Saturday, May 12, 2012

Predictably Irrational (Experimental Design)

Interesting TED Talk by Dan Ariely, a professor of Psychology and Behavioral Economics at Duke University, on irrationality in decision-making. I saw this a few years ago and found it excellent for starting conversations with Stats students about experimental design, bias, and (if later in the year) practice with hypothesis testing.

Here are some questions to start discussion:
  1. Regarding the data summarized in a bar chart regarding % of drivers donating organs in the various European countries, was it collected through a study or an experiment? Explain.
  2. According to the video, what was the difference between the countries with high percentage and the countries with low percentage of organ donors?
  3. Sketch a possible experimental design diagram for the experiment involving doctors and pain medication hip replacement.
  4. Sketch a possible experimental design diagram for the experiment involving subscription to the Economist. What was the population of interest? To what extent should the conclusion be drawn?
  5. Sketch a possible experimental design diagram for the experiment regarding ugly Tom and ugly Jerry. Was there a control group in this experiment (based on just the information provided)? Explain.
  6. What bias is being investigated in the video?

Here are some screenshots from the video to set context for question 4:
Treatment 1:

Treatment 2:

This is an AP Test style sample answer to question 4:

Lastly, check the conditions and perform an appropriate hypothesis test using the data from the video. Here's the screenshot with the results:

That's all I have come up with for now. I would appreciate suggestions for improvement.

This is one of the more memorable videos for kids, especially the ugly Tom and ugly Jerry's effect on Tom and Jerry respectively. Students got to see examples of experiments and how they were designed so that conclusions can be drawn. I think it would be even better if the students also got to read the original papers, especially if the papers use the statistical methods learned at the high school level.

Thursday, May 10, 2012

Steve Novak vs Tyson Chandler

The 2011-2012 season of the NBA has been one of the most memorable seasons, at least for this former and now renewed fan of the NBA. I stopped following the NBA after Michael Jordan's second retirement from the Chicago Bulls, partly because I didn't connect with any player or team. I'd catch a few games every now and then, but I haven't cared about any player or any team for quite some time... until Linsanity. The rise of Jeremy Lin has been a joy to witness. The impact of it all, especially on the Asian or just the Taiwanese community, will have to wait for another post. Today, we'll focus on a math question.

One thing about watching Jeremy Lin is that I also got to watch many of his teammates on the New York Knicks. Two players who had an impressive statistical season are Steve Novak and Tyson Chandler. One for having a league-leading 3-point field goal percentage and the other for league-leading field goal percentage. They were both very impressive, but who had the more impressive achievement (statistically speaking)?


To answer the question, we can choose to compare them with players from any time or just with players in this season. I've decided to compare them with the latter, it seems more reasonable to compare them to their peers. Rules of the game do change and player abilities also evolve over time.

A little digging and I find an great site for basketball statistics at basketball-reference.com. It took a little time to clean up the data and combine the stats for players who were traded mid-season. One other issue that popped up were outliers, players who had 0% because they attempted low number of shots and missed all of them and players who had a very high percentage because they made most or all of the few shots they had attempted. I decided (arbitrarily) to set the minimum number of attempts to 66 (the number of games in this shortened season) which is equivalent to about 1 shot attempt per game. Doing this eliminated these outliers and left a histogram that is approximately normal.



From the histograms above, Tyson Chandler seems to be further away from the pack than Steve Novak. In Statistics, there's a useful measure of the distance from the mean in terms of standard deviation that is useful here. It's the z-score. It gives us an idea about how typical or how extreme a value is relative to peers. Z-score is given by the quantity of the value (x) minus the mean (mu), divided by the standard deviation (sigma).
\[ \begin{align*}
z &= \frac{ x - \mu }{\sigma}
\end{align*} \]
Calculating the z-score for Steve Novak's 3-point field goal percentage, when compared to his peers, we get
\[ \begin{align*}
z_{novak} &= \frac{ 0.471631206 - 0.351442150 }{0.051557762}\\
&= 2.33115347
\end{align*} \]
Calculating the z-score for Tyson Chandler's field goal percentage, when compared to his peers, we get
\[ \begin{align*}
z_{chandler} &= \frac{ 0.678873239 - 0.441181706 }{0.060831197}\\
&= 3.9073953
\end{align*} \]
While both achievements are impressive, Chandler's field goal percentage is almost 4 standard deviations above the mean while Novak's performance is slightly higher than about 2 standard deviations. Chandler wins. It's not even close. One thing to keep in mind, even as we draw this conclusion, is that players who attempt a large number of 3-pointers are typically pretty good at it. Whereas, it's reasonable that even poor shooters will attempt 2-pointers so Tyson's "peers" may have a larger proportion of poor shooters thus lowering the average.

Now, if only I could get my hands on the data for the points scored in the first 5 career starts of a player. :)

PS: Apparently, Tyson Chandler's .679 field goal percentage is second (third?) only to Wilt Chamberlain's .7270 (1972-1973) and .6826 (1966-1967) on the all time list for highest field goal percentage in a single season. Unfortunately, Chandler did not meet the rate minimum requirements to show up on that list. For field goal percentage, the requirement is 300 FG. Chandler had 241 FG.

PPS: Chandler could also have made the true shooting percentage list. Once again, unfortunately, for true shooting percentage, the requirement is 700 PTS. Chandler had 699 (missed it by 1).

Saturday, May 5, 2012

Graph of a Function and its Inverse (VIDEO)

Using a technique from art class, this activity helps students visualize the reflection of the graph of a function about the line y=x when sketching a function and its inverse. Students who went on to calculus tell me this was one of the activities they remembered from my Pre-Calc class.

Be sure to turn on the annotations (if it is off) to see the directions and comments. This time, I decided to spare you from my voice and some random Creative Commons licensed background music I would usually add to videos.



I suppose you can also use this strategy in Geometry to help students visualize reflections. The novelty factor from doing this in a Pre-Calc class really helped students remember how a function and its inverse are connected graphically. This is especially useful when learning about logarithmic functions.

PS: This is my first time playing with annotations on youtube.

Friday, May 4, 2012

Mathy Comics - XKCD

More than just mathy, it's also sciency and geeky. XKCD is a must read! When you visit the site, be sure to mouse over the picture and hold. The alt text will soon appear with a hidden punch line and commentary. Here are two of my favorites on:

Correlation and causation

Extrapolation

Andromeda Yelton (@ThatAndromeda) keeps a fairly complete directory of XKCD comics by topic called DCKX - Directory of Curricular Knowledge in xkcd. Handy for those who teach math and science.

Mathy Comics
Mathy Comics - SMBC (Saturday Morning Breakfast Cereal)
Mathy Comics - XKCD

Thursday, May 3, 2012

AP Test Taking FAIL

... or "this is what happens when you don't read or follow directions"

I've told it several ways. This one seemed to work well with kids:

Me: ...So a long long time ago...

Students: How long?

Me: Hrm. When you could still hear Boyz II Men on the radio.

Students: No one listens to radio anymore!

Me: Ugh. Anyways. There was a boy who came to the United States with his family hoping for better opportunities and a better education. After a recommendation from a counselor, he decided to take AP Computer Science in his sophomore year, which back then was in a programming language called Pascal and students worked on Apple IIes with green monochrome monitors and floppy disks.

He did well in the class, usually getting one of the higher scores on quizzes and tests and he was certain he would do well on the AP test. On the AP test, he breezed through the Multiple Choice section and felt confident about his performance, at least until the Free Response Questions section.

The first question in FRQ that he worked on is a function called square root. He thought long and hard about what the question was really asking and about the kind of answer the function was expecting. A satisfactory answer wasn't coming to him as quickly as it previously had in class and in practice tests. As time passed, he began to feel that a solution would elude him. Throughout his academic career he was successful because of his persistence, so he persisted.

About half-an-hour, maybe 45 minutes later, he decided that at the rate he was working, he'd never finish the test. He wasn't one to give up easily, but after some hesitation and indecision, the anxiety about not finishing the test got to him. He finally decided to move on and work on the next question and revisit the first question if he had time later. He turned over the page and looked at the next question and noticed that it said #1. Puzzled, he turned back to the first question he worked for close to an hour to see what was going on. He was shocked that it had no number. Was this a typo? Did College Board make a mistake? Upon further examination, the question that he worked on before wasn't even a question, it was just an example. His heart sank.

He doesn't remember if he completed the test, probably just blocked out the memory. He did get a score of 3 on that test, and the following year he felt it necessary to redeem himself by doing better on an even more challenging test (Computer Science AB), which he did. Eventually, the boy grew up to become a teacher and to tell his students a cautionary tale about the pitfalls of not reading directions carefully on the AP test.

THE END

This is a story I tell my students every year about the importance of reading and following directions. It took a little digging, but here it is. With a little annotation, it is pretty self-explanatory. Feel free to share it with your students.