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:

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.

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.

Nix the Tricks – AP Stats Edition

For AP Stats teachers, this is the time of year where we move from innocent ideas like scatterplots and experimental design, and into uncharted waters; those topics which require sharper focus, and more time and reflection to develop properly. Sampling distributions, the Central Limit Theorem, confidence intervals and hypothesis testing…new and scary ideas.  With the crush to cover content before May, it’s easy to fall into traps where we shortchange discovery and real meaning and replace them with quick tricks.  Here I present one of my least favorite Statistics “tricks”, and hope you “Nix this Trick”!

Nix Header“Nix The Tricks” is a powerful, free document for math teachers of all grades; a crowdsourced collection of math shortcuts and well-intentioned devices teachers employ to assist students with math mechanics; devices which ultimately under-cut student understanding of mathematics.  Along with the tricks are suggestions for developing math concepts in your classroom without tricks; encouraging communication of ideas and language.  It’s a labor of love, compiled and edited by Tina Cardone, who I admire for her dedication to this project.  Some of my ideas from a blog post last year on phrases from math class which need to be expunged have been absorbed into Nix The Tricks, and I am thrilled to have had even a small part in building this document.  Share it with your math friends, and let the debates begin!

Back to Stats-world, and a phrase we need to Nix.  It’s time for hypothesis testing, a new world of strange symbols for the null and alternate hypothesis, lots of conditions and tests to think about, and making logical connections between computed values and real-life consequences.  Writing tight, meaningful conclusions takes practice, revision, and patience. But why struggle, when we have a cute shortcut?

When the P is low, reject the Ho!

This is the short version of the general argument that when we have a sufficiently low P-vale (below alpha), we have evidence against the null hypothesis, and in favor of the alternate. But why go through all of this meaning, when we can talk about Hos in math class!


So, what’s wrong with this catchy phrase?  Well, first, and probably most importantly, it’s damn offensive.  For teachers, talking about Hos in class, or even providing a “giggle” momnent about the idea, is out of bounds.  We all get that, right?  Good.

In stats-world, the problem with this phraise is that it provides students an excuse to not develop real understanding about the connection between P-Values, Alpha, and the null hypothesis.  As an AP Reader, I enjoy the opportunity to see how students craft their conclusions to a hypothesis test. In 2012, I read question 4, which was a full 2-proportion z-test.  It was fascinating to observe the clear differences between the written approaches to conclusions; which textbook they probably used, what mnemonic devices did their teacher push, how much attention was paid to written practice.  In addition, while many approaches relied upon a canned template, where students simply fill in blanks (with mixed success), I also enjoyed well-developed explanations which demonstrate clear evidence of understanding of the logic of hypothesis testing.

At last year’s AP Stats Reading “Best Practices Night” Luke Wilcox did a wonderful job explaining how he challenges his students to become clear communicators from day 1. You can download his presentation, and many other “best practices” resources, at the famous APStatsMonkey page.  Here’s a fantastic example from Luke’s class, which demonstrates clear understanding of the process:


In AP Stats, communication is essential.  Here are some thoughts and ideas to keep in mind:

  • A strong conclusion has linkage between a computed P-value and a defined significance level (alpha).  This is the computation piece.  The art of statistical writing is taking this numerical result and using it to reach a conclusion about our population.
  • My students write, write and write, and my boards are covered with samples, which we critique and revise.  I like to randomly assign students to work together (I often use playing cards for this), so that “group think” does not set in. I want students to debate language, and I can see from afar which groups are on-point by having them on boards around my room
  • My document camera is also a valuable resource here. As an opener, I’ll have students examine a homework problem, and write their conclusion on an index card. Random cards are selected and critiqued.

As many Stats teachers head toward their hypothesis testing units, let work together to Nix this Trick, and improve student writing!

My Favorite Teacher Circle: PASTA

Just got back from the fall meeting of my favorite local teacher circle, PASTA.  The Philadelphia-Area Statistics Teachers Association meets a few times each year to share best-practices in statistics teaching.  Many of this month’s presenters are AP Statistics readers, and the ideas are not specific only to stats…we just share great classroom action.  I gave a recap of our last meeting in the winter; enjoy the great ideas from our Fall meeting, and visit Beth Benzing’s website for materials from the meeting!

Daren Starnes, famous in the Stats-world as author of The Practice of Statistics, shared his first experience with Team Quizzes.  I have tried team quizzes before, mostly for quizzes where I knew students were having the most difficulties with material.  But Daren added some features I had not before considered:

  • Students are assigned to their teams at random.
  • Each team member received a copy of the quiz, and must complete the quiz.
  • In a quiz, one question is chosen randomly to be graded from each paper.  A student’s grade is a combination of the score they receive on the question, along with the average of the scores from the other papers in the team.

Daren also commented on the roles of introverts and extroverts in the teams, and how this method could empower introverted students to self-advocate.  He suggest the book Quiet: The Power of Introverts as a resource.

AdamAdam Shrager, famous as the social director and man-about-town at the AP readings, shared his movie-correlations activity.  This has become one of my favorite activities during the stats year.  Students are asked to fill out a movie-preference survey, which Adam then uses to compute peer-to-peer correltations in Excel.  (look for “correlation” in excel…you may need to activate the Stat Pack) Discussions regarding the interpretation of positive and negative correlations then occur.  Most importantly, mis-conceptions of the meaning of low or zero r-values are discussed with a context easily understood by students.


Leigh Nataro shared her “Pacing a Normal Distance” activity, where students walked between 3 different campus buildings using “meter-long” steps.  The data is then entered into Fathom, and is used to discuss variability, the 68-95 rule, and normal probability plots.  Fun discussions of outliers and error as well!


Our host, Beth Benzing from Strath Haven High School, shared a family income Fathom file which draws samples of various sizes from a clearly skewed distribution.  In addition to to having students record observations and work towards generalizations, Beth has worked to increase the rigor in her associated questions, using past AP items as her framework.  Some examples:

  • What is the probability that a sample of 5 families will have a combined income of over $500,000?
  • What is more likely: a sample of size 5 having a mean income of over $80,000, or a sample of size 25 having a mean income over $80,000?  You may recall a similar AP question from a few years ago regarding samples of fish.


Brian Forney shared ideas for bringing concepts from Sustainability to the AP Stats classroom.  In one example, Brian shared data on depths of ice sheets over time, with excellent opportunities to discuss cause and effect from scatterplots.  Check out Brian’s presentation on Beth’s website.

Finally, I was happy to share my recent lesson on Rock, Paper, Scissors and two-way tables.

The meeting concluded with some great ideas for making multiple-choice assessments more fair and effective.  There were a number of excellent ideas here, but I think I’ll look up some more info on alternate assessment methods and save it for another post…so stay tuned!

A Bowl of PASTA with Stats Friends

Today I attended the winter meeting for one of my favorite organizations: PASTA, the Philadelphia Area Statistics Teachers Association.  This group meets a few times a year to discuss best practices in statistics education, and includes a number of AP teachers, many of whom are AP exam readers.  As always, lots of interesting ideas today:

Joel Evans, from my home school, spoke on his first attempts to “flip” his AP Statistics class.  Based on feedback from his students, Joel realized that Powerpoints often dominate his classroom culture.  By flipping, Joel hoped to have students review material before class, then use class time to practice and discuss.   Follow Joel’s flipping story in the slides below.

It is always a pleasure to have Daren Starnes at our meetings.  Daren, one of the co-authors of the ubiquitous The Practice of Statistics textbooks, joins our group often to discuss his ideas for teaching statistics.  Today, Daren shared a presentation, “50 Shades of Independence”.

Daren asked us to think about all of the places where we encounter “independence” in AP Statistics:

  • probability of independent events
  • independent trials
  • independent random variables
  • independent observations
  • independent samples
  • independent categorical variables (chi-squared)

Man, that’s a lot of independence!

Which items from the list above deal with summarizing data?  Which are needed for inference?  How are they related?  How do we help our students understand the varied, and often misunderstood, meanings of independence.

Daren has a knack for leading conversations which invite participants to express and discuss their math beliefs. Daren   Many of the arguments concerning independence, according to Daren, are “overblown”, in that teaching them in a cursory manner often causes us to lose focus on the big picture. That’s not to say that we should discard them, but that, when teaching inference, we should have students focus on items which would cause a hypothesis test to be “dead wrong” if we didn’t mention them, i.e. randomness, justifying normality conditions.

penniesRuth Carver continued the presentations with some new tech twists on a lesson used by many stats teachers: analyzing sampling distributions by looking at the age of pennies.  A population graph of the ages of 1000 pennies hangs proudly in Ruth’s classroom.

After agreeing that the population is clearly skewed right, we move to the main event – drawing random samples from the population and analyzing the data we get from repeated samples of the same size.  Ruth has developed a lesson for the TI Nspire which generates the samples, and challenges students to think about the behavior of the sampling distributions, now considering the effects of sample size.  Ruth’s presentation allows students to experience and express the differences between:

  • Standard deviation of a population
  • Sample standard deviation
  • Standard deviation of a sampling distribution


Great job Ruth!  Looking forward to more PASTA with my stats friends!