Category: Statistics

Categories
Statistics

The NFL Draft: Shopping for Bargains!

Last week, the NFL player draft took place over 3 days in New York City, and now the annual exercise of “grading” each team based on their draft haul commences.  It’s a fun debate, with grades often based more on feel or perceived value, rather than any real analysis.

There are many ways to evaluate draft results, but from a purely mathematical standpoint, I like to look at value.  Which teams  got the best “bargains”, and which teams went out on a limb?  If you had the 20th pick in the draft, did you get the 20th best player?  Or did you draft a lower-ranked player.

I took all of this year’s 254 players drafted in the NFL draft, and compared them to their draft ranking, according to CBS Sports.  The only real reason I have for using CBS as opposed to the many other draft rankings out there, is that it was easy to pull their data out into a spreadsheet.  From there, I computed the “value” of each pick.  If a team drafted a player above his rank, this is negative value.  If a team drafted a player after his rank, this is a positive value.  Some examples:

Geno Smith was drafted with the 39th pick, but was ranked 21st by CBS Sports, so his value was +18

Meanwhile, E.J. Manuel was drafted with the 16th pick, but was ranked 40th, for a value of -24.  

Some players represented great values for the teams which picked them:

Cornelius Washington, Chicago Bears (pick 188, ranked 82, +106)

Andre Ellington, Arizona Cardinals (187, 88, +99)

Jordan Poyer, Philadelphia Eagles (218, 119, +99)

While other players could be considered “reaches”:

B.J. Daniels, SF 49ers (pick 237, ranked 818, -581)

Jon Meeks, Buffalo Bills (143, 834, -691)

Ryan Seymour, Seattle Seahawks (220, “1000”, -780).  Ryan is the only drafted player who did not appear in CBS’s top 1000, so I just assigned him #1000.

There is a bit of un-fairness here, as many teams will use later picks on “projects”, players who have little expectation of making the team, but who seem to have a particular upside, so there was much volatility in the later round values.

From there, I simply added up the value scores for the players drafted by each team, and found an overall value score.  So, which teams earn the best grades?  Only 3 teams earned overall positive scores.  This is understandable, as it is much easier to earn negative scores than positives, especially in the later rounds.

THE TOP 3:

Minnesota Vikings (+187)

Chicago Bears (+51)

Philadelphia Eagles (+25)

THE BOTTOM 3:

Buffalo Bills (-836)

SF 49ers (-1097)

Seattle Seahawks (-1571)

For math class, have your students think of other ways to measure draft success.  Is the value measure here valid?  How can the method be adjusted?  How do some of the huge negative numbers in this data influence results?  Feel free to download and toy around with the data in my draft value tracker, and let me know what you come up with!

Categories
Statistics

Monopoly Math

The big Monopoly battle is coming near its end, and the iron and racecar are battling for Monopoly supremacy.

Monopoly Board

Both players own properties on the next block, and have some spaces they’d like to avoid.  For the car, here are the spaces he’d like to avoid.

Car Spots

And for the iron, there are a few spaces to avoid.

Monopoly - Iron 1

Since there some houses and hotels on some of the spaces, they are worth different amounts.  Below, here is how much each player will have to pay if they land on the “bad” spaces.

Car Board B

Monopoly - Iron 2

So, here’s the question:  which player is in “worse” shape?  Which player should be more worried about their upcoming turn?

Let this stew with your classes, and would enjoy hearing some class reflections.  The big reveal will come in a few days.

Categories
Statistics

Expected Value and Analyzing Decisions, part 1

As an A.P. Statstics reader, I am excited to see the increased emphasis on statistics and probability in the Common Core standards.  Ever better, the standards specifically ask our students to be able to reach conclusions based on data:

CCSS.Math.Content.HSS-MD.B.5 (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

CCSS.Math.Content.HSS-MD.B.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

This is a great start, and requires that we move beyond just flipping coins and drawing beads from bags.  We need to get our students writing about what they see, and provide strategies for developing clear, statistical thinking.

In this first post, we’ll look at a famous game show, and examine possible decisions.  Next, the game of Monopoly will provide a more complex argument in expected value.  Finally,  in a third post, we’ll look at past Advanced Placement Statistics items and adapt them for use in non-AP clasrooms.

The game show “Let’s Make a Deal” provides a simple example in decision-making based on values and probability.  At the end of the show, contestants who won prizes during the show are asked if they would like to give up their prizes for a chance at a bigger prize by choosing one of three doors:

Should a player give up their prizes?  How much does the “big deal” need to be worth in order for a player to be tempted to give up what they won?  How about the other two doors…do they matter?  Show the first video to your class, and let them debate whether they would go for it, and take a class vote.

In the next stage, we can show what all 3 doors have behind them, and begin to consider the game as a whole:

Does this new information change any decisions?  How willing are you to risk your prizes if you know there are some other nice prizes to be won, which may or may not have the same value of what you already have?  The slides below let you walk through a discussion with your class.  Use dry erase boards or even Google Forms to have students share their ideas.

There are a number of approaches which can be taken here.  Hopefully your students develop one of these on their own, or have creative, new ideas.

  • In this example, 1/3 of the doors hold the Big Deal, but another door has a prize of essentially equal value, the “medium” prize.  It may be difficult for us to predict the value of the middle prize, but it seems plusible that 2/3 of the doors will have value at least equal to what we are giving up.
  • In the worst-case scnenario, the non “Big Deal” doors have minimal value to us, essentially zero.  So, we have 1/3 chance of selecting the Big Deal, and 2/3 chance of winning nothing.  We can compute the expected value:

  • In the video above, we have some insight into the three doors on the show, and there always seems to be a “middle” prize and a “small” prize.  We can compute the expected value based on this information:

In any case, it seems reasonable for us to consider giving up the $7,000, but next we can think about the limits of our decisions.  What is the most you would give up to go for the Big Deal?  Can students create a general rule for making the decision?

And finally, a few parting shots for discssion:

  • It’s easy to think of the prizes as a cash value only, but does the prize you are giving up matter?  What if you always wanted to go to Paris?  Does that add to the value?  Or are there prizes you would never give up?
  • Expected value gives a nice summary of what we should expect to see in the long run, after many, many trials.  But on Let’s Make a Deal, you only have one shot…the chance of a lifetime.  Does this change your approach to the decision-making?

Would enjoy hearing your class experiences in sharing the Let’s Make a Deal examples.  Stay tuned for the next post on Monopoly.