Of course, the sample size is naturally restricted by the small number of Super Bowls, but if the author (Mark Pesavento) had really been interested in the question of whether or not the coin toss is correlated with the winner in a football game, he could’ve easily collected data over a couple of seasons and obtained an answer to the question.  At the very least, he could’ve owned up to the fact that his analysis is worthless, but instead, to the critics he offers only the following rebuttal: “because of the small sample size, some statisticians argue that the win-loss record of coin-toss winners is statistically insignificant.”

This is completely disingenuous, because it suggests that there would be a debate among statisticians about the significance in the data Pesavento uses, when no such debate exists.  Anyone with even a rudimentary background in statistics would understand that the sample size here would be too small to draw the conclusion he draws.

Moreover, Pesavento falls for one of the most common traps in statistics: mistaking correlation for causation.  Even if the data was much stronger in indicating that the coin toss winner is at a disadvantage, this would not imply that Brees should hope to lose the toss.  A correlation between these two effects does not imply a causal relationship between the two.  I feel like I’ve discussed this before, but just in case, here’s a thorough discussion of this misconception.

Here this point is moot, since we don’t even have a correlation.  I thought no one would need to point out that “No correlation does not imply causation,” but apparently we do.

Thankfully, most of the comments on Pesavento’s post are scathing in regards to his methods.  But that’s cold comfort in light of the fact that the article was deemed fit for posting on the front page of Yahoo.

In the first part of this discussion, we introduced a new way to conduct the pool: rather than looking at the last digit of a team’s score, we looked instead at the digital root of the team’s score.  Recall that the digital root of a team’s score is obtained by adding the digits in their score.  If that sum is between 1 and 9, we stop – if it is larger than 9, we compute the digital root again, until we get a digit between 1 and 9.  For example, the digital root of 14 is 1 + 4 = 5, while the digital root of 38 is 2, since 3 + 8 = 11, and 1 + 1 = 2.  We then analyzed the distribution of scores, and found that the digital root of a team’s score is more evenly distributed between 1 and 9 than the last digit of a team’s score is evenly distributed between 0 and 9 (this is subject to the convention that we assign 0 a digital root of 9, since 0 is the only number with digital root equal to 0).

In the second part of the discussion, we tackled questions of independence.  Namely, we asked whether the last digit in one team’s score is independent of the last digit of the other team’s score, and similarly we asked whether the digital root in one team’s score is independent of the digital root of the other team’s score.  In both cases, we found the answer to be negative.

The subject of this article is based on the following observation: when you have wagered in a traditional football pool, it’s not uncommon for a small number of squares to be hit with high frequency during the course of a game.  For example, suppose you watch a game in which one team scores 7 points, then 3, then 7, then 3, while the opposing team never scores.  This means that the game’s score will go from 0-0, to 7-0, to 10-0, and then to 17-0.  So, while there are four unique scores in the game, with the usual football pool, only two squares will be hit: the 0-0 square, and the 7-0 square.  However, with the digital root pool, four squares will be hit: again using the convention that we assign 0 a digital root of 9, the squares will be 9-9, 7-9, 1-9, and 8-9.

The reason the digital root pool hits more squares in this case is because whenever one team increases its score by 10, the last digit of their score will return to a previous value.  However, with the digital root method, if a team increases its score by 10, the digital root increases by 1.  Because a score increase of 10 is a relatively common occurrence in football (all one needs is a touchdown, extra point, and field goal), one may therefore guess that using the digital root pool, more squares should be hit during the course of the game.

Whether one would want more squares to be hit or not is up for debate, but I see certain benefits.  For example, if more squares are hit during the game, then more people will have something invested in the game as it airs.  If you are sitting on the square that represents the current score, you want the score to remain the same through the end of the quarter so that you can reap the rewards – but if the winning squares keep bouncing around between a small number of people, there may be fewer people actively invested in the score as the game progresses.  This is especially true in Super Bowl parties, when many of the attendees are less interested in the game than they should be.

In other words, I’m of the belief that if more squares are hit, it’s a good thing.  It therefore becomes natural to ask whether or not the digital root pool actually does hit more unique squares than the traditional pool.  Thankfully, we have a wealth of data which we can use to answer this question.

I looked at all the games from this current season, and counted the number of boxes that would have been hit in each game using the traditional pool and the digital root pool.  Averaged over 331 games (this includes preseason and postseason), the number of squares hit using the traditional pool is approximately 6.84.  By comparison, the number of squares hit using the digital root pool is 8.43 – an increase of 1.59 boxes, or an increase of about 23%.  This effect is amplified when one considers the fact that the digital root pool uses only 81 squares, as opposed to the traditional pool’s 100.  This means that as a proportion of the total number of squares, the traditional pool hits about 6.84% of its squares, while the digital root pool hits 10.4% – here we have an increase of over 50%!

This is strong evidence that the digital root pool hits more squares than the tradition pool.  In fact, the data shows that an average game will have a change in the score approximately 8.73 times, which is only a bit higher than the average number of boxes hit by the digital root pool.  This makes sense when we slice the data another way: of the 331 games analyzed, in 252 of them the number of squares hit with the digital root pool was equal to the number of changes of score, meaning that no square got hit more than once.  The same cannot be said of the traditional pool – in this case, the number of games in which no square got hit more than once was only 62.

The data has convinced me that the digital root pool may be better suited for festive gathering, where wagering on football will be but one of many activities designed to induce merriment.  At the very least, it’s hard to argue that the traditional pool will hit as many squares as the digital root pool. Some may balk at a break from football pool tradition, but that’s ok.  I won’t watch football games with them anyway.

1) Arguing that students shouldn’t learn calculus because they may not use it in their everyday life is specious. By this reasoning, I should never have taken any courses in history, biology, or chemistry. The purpose of high school education in this country seems to be not only determining what educational avenues students want to pursue further, but also what avenues they don’t want to pursue. If you want to argue that students should only be learning things that they can apply to their everyday lives, then you are arguing for a much more sweeping reform of education.

I do acknowledge that there is an opportunity cost at work when we spend a year teaching a student calculus rather than statistics, and certainly the average student will find more use later in life for the latter. But there’s also an opportunity cost at work when we spend a year teaching a student statistics rather than calculus, especially for students who aren’t sure in what direction their academic future will head. If anything, this seems to be an argument for offering both statistics and calculus for students, rather than forcing them into one option or the other.

2) About 2/3rds of the way through the talk, Benjamin asserts that “if our students, if our high school students, if all of the American citizens knew about probability and statistics, we wouldn’t be in the economic mess we’re in today.” This is met with some cheers from the audience, but is it actually true?

The answer depends on your definition of “knowing” probability and statistics. I agree that having some knowledge of statistics is a good thing for the population at large, and there are no doubt many fundamental principles that could be taught at a high school level – for example, the idea that correlation does not imply causation, or the ways in which one can manipulate data or graphs of data. These topics, among others, are discussed in the book How to Lie with Statistics, which would be a great required reading book for any teacher trying to impart intuition and a healthy dose of skepticism onto his or her students, and is written for a general audience.

However, even if everyone in America had this basic level of knowledge, it’s not at all clear that this would have somehow saved us from economic catastrophe. If you work in finance, odds are pretty good that you already have a knowledge of statistics that goes beyond a high school level, but this didn’t stop the economy from tanking.

Nassim Nicholas Taleb wrote an excellent article last year on the limits of statistics, which is well worth a read if you can spare the time. One of the arguments he makes is that part of the reason the financial models caused such an economic implosion was that these models are necessarily unable to predict black swan events, which can have a tremendously negative impact, but are also tremendously rare. In fact, he argues that statistics is actually quite poor at trying to predict what will happen in extremely complex systems where rare extreme events can have a profound effect on the system.

However, this is not what one learns in a high school or even undergraduate class on statistics. Most problems at this level involve simple systems (games of chance, for example). In other words, studying statistics at a low level does not expose one to the subtleties and limitations of the subject – in particular, I don’t think it’s feasible to say that if every high school graduate had taken a course in statistics, somehow we would have prevented the current economic catastrpohe. To do so would have required a much deeper understanding of statistics among those applying the financial models than can be supplied at the high school level.

This brings me to my third point…

3) To have a good understanding of statistics, one must already have a working knowledge of calculus. There is a limit to the amount of depth a probability or statistics course can explore when calculus is not a prerequisite, and because of this many results (such as the Central Limit Theorem) are stated without proof. This is fine if you are simply trying to expose students to some of the standard tools in the subject, but if you can’t go deeper, there really is a limit to the level of understanding a student can achieve.

I agree that there is a great deal of value in teaching statistics to high school students, even at the level of pre-calculus. One can still impart a significant amount of intuition at this level. However, for students who plan to use statistics in any significant capacity, it’s important that they develop a working knowledge of calculus as well.

If you’re not planning on going into a technical field, certainly you’ll get more value out of a basic statistics class than you will a calculus class. But students who dislike math in general will probably still dislike statistics, even though there’s more to like for someone who’s not interested in math than there is in a calculus course.

This, in turn, brings me to my next point…

4) In the larger debate over the failings of math education in America, the choice of whether to teach statistics or calculus in secondary school misses the point entirely. By the time students reach the later years of their high school career, most already have a pretty well developed sense of their relationship to mathematics – either they had good teachers and enjoyed the subject, or through a series of misfortunes which may have been out of the student’s control, they feel like math is a subject they will never understand, and will struggle with until they have the freedom to not take a math class, and are finally free from its iron grip.

Sadly, the problems with math education in this country run much deeper, and swapping out calculus for stats at higher levels won’t alleviate the fundamental problems students have with mathematics. When I grade papers in a calculus class, students make just as many (if not more) algebra mistakes as they do calculus mistakes. In other words, many students leave the year without having mastered the math concepts presented to them during that year. Compound this over several years, and it doesn’t matter if you give them a calculus book or a statistics book – they will have trouble because they haven’t mastered the prerequisites.

Certainly one can argue that there are fewer prerequisites in a statistics class, but prerequisites are still present, and algebra is certainly one of them. If a student has a poor understanding of algebra, it’s reasonable to assume he will have significant gaps in his understanding of statistics, and if the goal is to give students an intuition for randomness and understanding data that can help them in their everyday lives, gaps in statistical understanding are significant problems. Therefore, achieving this goal isn’t as simple as making sure every high school senior has taken a statistics class – we really need to insist that every student first has a working knowledge of algebra. This is a problem we have already, and is not resolved by Benjamin’s proposal.

5) This is a small point, but important. What really bothers me about this talk is when Benjamin makes the statement that, “If [probability and statistics] is taught properly, it can be a lot of fun!” Well, yes, but this is true of any subject. Implicit here seems to be the idea that calculus cannot be fun, even if taught well. I’m sure this isn’t Benjamin’s intention, but it’s easy to misinterpret, especially if you are someone who has never taken a calculus class, or has fallen victim to the commonly held opinion that calculus is some kind of black magic whose secrets only a chosen few can hope to unravel.

The truth (and one that Benjamin knows) is that any math class can be fun if taught properly. A more accurate statement might be “it’s easier to make probability and statistics fun for students,” because of the vast applicability to everyday life, from games of chance to calculating the probability that someone in a family is colorblind. But to suggest that statistics is inherently more fun for students than calculus does a disservice to all the great teachers of calculus. Either class can be fun and valuable if taught well, or traumatizing if taught poorly.

Mr. Prezbo, using probability to make math fun. No doubt he could work this magic on other math subjects as well.
I understand what Benjamin is saying, and I also understand its appeal. The argument works well as a 3 minute sound clip, but upon further reflection, there are some significant questions that need to be addressed. There are many problems with math education in this country, and I’m not sure which, if any, are solved by this proposal.

From my own experience, no students in my high school were forced to take either calculus or statistics, although both courses were offered. Preparing students exclusively for either one or the other will of course do a disservice to some, so perhaps putting both on the table is the best compromise, although this becomes a problem for schools with limited resources. I am confident, however, that simply putting statistics on the pedestal currently occupied by calculus doesn’t do a whole lot in terms of fixing everything that’s broken.