Power and Virtual Coins

This activity was inspired by the article “Innocent Until Proven Guilty”, by Catherine Case and Doug Whitaker. NCTM Mathematics Teacher, Volume 109, Issue 9 (May 2016)

Around February each year, the AP Statistics message boards come alive with new and veteran AP Statistics teachers seeking ideas to help students understand the concept of statistical power. While Power is a “minor league” topic in the AP Stats curriculum, a robust discussion of the concept can help tie together the logic of statistical inference: P-values, error and sampling variability. I’ve developed a few activities to try to bring Power to life (see here and here). And while each was satisfying in their own way, none of them really met one of my overarching classroom goals – to have students identify and express a new idea with their groups before I provide clarification. This year’s activity worked nicely as it allowed students to experience statistical power and generate meaningful conversation. Download the student version below, then read to learn how it works.

In this activity students will investigate the “fairness” of 3 virtual coins through a Desmos graph, using 3 different sample sizes to compile evidence. For each sample, students use their graphing calculator to compute a P-value and then reach a statistical conclusion. For coin A, I led students through the steps for n=10 and encouraged them to work through the next two sample sizes using their group-mates as a support system.

As students completed all three columns for coin A, I asked them to make a final decision regarding the fairness of coin A – is there convincing evidence that coin A is unfair? Students discussed findings with their groups and thoughts about how each column provided convincing evidence. Here is what the class-wide vote and conversation revealed:

  • Of my 42 total students (2 classes), only 1 student concluded that coin A was unfair.
  • All groups agreed that the larger sample size (n=100) was more useful in reaching a decision about the coin.

Spoiler alert: coin A is unfair! If you take a peek under the Desmos hood, you will find that coin A is “programmed” as 48% heads, 52% tails. I didn’t reveal the true proportion until the end, but we are off to a good start here: small differences between the null and “truth” are less likely to be detected.

Groups then tackled coin B with little assistance from me. Working through each column, then the follow-up conversation and decision, took about 5 minutes. This time about 60% of the students concluded that coin B was unfair.

Finally, coin C. Many students quickly concluded that coin C was unfair (it is!) but worked through each of the columns and sample sizes. In the end, there was class-wide agreement that coin C is an unfair coin.

At this point I revealed the truth about each coin:

  • Coin A: 45% heads
  • Coin B: 40% heads
  • Coin C: 25% heads

So, what do our finding show us about hypothesis testing and decision-making as a whole? I was thrilled when one of my students who does not volunteer often raised his hand to offer the following: “If there is a big difference between the null and the truth, it’s easier to reject the null.”

Yes! That’s a big part of power. What else?

Larger sample sizes are more likely to detect a difference when one exists.

Yes! And now we have a nice framework for power. From here I shared a working definition of power and included thoughts on alpha, which are not part of this activity now but could be in a later version.

EmPower your students to develop statistical ideas!


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:

FullSizeRender - 3

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.

Statistics Technology

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!