When Super Bowl XLIX Starts to Bore You…

I appreciate that the Super Bowl unites all Americans in their inability to read Roman numerals.

(Caution — oxymoron ahead!) If you’re a smart football fan, then you’ve invited folks to your house to watch the game, so you don’t have to drive home drunk on a cold Sunday night in February.

If this year’s Super Bowl is like last year’s, your guests will be bored by halftime, with the Seahawks leading 22‑0.

Or if deflated balls are used, the Patriots could be leading 38‑7 at the end of the third quarter, like when they played the Colts two weeks ago.

In either case, you’ll need something to keep your guests entertained between commercials and after the Katy Perry halftime show. I suggest the following problem.

Imagine that the NFL has eliminated divisions, and there are just two conferences with 16 teams each. To simplify things, every team plays the other 15 teams in their conference exactly once each. At the end of the regular season, what is probability that every team within a conference has a different record? (Assuming, of course, that each team is equally likely to win on any given Sunday, and assuming that ties are not possible.)

It’s not a trivial problem, so I’ll give you some time to think about it. The answer will be revealed on Monday, February 2, 2015; that is, after the game.

Results of a Wonderlic-SAT Comparison

Eli Manning and Tom Brady are arguably the smartest pair of quarterbacks to face each other in a Super Bowl. That’s not just hyperbole; there’s data to support it. Manning scored a 39 on the Wonderlic test, and Brady scored a 33, giving them an average score of 36. That’s the highest average ever for the starting quarterbacks in a Super Bowl.

The two starting quarterbacks for Super Bowl XLVII, Colin Kaepernick and Joe Flacco, are no intellectual slouches, either. Flacco scored a respectable 27 on the Wonderlic, and Kaepernick rocked the test with a 37, placing him two standard deviations above the norm. That puts him in the 97th percentile. If he wins the Super Bowl this Sunday, he’ll be the second-smartest quarterback to do so.

Last week, I asked readers to supply me with data for a research project. The Wonderlic test is used by the National Football League to measure the problem-solving abilities of prospective players. The SAT (and the ACT) have long been used as college entrance exams, and both claim to predict college success. My hypothesis is that the Wonderlic — a 12-minute, 50-question test — would be equally good at predicting college success.

The following presents the (I) results, (II) limitations of the research, and (III) some notes about the methodology. (Sorry, I don’t mean to be pretentious or to imply false erudition by using Roman numerals. I just know that some folks are interested in (I) but could give a rat’s butt about (II) or (III), so I thought dividing this post into sections might be helpful. Hopefully by using the phrase “rat’s butt,” I’ve removed all sense of pretense.)

I. Results

Neither the SAT nor the Wonderlic are good at predicting college success, but to my surprise, the SAT is better than the Wonderlic.

The following correlation coefficients resulted when three pair-wise correlations were performed:

• Wonderlic and GPA: r = 0.0086
• SAT and GPA: r = 0.0506
• Wonderlic and SAT: r = 0.2897

When comparing the Wonderlic and college GPA (n = 46), the correlation coefficient was r = 0.0086, meaning that roughly 9% of the variance of college GPAs can be explained by Wonderlic scores.

When comparing the SAT and college GPA (n = 41), the correlation coefficient was r = 0.0506, meaning that roughly 22% of the variance of college GPAs can be explained by SAT scores.

When comparing the Wonderlic and SAT (n = 44), the correlation coefficient was r = 0.2897, meaning that roughly 54% of the variance of college GPAs can be explained by Wonderlic scores.

Though not quite as strongly, these results corroborate my previous findings that neither the SAT nor the Wonderlic is a very good predictor of college success, but both are pretty good predictors of scores on other standardized tests.

II. Limitations

A number of factors discredit the validity of this research, among them:

• Voluntary Response Bias. The majority of respondents were above average in all categories. Additional data is needed from individuals who scored poorly on the SAT/ACT or Wonderlic or who had below-average college GPAs.
• Sample Size. It is difficult to draw conclusions from a sample of just 54 individuals.
• Timing. Those who responded often took the Wonderlic many years after taking the SAT. This is an issue with data from NFL prospects, too; they take the SAT prior to entering college, but they take the Wonderlic at least three years later. Certainly, those years of experience would influence the results.
• Consistency. College GPA is not transferrable. Without a doubt, earning a 3.1 GPA at Harvard University is more impressive than holding a 3.9 at the Univerity of the District of Columbia. Even within the same university, there can be discrepancies; it’s likely more difficult to hold a high GPA if your major is electrical engineering than, say, parks and recreation. Unfortunately, it’s one of the only means of comparing two students from different schools, apart from reputation of the issuing institution.

Consequently, this research should be taken in the spirit it was intended. It it not academic research. It was merely a tongue-in-cheek attempt to show that neither the SAT/ACT nor the Wonderlic test are terribly good at predicting college success.

That said, this analysis could serve as the impetus for an academic research project. By gathering Wonderlic scores from high school students at the same time that they take the SAT, and then tracking them to determine their success in college, the viability of the Wonderlic test as a college entrance exam could be determined. (It should be noted that the Wonderlic Personality Test (WPT) is used by the NFL when evaluating prospective players, but scores on the Wonderlic Basic Skills Test (WBST) are already accepted by some colleges.)

III. Notes About the Method

For consistent comparison, all college exam scores were converted to a scale based on the old SAT (out of 1600). ACT scores were converted using results of a concordance study conducted by the ACT and the College Board. Converting scores from the new SAT to the old SAT used the method described below.

Because the maximum score on the new SAT is 2400 and the maximum score on the old SAT was 1600, the following conversion formula might seem reasonable:

2/3 × new SAT = old SAT

However, there are two reasons that won’t work. First, in addition to covering the same math topics as the old SAT, the new SAT also covers Algebra II. Second, the writing section has proven to be the hardest part of the new test; the average score on the writing section is 493, since its inception in 2005; by comparison, the average scores for math and reading are 516 and 501, respectively, during the same time period.

Using scores from 2000-11, it seems that approximately 67.3% of a student’s score on the new SAT comes from the math and reading sections; the writing section only accounts for about 32.7% of the student’s total score. Second, the average score on the old SAT from 2000-05 was 1024, whereas the average combined score for the math and reading sections on the new SAT from 2005-11 was 1017, which means that the average score on the old SAT was about 0.7% higher than the average combined score on the math and reading sections of the new SAT.

Consequently, for any respondent who listed a new SAT score, I multiplied their score by 0.673 to find their score on just the math and reading sections, and then I multiplied by 1.007 to account for the higher average score on the old SAT. This is obviously an imperfect system. That said, one of the respondents told me that his combined math/reading score on the new SAT was 1390, and this formula yielded an old SAT estimate of 1410. Since the old SAT score should be slightly higher, it seems that the formula is reasonable. I therefore used this formula for all respondents who listed a new SAT score, of which there were only two.

No changes were made to the college GPAs, despite the inherent flaws described above.

Once the data was in comporable form, my good friend Excel was used to perform a linear regression and determine the correlation coefficient.

Super Bowl Math (and Results of the Super Bowl Squares Online Contest)

There were a lot of interesting mathematical things that happened tonight.

First things first: Big props to Valerie Strauss of The Answer Sheet, who asked, “Tom Brady vs. Eli Manning: Who’s Smarter?” and then was smart enough to link to one of my previous posts when trying to answer the question.

Math Incident #1

Within the first five minutes of coverage, it was announced that the NFC had won 14 consecutive coin tosses. Posted on the screen:

Odds: 1 in 16,384

I was a little bummed that it didn’t say, “Odds: 1 in 2¹⁴.” But I can’t complain. It’s not every day that probability gets international publicity.

Math Incident #2

With just under 4:00 left in the game, Wes Welker dropped a pass from Tom Brady. During the replay, Cris Collinsworth said that it was a pass that Welker makes “100 times out of 100.” Um, Cris, in case you missed it… I don’t know how many passes just like this that Wes Welker has caught, but he missed this one, so we have at least one data point showing that, in fact, he doesn’t always catch this pass. If you want to revise your statement to “99 out of 100,” I could live with that.

Super Bowl Squares Online Contest

The results of the Super Bowl Squares Online Contest have been posted at http://mathjokes4mathyfolks.com/super-bowl-squares-results.html. But allow me to spoil some of your fun before you click that link:

• There were no winners for the first quarter score (Patriots 0, Giants 9; winning square, 9‑0).
• There were no winners for the second quarter score (Patriots 10, Giants 9; winning square, 9‑0).
• There were no winners for the third quarter score (Patriots 17, Giants 15; winning square, 7‑5).
• There were two winners in the fourth quarter (Patriots 17, Giants 21; winning square, 7‑1).

That means that Ben Morris and Tom Coffin were the only winners of the 31 participants, so they split the $155 pool of Monopoly money, each receiving $77.50.

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.

Super Bowl Squares Contest

Laurence Tynes, the hero; Billy Cundiff, the goat. And so we head to Super Bowl XLVI with a rematch of the game four years ago. One can only hope that this game will be half as exciting as that one.

Your math/football trivia for the day? Super Bowl XLVI is the second to require each of the first four Roman numerals (I, V, X, L); the first was Super Bowl XLIV two years ago. [Thanks to Eric Langen for pointing out my previous error.] Personally, I’m looking forward to Super Bowl LXVI, when the first four Roman numerals will occur in decreasing order. A real treat will occur in 3532, when Super Bowl MDLXVI will be played, wherein all six of the Roman numerals will appear in decreasing order. While I’m fairly certain I won’t be around to see that one, I hold out hope that I am reincarnated as a star football player who earns that game’s MVP honors; though it’s far more likely that I will return as a football to be used by adolescents in a backyard game.

Buoyed by the success of the online version of my favorite game, I’ve decided to run another online contest. This one relates to Super Bowl XLVI, and you’re asked to predict the units digit of each team’s score at the end of each quarter when the Patriots and Giants square off on Sunday, February 5.

Probably the most common type of office betting pool is a square football pool, which is often referred to as just The Squares. The pool is played on a 10 × 10 grid, and contestants can buy squares within the grid for a certain amount of money. After all 100 squares have been purchased, the numbers 0‑9 are randomly assigned to each row and column. The numbers for each row represent the units digit of the score for one team, and the numbers for each column represent the units digit of the score for the other team. The winners are the four people whose squares correspond to the units digit of the actual score of the game at the end of the 1st, 2nd, 3rd, and 4th quarters.

Feel free to use this Excel spreadsheet if you’d like to run your own version of this game. (Though be sure to check all applicable laws, to ensure that you’re not in violation of local or state gaming laws.)

The difference between the typical version of this game and the version I’m running here is that you get to pick which pairs of numbers you want. Consequently, winning isn’t solely a matter of random luck. But there’s a catch — you can pick the most likely number pairs, but chances are other folks will pick those numbers, too, and the winnings are divided among everyone who picked that pair. So, should you pick 0‑0 and divide the pot with a thousand others; or should you pick the highly unlikely 5‑2 and have the winnings all to yourself?

Please note that the game I’m running is for entertainment only. No money is required to play, and there will be no pay-out to the winners. If all goes well this year, perhaps next year there will be a real version that allows you to wager your hard-earned money in such a silly manner — assuming, of course, that I can find a way to skirt the myriad state gaming laws that would prevent me from running such a contest.

In case you’re wondering, “Why are you doing this?” remember that I’m the author of a math joke blog. Why do I do any of the things I do? For fun, mainly, and because I’m a certifed math geek. I like the math psychology of this game, and I’m just interested in the numbers that people will pick.

Here are the official rules:

• Imagine that you have $5, and each square costs $1, so you can buy up to five squares. It’s your money, spend it how you like — if you want to choose the same pair of numbers for all five bets, go ahead, knock yourself out. And what the hell do I care? Enter as often as you like; if you’ve got nothing better to do with your time than repeatedly submit entries for this contest, well, that’s your problem.
• All money bet will be divided equally among the four quarters, so the total amount will be equal to $5n, where n is the number of contestants. (Should a contestant enter fewer than five choices, the last entered choice will be repeated multiple times to get the total to five.)
• If you pick a winning square, you will share the winnings with everyone else who picked the same square. (For example, if 200 people play this game, there will $1,000 in the pot, so the winning amount for each quarter will be $250. If ten people choose 7-3 and it hits for one quarter, each person will receive $25.)
• Enter your five choices as two-digit numbers, where the tens digit represents the Patriots’ score and the units digit represents the Giants’ score. (For instance, if you want Patriots 7, Giants 3, enter 73; but if you want Patriots 0, Giants 7, enter 07.)

That’s it. Access the form via the link below:

Super Bowl Squares Contest

My friends Andy and Casey Frushour have been keeping data about which pairs of numbers occur most often. Before making your picks, you might want to check out their analysis of data from six years of NFL games as well as from all 45 Super Bowls.

Bets will be accepted until 11:59 p.m. ET on Saturday, February 4, and an image showing the number of times each square was chosen will be posted at:

Super Bowl Squares Contest – Summary of All Bets

The complete results for this contest will be posted on Monday, February 6, at the URL below. (But note that this link will return a “404 Error – File not Found” message prior to February 6.)

Super Bowl Squares Contests – Results

Good luck!

Smart Quarterbacks, the Super Bowl, and SAT Scores

This weekend, when the Pittsburgh Steelers take on the Green Bay Packers in Super Bowl XLV, it’ll be a match-up pitting a very smart quarterback against, well, a guy who’s not exactly the sharpest knife in the drawer.

If you’re like most of the world, you probably don’t perceive Ben Roethlisberger to be very smart.  He attended Miami University, but information about what he studied is considerably harder to find, and few would call him intelligent. After all, he rides his motorcycle without a helmet, frequently fraternizes with underage co‑eds, and associates with people who occasionally urinate in public. So it will come as no surprise that Roethlisberger scored lower on the Wonderlic test — the 50‑question, 12‑minute exam administered by the National Football League to measure the problem-solving ability of players who will enter the draft — than Aaron Rodgers.

The maximum possible score on the Wonderlic test is 50. Roethlisberger scored 25, Aaron Rodgers scored 35. (Wanna know how you compare? Try a sample Wonderlic test for yourself.)

So, does this mean that Rodgers has an advantage in Sunday’s game? Not necessarily.

Below is data from the last ten Super Bowls. The winning quarterback is listed first, and his Wonderlic score is given in parentheses. (Sorry, I couldn’t locate the Wonderlic score of Brad Johnson.) But for the other nine games, the team whose quarterback had a higher Wonderlic score won four times, the team whose quarterback had a lower Wonderlic score won four times, and last year, the two quarterbacks had the same score.

Super Bowl XXXV – 1/28/01
Trent Dilfer, Baltimore Ravens – Fresno State (22)
Kerry Collins, New York Giants – Penn State (30)

Super Bowl XXXVI – 2/3/02
Tom Brady, New England Patriots – Michigan (33)
Kurt Warner, St. Louis Rams – Northern Iowa (29)

Super Bowl XXXVII – 1/26/03
Brad Johnson, Tampa Bay Buccaneers – Florida State (unavailable)
Rich Gannon, Oakland Raiders – Delaware (27)

Super Bowl XXXVIII – 2/1/04
Tom Brady, New England Patriots – Michigan (33)
Jake Delhomme, Carolina Panthers – Louisiana-Lafayette (32)

Super Bowl XXXVIX – 2/6/05
Tom Brady, New England Patriots – Michigan (33)
Donovan McNabb, Philadelphia – Syracuse (14)

Super Bowl XL – 2/5/06
Ben Roethlisberger, Pittsburgh Steelers – Miami, Ohio (25)
Matt Hasselbeck, Seattle – Boston College (29)

Super Bowl XLI – 2/4/07
Peyton Manning, Indianapolis Colts – Tennessee (28)
Rex Grossman, Chicago Bears – Florida (29)

Super Bowl XLII – 2/3/08
Eli Manning, New York Giants – Ole Miss (39)
Tom Brady, New England Patriots – Michigan (33)

Super Bowl XLIII – 2/1/09
Ben Roethlisberger, Pittsburgh Steelers – Miami, Ohio (25)
Kurt Warner, Arizona Cardinals – Northern Iowa (29)

Super Bowl XLIV – 2/7/10
Drew Brees, New Orleans Saints – Purdue (28)
Peyton Manning, Indianapolis Colts – Tennessee (28)

As it turns out, the average Wonderlic score of an NFL player is 20, while the average score of an NFL quarterback is 24. Only one Super Bowl quarterback in the past ten years had a Wonderlic score below the league average. That was Donovan McNabb (14) in 2005. So while a higher Wonderlic score may not imply Super Bowl success, it does seem that quarterbacks who make it to the Super Bowl have above average scores.

Of course, a football team has more than just one player, so it might be more informative to look at the Wonderlic scores for every player on a team. Sadly, I don’t have that kind of time, but such an analysis was done at least once. The Denver Broncos defeated the Green Bay Packers in Super Bowl XXXII; the average Wonderlic score for the Broncos was 20.4, while the average score for the Packers was 19.6.

The Wonderlic test fascinates me. While it may not be the best predictor of success in the NFL, many companies use it to assess prospective employees’ problem-solving abilities. And it got me to thinking — if the Wonderlic test is adequate to predict job success, could it also be used to predict college success?

Consequently, I sought to answer the following question: Could the Wonderlic test be as good a predictor of college success as the SAT?

Unfortunately, acquiring data to analyze this question is no small task. Wonderlic scores of many NFL players are readily available online, but other companies aren’t willing to release the scores of their employees. (Truth be known, the NFL isn’t really willing to release its employees’ scores, either, but players’ scores are interesting trivia for the public, so sports reporters find ways to uncover them.) In a quick search, I was able to locate the Wonderlic scores of scads of NFL players. However, unearthing the college GPA and SAT scores of those players was exorbitantly difficult. I found all three numbers for just six players online (see table below). I tried to acquire the numbers for other players over the phone, but I met with limited success. A typical conversation went something like this:

Woman in Registrar’s Office at University of Virginia: Hello.
Me: Uh, good afternoon, ma’am. I’m trying to locate the GPA and SAT scores of one of your former students.
Woman: Whose information are you looking for, sir?
Me: Matt Schaub.
Woman: And you are?
Me: Patrick Vennebush.
Woman: Are you related to Mr. Schaub?
Me: Um, no, ma’am.
Woman: Are you a prospective employer?
Me: No, ma’am.
Woman: So… why do you need Mr. Schaub’s information?
Me: Well, see, I’m comparing professional football players’ scores on the Wonderlic test…
Woman: The what?
Me: The Wonderlic test. It’s a test they give to professional football players to determine their problem‑solving ability.
Woman: Hold on — Mr. Schaub is a professional football player?
Me: Yes, ma’am. He played quarterback for the University of Virginia from 1999 to 2003, and now he plays for the Houston Texans.
Woman: So, why do you need Mr. Schaub’s GPA and SAT scores?
Me: Well, I’m trying to determine if the Wonderlic test could be used as a predictor of college success. I need Mr. Schaub’s GPA and SAT scores to see if the Wonderlic test was as accurate as the SAT in predicting how well he did in college.
Woman: Well, I can’t just go around giving out information about former students to total strangers.
Me: Yes, I understand, ma’am, but I’m not going to publicize the information. I just want to analyze it.
Woman: And what will you do with your analysis?
Me: Well, I was planning to post the results on my blog.
Woman: So, you write a sports blog?
Me: Well, no, ma’am. It’s actually a math blog.
Woman: A math blog that focuses on sports?
Me: Um, well, no.
Woman: Then what kind of math blog is it?
Me: Well, actually, it’s a blog about math jokes.
Woman: About what?
Me: Math jokes.
Woman: [click]

Several other calls met a similar fate. Consequently, I only have Wonderlic, GPA and SAT scores for six players. But, whatever. Let’s roll with it and see what happens. The three numbers for each player are shown below.

Player	College	Wonderlic	GPA	SAT
Tim Tebow	Florida	22	3.66	890
Brady Quinn	Notre Dame	29	3.00	1030
Peyton Manning	Tennessee	28	3.61	1030
Aaron Rodgers	California	35	3.60	1300
Myron Rolle	Florida State	33	3.75	1340
Ryan Fitzpatrick	Harvard	48	3.20	1580

From this limited sample, three pair-wise correlations were calculated:

• SAT and GPA: r = ‑0.14
• Wonderlic and GPA: r = ‑0.36
• Wonderlic and SAT: 0.95

There’s not a very strong correlation between SAT and GPA. But here’s the thing: the correlation between SAT and GPA for this set of six football players isn’t that much worse than the correlation between SAT and GPA reported in Validity of the SAT for Predicting First-year College Grade Point Average, a study of 151,316 students at 726 four‑year institutions undertaken by the College Board; in that study, r = 0.29.

There’s not a very strong correlation between Wonderlic and GPA, either, but it’s stronger than the correlation between SAT and GPA for the six football players above and for the 151,316 students in the College Board study.

There is, however, a very strong correlation between Wonderlic and SAT, which is perhaps just another way of saying that both tests are equally lousy at predicting college success.

Of course, there are all kinds of reasons that this analysis might be invalid:

• the sample is too small;
• it is difficult to compare GPA from school to school, since it might be more difficult to earn a 3.20 at an Ivy League college than at a public university;
• it is difficult to compare GPA between students within a school, since it might be more difficult to earn a 3.20 in electrical engineering than, say, in parks and recreation;
• and, the grades of college football players may be artificially inflated.

Still, I think I’m onto something here. Wouldn’t it be great if we could replace the four‑hour SAT with the 12‑minute Wonderlic test? The marketing of it would be easy. For school administrators, simply tout a stronger correlation to college success than the SAT, and mention significantly lower costs. For students, simply state, “You can finish the Wonderlic in 5% of the time it takes to complete the SAT! You won’t have to give up your entire Saturday!” Now, wouldn’t that be grand?