Single-Digit NFL scores


Stay Focused

From Kirk Cousins, quarterback of the Washington Redskins:

Sometimes our guests ask why I have this hanging above my desk. It’s an old high school math quiz when I didn’t study at all and got a C+… just a subtle reminder to me of the importance of preparation. If I don’t prepare I get C’s!


Applying Science to Speed Training

I enjoyed this surprising (well, surprising to me) application of exponential functions: training sprinters and other runners.

College Basketball’s Numbers Game

This is a very interesting (and readable) article about the blending of mathematics and coaching college basketball:

The Running Nerd: The US Marathoner Who Is Also a Statistics Professor

I loved these articles about Jared Ward, an adjunct professor of statistics at BYU who also happens to be a genuine and certifiable jock… he finished the 2016 Olympic marathon in 6th place with a time of 2:11:30.

Ward started teaching at his alma mater after graduating from BYU with a master’s degree in statistics in April 2015…

Ward wrote his master’s thesis on the optimal pace strategy for the marathon. He analyzed data from the St. George Marathon, and compared the pace of runners who met the Boston Marathon qualifying time to those who did not.

The data showed that the successful runners had started the race conservatively, relative to their pace, and therefore had enough energy to take advantage of the downhill portions of the race.

Ward employs a similar pacing strategy, refusing to let his adrenaline trick him into running a faster pace than he can maintain.

And, in his own words,

[A]t BYU, on our cross-country field, on the guys side, there were maybe 20 guys on the team; half of them were statistics or econ majors. There was one year when we thought if we pooled together all of the runners from our statistics department, we could have a stab with just that group of guys at being a top-10 cross-country team in the nation…

To be a runner, it’s a very internally motivated sport. You’re out there running on the road, trying to run faster than you’ve ever run before, or longer than you’ve ever gone before. That leads to a lot of thinking and analyzing.We’re out there running, thinking about what we’re eating, what we need to eat, energy, weightlifting, how our body feels today, how it’s going to feel tomorrow with how much we run today. We’re gauging all of these efforts based on how we feel and trying to analyze how we feel and how we can best get ourselves ready for a race. As opposed to all the time on a soccer field, you’re listening to do a drill that your coach tells you to do, and then you go home.

I think we have a lot of time to think about what we are doing and how it impacts our performance. And statistics is the same way. It’s thinking about how numbers and data lead to answers to questions.

Yes, I think there’s probably some sort of connection there to nerds and runners.

Sources: and

Exponents and the decathlon

During the Olympics, I stumbled across an application of exponents that I had not known before: scoring points in the decathlon or the heptathlon. From

Decathlon, which at the Olympics is a men’s event, is composed of 10 events: the 100 meters, long jump, shot put, high jump, 400 meters, 110-meter hurdles, discus throw, pole vault, javelin throw and 1,500 meters. Heptathlon, a women’s event at the Olympics, has seven events: the 100-meter hurdles, high jump, shot put, 200 meters, long jump, javelin throw and 800 meters…

As it stands, each event’s equation has three unique constants — $latex A$, $latex B$ and $latex C$— to go along with individual performance, $latex P$. For running events, in which competitors are aiming for lower times, this equation is: $latex A(BP)^C$, where $latex P$ is measured in seconds…

B is effectively a baseline threshold at which an athlete begins scoring positive points. For performances worse than that threshold, an athlete receives zero points.

Specifically from the official rules and regulations (see pages 24 and 25), for the decathlon (where P is measured in seconds):

  • 100-meter run: 25.4347(18-P)^{1.81}.
  • 400-meter run: 1.53775(82-P)^{1.81}.
  • 1,500-meter run: 0.03768(480-P)^{1.85}.
  • 110-meter hurdles: 5.74352(28.5-P)^{1.92}.

For the heptathlon:


  • 200-meter run: 4.99087(42.5-P)^{1.81}.
  • 400-meter run: 1.53775(82-P)^{1.88}.
  • 1,500-meter run: 0.03768(480-P)^{1.835}.

Continuing from FiveThirtyEight:


For field events, in which competitors are aiming for greater distances or heights, the formula is flipped in the middle: $latex A(PB)^C$, where $latex P$ is measured in meters for throwing events and centimeters for jumping and pole vault.

Specifically, for the decathlon jumping events (P is measured in centimeters):

  • High jump: 0.8465(P-75)^{1.42}
  • Pole vault: 0.2797(P-100)^{1.35}
  • Long jump: 0.14354(P-220)^{1.4}

For the decathlon throwing events (P is measured in meters):

  • Shot put: 51.39(P-1.5)^{1.05}.
  • Discus: 12.91(P-4)^{1.1}.
  • Javelin: 10.14(P-7)^{1.08}.

Specifically, for the heptathlon jumping events (P is measured in centimeters):

  • High jump: 1.84523(P-75)^{1.348}
  • Long jump: 0.188807(P-210)^{1.41}

For the heptathlon throwing events (P is measured in meters):

  • Shot put: 56.0211(P-1.5)^{1.05}.
  • Javelin: 15.9803(P-3.8)^{1.04}.

I’m sure there are good historical reasons for why these particular constants were chosen, but suffice it to say that there’s nothing “magical” about any of these numbers except for human convention. From FiveThirtyEight:

The [decathlon/heptathlon] tables [devised in 1984] used the principle that the world record performances of each event at the time should have roughly equal scores but haven’t been updated since. Because world records for different events progress at different rates, today these targets for WR performances significantly differ between events. For example, Jürgen Schult’s 1986 discus throw of 74.08 meters would today score the most decathlon points, at 1,384, while Usain Bolt’s 100-meter world record of 9.58 seconds would notch “just” 1,203 points. For women, Natalya Lisovskaya’s 22.63 shot put world record in 1987 would tally the most heptathlon points, at 1,379, while Jarmila Kratochvílová’s 1983 WR in the 800 meters still anchors the lowest WR points, at 1,224.

FiveThirtyEight concludes that, since the exponents in the running events are higher than those for the throwing/jumping events, it behooves the elite decathlete/heptathlete to focus more on the running events because the rewards for exceeding the baseline are greater in these events.

Finally, since all of the exponents are not integers, a negative base (when the athlete’s performance wasn’t very good) would actually yield a complex-valued number with a nontrivial imaginary component. Sadly, the rules of track and field don’t permit an athlete’s score to be a non-real number. However, if they did, scores might look something like this…



This Is Why There Are So Many Ties In Swimming

From the excellent article “This Is Why There Are So Many Ties In Swimming“, ties in swimming are allowed by the sport’s governing body because of the inevitability of roundoff error.

In 1972, Sweden’s Gunnar Larsson beat American Tim McKee in the 400m individual medley by 0.002 seconds. That finish led the governing body to eliminate timing by a significant digit. But why?

In a 50 meter Olympic pool, at the current men’s world record 50m pace, a thousandth-of-a-second constitutes 2.39 millimeters of travel. FINA pool dimension regulations allow a tolerance of 3 centimeters in each lane, more than ten times that amount. Could you time swimmers to a thousandth-of-a-second? Sure, but you couldn’t guarantee the winning swimmer didn’t have a thousandth-of-a-second-shorter course to swim. (Attempting to construct a concrete pool to any tighter a tolerance is nearly impossible; the effective length of a pool can change depending on the ambient temperature, the water temperature, and even whether or not there are people in the pool itself.)

Tennis and best 2-out-of-3 vs. best 3-out-of-5

I recently read a very interesting article on regarding men’s and women’s tennis that reminded me of the following standard problem in probability.

Player X and Player Y play a series of at most n games, and a winner is declared when either Player X or Player Y wins at least n/2 games. Suppose that the chance that Player X wins is p, and suppose that the outcomes of the games are independent. Find the probability that Player Y wins if (a) n = 3, (b) n = 5.

The easiest way to solve this is to assume that all n games are played, even if that doesn’t actually happen in real life. Then, for part (a), we can use the binomial distribution to find

  • P(X = 0) = P(Y = 3) = (1-p)^3
  • P(X = 1) = P(Y = 2) = 3p(1-p)^2
  • P(X = 2) = P(Y = 1) = 3p^2(1-p)
  • P(X = 3) = P(Y = 0) = p^3

Adding the first two probabilities, the chance that Player Y wins is (1-p)^3 + 3p(1-p)^2 = (1-p)^2 (1+2p).

Similarly, for part (b),

  • P(X = 0) = P(Y = 5) = (1-p)^5
  • P(X = 1) = P(Y = 4) = 5 p (1-p)^4
  • P(X = 2) = P(Y = 3) = 10p^2 (1-p)^3
  • P(X = 3) = P(Y = 2) = 10 p^3 (1-p)^2
  • P(X = 4) = P(Y = 1) = 5 p^4 (1-p)
  • P(X = 5) = P(Y = 0) = p^5

Adding the first three probabilities, the chance that Player Y wins is (1-p)^5 + 5p(1-p)^4 + 10p^2(1-p)^3 = (1-p)^3 (1+3p+6p^2).

The graphs of (1-p)^2 (1+2p) and (1-p)^3 (1+3p+6p^2) on the interval 0.7 \le p \le 0.9 are shown below in blue and orange, respectively. The lesson is clear: if p > 0.5, then clearly the chance that Player Y wins is less than 50%. However, Player Y’s chances of upsetting Player X are greater if they play a best 2-out-of-3 series instead of a best 3-out-of-5 series.

best2outof3Remarkably, this above curve has been observed in real-life sports: namely, women’s tennis (which plays best 2 sets out of 3 — marked WTA below) and men’s tennis (which plays best 3 sets out of 5 in Grand Slams — marked ATP below). The chart indicates that when two men’s players ranked 20 places apart play each other in Grand Slams, an upset occurs about 13% of the time. However, the upset percentage is only 5% in women’s tennis. (This approximately matches the above curve near p = 0.8.)

However, in tennis tournaments that are not Grand Slams, men’s tennis players also play a matches with a maximum of 3 sets. In those tournaments, the chances of an upset are approximately equal in both men’s tennis and women’s tennis.

However, since the casual tennis fan (like me) only tunes into the Grand Slams but not other tennis matches, this fact is not widely known — which gives the misleading impression that top women’s tennis players are not as tough, skilled, etc. as men’s tennis players.

The FiveThirtyEight article argues that top women’s tennis players don’t make it to the latter stages of Grand Slam tournaments than top men’s players because of the two tournaments are held under these different rules, and that women’s tennis would be better served if their matches were also played in a best-3-out-of-5 format.



Mathematics that Swings: The Math Behind Golf

From the YouTube description:

Mathematics is everywhere, and the golf course is no exception. Many aspects of the game of golf can be illuminated or improved through mathematical modeling and analysis. We will discuss a few examples, employing mathematics ranging from simple high school algebra to computational techniques at the frontiers of contemporary research.

New England Patriots Cheat At the Pre-Game Coin Flip? Not Really.

Last November, CBS Sports caused a tempest in a teapot with an article with the sensational headline “Patriots have no need for probability, win coin flip at impossible rate.” From the opening paragraphs:

Bill Belichick is never unprepared. Or at least that’s the perception. When other coaches struggle with when to use timeouts or how to manage the clock, the Patriots coach, almost effortlessly, always seems to make the right decision.

Belichick has also been extremely lucky. The Pats have won the coin toss 19 of the last 25 times, according to the Boston Globe‘s Jim McBride.

For some perspective: Assuming the coin toss is a 50/50 proposition, the probability of winning it at least 19 times in 25 tries is 0.0073. That’s less than three-quarters of one percent.

As far as the math goes, the calculation is correct. Using the binomial distribution,

\displaystyle \sum_{n=19}^{25} {25 \choose n} (0.5)^n (0.5)^{25-n} \approx 0.0073.

Unfortunately, this is far too simplistic an analysis to accuse someone of “winning the coin flip at an impossible rate.” Rather than re-do the calculations myself, I’ll just quote from the following article from the Harvard Sports Analysis Collective. The article begins by noting that while the Patriots may have been lucky the last 25 games, it’s not surprising that some team in the NFL was lucky (and the lucky team just happened to be the Patriots).

But how impossible is it? Really, we are interested in not only the probability of getting 19 or more heads but also a result as extreme in the other direction – i.e. 6 or fewer. That probability is just 2*0.0073, or 0.0146.

That is still very low, however given that there 32 teams in the NFL, the probability of any one team doing this is much higher. To do an easy calculation we can assume that all tosses are independent, which isn’t entirely true as when one team wins the coin flip the other team loses. The proper way to do this would be via simulation, but assuming independence is much easier and should yield pretty similar results. The probability of any one team having a result that extreme, as shown before, is 0.0146. The probability of a team NOT having a result that extreme is 1-0.0146 = 0.9854. The probability that, with 32 teams, there is not one of them with a result this extreme is 0.985432 = 0.6245998. Therefore, with 32 teams, we would expect at least one team to have a result as extreme as the Patriots have had over the past 25 games 1- 0.6245998 = 0.3754002, or 37.5% of the time. That is hardly significant. Even if you restricted it to not all results as extreme in either direction but just results of 19 or greater, the probability of one or more teams achieving that is still nearly 20%.

The article goes on to note the obvious cherry-picking used in selecting the data… in other words, picking the 25 consecutive games that would make the Patriots look like they were somehow cheating on the coin flip.

In addition the selection of looking at only the last 25 games is surely a selection made on purpose to make Belichick look bad. Why not look throughout his career? Did he suddenly discover a talent for predicting the future? Furthermore, given the length of Belichick’s career, we would almost expect him to go through a period where he wins 19 of 25 coin flips by random chance alone. We actually simulate this probability. Given that he has coached 247 games with the Patriots, we can randomly generate a string of zeroes and ones corresponding to lost and won con flips respectively. We can then check the string for a sequence of 25 games where there was 19 or more heads. I did this 10,000 times – in 38.71% of these simulations there was at least one sequence with 19 or more heads out of 25.

The author makes the following pithy conclusion:

To be fair, the author of this article did not seem to insinuate that the Patriots were cheating, rather he was just remarking that it was a rare event (although, in reality, it shouldn’t be as unexpected as he makes it out to be). The fault seems to rather lie with who made the headline and pubbed it, although their job is probably just to get pageviews in which case I guess they succeeded.

At any rate, the Patriots lost the coin flip in the 26th game.