bowl games… but surely the Golden Eagles of Southern Miss (0-12) and the Akron Zips (1-11) are available, no?
Math on Monday Night Football
Tonight’s Eagles-Panthers game had several mathematical incidents.
A Counting Problem
During pre-game warm-ups, a voice-over quoted one of the player’s thusly:
You want me to describe playing on Monday night in one word? Prime time.
Computer scientists start counting at 0. Apparently pro football players start counting at 2.
Speaking of counting… did you notice that the sentence above had three hyphenated words? That’s just crazy.
Flippin’ Out
During the opening kick-off, Mike Tirico mentioned that the Carolina Panthers lost the coin toss. That alone is not exceptional, but it was the eleventh straight game that they had lost the flip. The odds of a team being that unlucky? How about 2,047 to 1?
Perhaps they can blame bad luck for their eight losses this season, too.
Numerically Interesting Milestone
Wide receiver Steve Smith of the Panthers caught a pass in the first half that took him to 745 career receptions for a total of 11,011 yards. What a cool number! First, it’s a palindrome. Second, 11011_{2} = 27_{10}, and he currently ranks 27th among wide receivers in career receiving yards. That’s a pretty fun coincidence.
Football Math for Super Bowl Week
Super Bowl week seems an appropriate time to share some jokes that involve football and math.
[Super Bowl Squares Online Contest]
What is this?
B
BA
BACK
Here’s another one involving fractions. (And that lead-in should be a hint if you had trouble with the question above.)
What do you call a Patriots fan with half a brain?
Gifted!
And just to be an equal opportunity offender…
What did the average Giants player get on his Wonderlic test?
Drool.
There are several one-liners involving football and math (sort of).
Pro football players are so huge, it takes only four of them to make a dozen.
Their nickel defense is only worth 3¢.
His uniform number was 29, which was also his house number. He wore it to make sure he remembered where to go after the game.
That last one reminded me of a mathy football joke involving dumb people…
By the time Bubba arrived to the football game, the first quarter was almost over. “Why are you so late?” his friend asked.
“I tossed a coin to decide between going to church or coming to the game.”
“I don’t understand. How long could that have taken?”
“Well,” Bubba said, “I had to toss it 14 times.”
For a similar, non-football coin-tossing joke, read the one about the student at the final exam.
The Book of Tebow
Editor’s Note: The following post is more about (American) football than math, but it does contain some humor (or, perhaps more correctly, it contains material similar to the other material that poses as humor on this blog). Just be forewarned. Read at your own peril.
I fell in love with Denver on a family trip in 1982. My favorite colors as a kid were blue and orange. So I was already a fan of the Denver Broncos when they acquired my favorite college player, John Elway, on May 2, 1983. (Ironically, my wife and I acquired our twin sons on May 2, also, albeit more than two decades later.) On Sunday afternoons growing up, I’d watch my hometown Pittsburgh Steelers at one o’clock, and I’d hope that NBC would show Elway and the Broncos during the late game.
So this whole Tebow thing? Yeah, I feel a little like I’m jumping on a bandwagon. Then again, I’ve been a fan of the Broncos for 28 years, so cut me a little slack.
Plus, it’s just so damned compelling. Any quarterback can win football games, but it takes a rare talent to repeatedly perform miracles. You better believe that I have already set our DVR to record tomorrow’s Broncos-Patriots game.
I also love the hype and the humor. The nickname “God’s Quarterback” seems to have stuck, and this great joke has surfaced:
And on the seventh day, God rested so he could watch his son play quarterback for the Denver Broncos.
A few days ago, a headline in The Christian Post caught my eye:
Tim Tebow ‘God’s QB,’ But Does God Care About Football?
I have a Speed Bump cartoon on the door to my office suggesting, in fact, that He does:
I look forward to all that will follow. The legend of Tim Tebow continues to grow, and no doubt organized religion will begin to take advantage of the publicity. I suspect a rewrite to the Good Book before too long…
The Gospel According to Tebow 1 God created Tebow in his own image, in the image of God created He him. And God said unto him, Go forth, and run and score, and replenish hope in the city of Denver: and have dominion over the dolphins of the sea; and over the fowl of the air, over cardinals and eagles and ravens; and over cowboys and redskins and titans; and over every living thing that moveth upon the turf. 2 And lo, He made a great arena, called as the Stadium of the Authority of Sports, which was ten-thousand cubits from the one rim to the other. It was round all about, and a line of thirty-one thousand, four-hundred fifteen and nine-hundred twenty-six thousandths cubits (approximately) did compass it round about. And the incorrect approximation of pi previously appearing in scriptures was thus smote, and it was good. 3 He placed the stadium above the water five-thousand two-hundred eighty feet, providing a wonderful number with which to demonstrate the law of divisibility by eleven. 4 And He bade him, play your best, and do not be discouraged in half the first, or by thine rating of eighty-three-point-four, or by trailing your opponents at the end of quarter third; play well when the end is nigh, and best your enemy after regulation time has expired. 5 Lastly God said unto him: kneel before me, with but one knee upon the earth and a clenched fist upon thine brow, and let photographers take pictures; and all the peoples of the earth shalt imitate thee and post their pictures at www.tebowing.com, thus begetting an international phenomenon. 6 And Tebow did as commanded, and it was good. |