Tag Archives: AP Statistics

Statistics Arts and Crafts

The Chi-Squared chapter in AP Statistics provides a welcome diversion from the means and proportions tests which dominate hypothesis test conversations. After a few tweets last week about a clay die activity I use, there were many requests for this post – and I don’t like to disappoint my stats friends! I first heard of this activity from Beth Benzing, who is part of our local PASTA (Philly Area Stats Teachers) group, and who shares her many professional development sessions on her school website. I’ve added a few wrinkles, but the concept is all Beth’s.

ACTIVITY SUMMARY: students make their own clay dice, then roll their dice to assess the “fairness” of the die. The chi-squared statistic is introduced and used to assess fairness.

clayYou’ll need to go out to your local arts and crafts store and buy a tub of air-dry clay. The day before this activity, my students took their two-sample hypothesis tests.  As they completed the test, I gave each a hunk of clay and instructions to make a die – reminding them that opposite sides of a die sum to 7. Completed dice are placed on index cards with the students names and left to dry. Overnight is sufficient drying time for nice, solid dice, and the die farm was shared in a tweet, which led to some stats jealousy:

The next day, students were handed this Clay Dice worksheet to record data in our die rolling experiment.

In part 1, students rolled their die 60 times (ideal for computing expected counts), recorded their rolls and computed the chi-squared statistic by hand / formula. This was our first experience with this new statistic, and it was easy to see how larger deviations from the expected cause this statistic to grow, and also the property that chi-squared must always be postivie (or, in rare instances, zero).

Students then contributed their chi-squared statistic to a class graph. I keep bingo daubers around my classroom to make these quick graphs. After all students shared their point, I asked students to think about how much evidence would cause one to think a die was NOT fair – just how big does that chi-squared number need to be? I was thrilled that students volunteered numbers like 11,12,13….they have generated a “feel” for significance. With 5 degrees of freedom, the critical value is 11.07, which I did not share on the graph here until afterwards.


In part 2, I wanted students to experience the same statistic through a truly “random” die. Using the RandInt feature on our calculators, students generated 60 random rolls, computed the chi-squared statistic, and shared their findings on a new dotplot.  The results were striking:

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In stats, variability is everywhere, and activities don’t often provide the results we hope will occur. This is one of those rare occasions where things fell nicely into place. None of the RandInt dice exceeded the critical value, and we had a number of clay dice which clearly need to go back to the die factory.

Desmos + Statistics = Happiness

Sunday – a quiet evening before President’s Day – checking out twitter – not looking for trouble – and then,

Wait..what’s this?  Standard Deviation?  It was my birthday this past Saturday, and the Desmos folks knew exactly what to get me as a present.  Abandon all plans, it’s time to play.  A lesson I picked up from Daren Starnes (of The Practice of Statistics fame) is a favorite of mine when looking at scatterplots.  In the past, Fathom had been the tool of choice, but now it’s time to fly with Desmos.  There are a few nuggets from AP Statistics here, and efforts to build conceptual understanding.


Click the icon to the right to open a Desmos document, which contains a table of data from The Practice of Statistics.  In you are playing along at home, this data set comes from page 194 of TPS5e and shows the body mess and resting metabolic rate of 12 adult female subjects. One of the points is “moveable” – find the ghosted point, give it a drag, and observe the change in the LSRL (least-squares regression line) – explore changes and think about what it means to be an “influential” point.

Next, click the “Means” folder to activate it.  Here, we are given a vertical line and horizontal line, representing the means of the explanatory (x) and response (y) variables. Note the intersection of these lines.  Having AP students buy into the importance of the (x-bar, y-bar) point in regression beyond a memorized fact is tricky in this unit.  Drag the point, play, and hopefully we can develop the idea that this landmark point always lies on the LSRL.

Another “fact” from this unit which can easily wind up in the “just memorize it” bin is this formula which brings together slope, correlation, and standard deviation:

The formula is given on the exam, with b1 acting as the slope, so even memorizing it isn’t required, but we can develop a “feel” for the formula by looking at its components.

Click the “Means plus Std Devs” Folder and two new lines appear. we have moved one standard deviation in each direction for the x and y variables. Note that the intersection of these new lines is no longer on the LSRL. But it’s pretty close…seems like there is something going on here.

Ask students to play with the moveable point, and observe how close the rise comes to the intersection point. Can it ever reach the intersection? Can we ever over-shoot it? In the “Rise Over Run” folder, we can then verify that the slope of the LSRL can be found by taking a “rise” of one standard deviation of y, dividing by a “run” of one standard deviation of x, and multiplying by the correlation coefficient, r.

There’s other great stuff happening in the Desmos universe as well.

1.  This summer brings the 4th edition of Twitter Math Camp, to be held at Harvey Mudd College in California. I’m thrilled to have latched onto a team leading a morning session on Desmos. Consider coming out for the free PD event, and join myself, Michael Fenton, Jed Butler, and Glenn Waddell for what promise to be awesome mornings. To be honest, I feel the Ringo of this crew….

2. Can’t make it to the west coast this summer? Join me at the ISTE conference in Philadelphia, where I will present a learning session: “Rethink Math Class with the Desmos Graphing Calculator“. Bring your own device and join in the fun!

3. Are you new to the world of Desmos? Michael Fenton has organized an outstanding series of challenges, with 3 difficulty levels, to help you learn by doing. Try them out – they promise to get you think about how you and your students approach relationships.

4. Merry GIFSmos everybody!  The team at Desmos has developed GIFSmos to let you build your own animated gifs from Desmos files. EDIT – as Eli noted in the comments, credit for GIFSmos goes to Chris Lusto.  Thanks for being so awesome, Chris!

AP Statistics “Best Practices” 2014

Last week, I arrived home after 8 days in Kansas City, where I participated in the AP Statstics Exam reading. It’s hard work, filled with long days of grading papers. But all the readers seem to take some sadistic delight in this work, and the professional connections made through the week are outstanding.

One of the highlights of the week is Best Practices Night, organized by my friend Adam Shrager. This year, 20 or so different folks presented 5-minute looks into their classrooms.  Below are summaries of some of my personal favorites. You can check out all of the presentations on Jason Molesky’s StatsMonkey site


You’ll find that AP Stats teachers enjoy candy….too much so at times my doctor tells me. Last year, Kevin shared his data collection activity with stomp rockets.  This year, Kevin upped the ante, with an activity where students launch Gummy Bears, Gummy Worms and other candies using catapults.  Which type of candy flies farthest? What can we say about the consistancy of the launches? I’m looking to incorporate this into my 9th grade class as an introduction to variability and estimation.


Kevin’s presentation on the StatsMonkey site is Keynote. I have converted it here to Powerpoint for us non-Keynote users.


Stats teachers have many data collection activities in their arsenal, but this idea from Brianna wins the prize for most off-beat concept. In this activity, students are asked to estimate life expectancy in a population. To collect data, the class uses something readily avilable every day: the obituaries. This presentation was one of the clear highlights of the evening, with many in attendance wondering what a class taught by the hysterically entertaining Brianna would be like!  Visit StatsMonkey for her activity worksheet, and use the dead as data!


Jigsaw puzzles make for great reviews in just about any math class.  Here, Christine shares puzzles she uses to review the Normal Distribution. Cut out the pieces, find the probabilities and solve the puzzle!  Template included.


Paul is part of the AP Stats Test Development Committee, and always has great ideas for the Stats Classroom. At the reading, Paul shared his sampling activity, using Air Gun ammo of different colors (and slightly different sizes) to draw small samples from a large population. Using a paddle made from pegboard, random samples can be drawn, leading to a first discussion on inference. Paul promises to share the plans for building your own sampling paddle, so check back on StatsMonkey often!

UPDATE: Paul’s presentation has been uploaded to the StatsMonkey Site, along with plans for making your own sampling paddles.


I appreciate presentations where speakers attempt to de-tangle a tricky concept in math class. Having students move beyond a “canned” understanding of the coefficient of determination and towards a real understanding of predictive improvement based on an explanatory variable is a worthwhile lesson. In his activity, Doug Tyson challenges students to grab as many Starburst candies (see…I told you Stats folks like cnady) as possible in their hand, then examines the predictive value of using hand size to estimate the number of grabbed candies.  How much are our predictions improved by thinking about hand size, as opposed to thinking about the mean?

There’s so much more sharing goodness on the StatsMonkey site, including:

  • A review of Geddit, for formative assessment
  • A QR code scavenger hunt
  • Hershey Kisses and Confident Intervals, which I used in my class this year

Soon, I will post more resources shared by Chris Franklin, who gave a brief history of stats education during her Professional Night presentation.