Category Archives: Statistics

8 Take-Aways From USCOTS

This weekend, I spend two days at USCOTS, the United States Conference on Teaching Statistics at Penn State University. The opportunity to connect with old friends, share ideas, and reflect my own practices was exciting. Here are just a sampling of my experiences, many of which could be their own blog post.  You can find many speaker presentations and more resources on the CauseWeb site. Hope you enjoy!

MOVING FROM SANITIZED DATA SETS – a Keynote by Shonda Kuiper of Grinnell College noted that while interest in the study of statistics is at an all-time high, are we really preparing students to apply statistical concepts in a realistic manner? Shonda challenged the group to move from canned textbook data sets and let large, real data sets drive coursework.  Her Stats2Labs website is a treasure chest of activities and data sets, from which she shared a rich set of NY Stops and Arrests, and Shonda shared her methods for using the set to faciltate discussions.  For my high school classes, I am most looking forward to using the Tangram Game applet, which collects many variables on gameplay.

I often tell my students that statistics is often about telling a story, and was thrilled to hear a college professor share this theme as well!

Photo May 30, 9 40 44 AMSTUDENT POSTERS – the poster sessions were interesting to me as a high school teacher, as my own students are preparing for “Stats Fair” next week.  How awesome to show my students that the presentations they are about to share are not too dissimilar from those they may encounter later in their academic careers, just in the sophistication of the studies and methods. You can’t beat having a small-group discussion with Beth Chance regarding the Rossman-Chance stats applets, which should be a part of every Stats class.  Posters on HS integrated math programs, flipped learning, and formative assessments provided info to think about for next year.

EGG ROULETTE – a session by James Bush and Jen Bready led to a fun “hook” I hope to try with my 9th graders next year.  James and Jen are masters of using video and pictures from the media to engage learners and grow discussion. Here, James chose Doug Tyson and I as “volunteers” to participate in a game. I quickly became worried when it was advertised as “Egg Russian Roulette”, and a clip from the Jimmy Fallon show was played:

me eggsWhat have I gotten myself into here?

The plot thickens when an egg carton makes an appearance…but filled with plastic eggs, some containing packing peanuts. I lost after 3 picks, and a simulation of the activity ensued from the group. Is there an advantage to being first? James alleges the person playing first loses 5/9 of the time. Try a simulation with your classes and find out!

CATCHING UP WITH OLD FRIENDS – Ruth Carver teaches high school about 20 minutes from me, and I cherish the times we find to trade stories.  By the time arrived at the conference, Ruth was already gushing over the many great sessions she had attended, and shared a quote from Dick DeVeaux which applies nicely to all classrooms:

Students like uncomfortable learning less and less.  They like things clear as a bell with no sweat, no thinking, no neurons firing.  They are confusing easy and comfortable with learning. To use a sports analogy, “Is that what you want from your sports practices – easy, comfortable, they didn’t break a sweat?  Well it should be the same with your classes; they should be sweating afterwards.  It’s hard stuff; they should be thinking hard.

What are we all doing to make sure our students sweat in math class?

Photo May 29, 6 04 25 PMIn return, I shared one of my new favorite ways to collect fun data: the website How well does it predict your age? What if you smile? What it I wear a hat? You will be toying with this and your friends at your next gathering.

SIMULATION-BASED INFERENCE –  This has become a hot topic in the stats world – I have come to use the StatKey site more often in my classes to have students simulate distributions – and was eager to learn how to leverage simulations with traditional hypothesis testing methods. Robin Lock and Kari Lock Morgan  shared examples where simulations allowed us to compute simulation distributions, but then move those results into traditional distributions and test statistics. My AP classes have generally been “successful” in that my AP passing rate is quite good, so it becomes tricky to want to ditch old methods. But the experiences and communication gained from simulation methods are too rich to be ignored. Infusing my classroom with more simulation-based inference could dominate much of my planning for next year.

CONNECTING – Connection was the theme of the conference, and a part of all of the talks. I strengthed bonds with old friends (many of whom I will see in 2 weeks for the AP Stats reading), and appreciate the many new folks I met for the first time. Doug Tyson’s silly selfie challenge gave me the courage to say hello to many people I wouldn’t normally have approched.  And though Doug won the challenge with a late-Friday “get” of Jessica Utts, the new AP Stats Cheif Reader – which I came so close to photo-bombing, I’ll take my photos with Allan Rossman and Roxy Peck as a well-deserved second-place.

Photo May 29, 6 17 19 PMPhoto May 29, 6 48 08 PM

DISCONNECTING – The saturday lunch-time talk by Michael Posner of Villanova University inspired the group by sharing the many connections he has made with the Stats community over his career. Michael often shares at our local PASTA (Philly-Area Stats Teachers Association) meetings, and I appreciate his desire to connect with high school teachers.

While explaining the power of connections he has made, Michael also challenged the group to disconnect, and reflect upon their teaching.  In particular, are we using our Stats expertise to clearly measure the efficacy of our teaching methods?  And while sharing ideas at conferences can be energizing, how do we personalize what we have gathered to work for our classroom?  Such great themes to consider at the close of a conference.

AFFIRMATION AND REFLECTION – When I first started teaching AP Stats, I was cautioned that stats teachers are often the lonliest people in their departments. Walk into a high school math planning room, bring up methods for solving quadratic functions, and you may soon have a full group conversation.  But try to start a discussion of two-sample t-tests?  Crickets…. This conference was attended by about 450 passionate stats people, with only about 10% being high school folks. But the college crowd could not have been nicer or more accomodating in wanting to share their ideas.  The entire experience left me energized that I am headed in (mostly) the right direction in what I do to encourage stats study, and with plenty of resources and connections for improving my practice.  Looking forward to USCOTS 2017!


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!

Adding Distributions of Simulated Data

The current chapter on expected value and combining distributions in AP Statistics is one of my favorites for a number of reasons.  First, we have the opportunity to play games and analyze them…if you can’t make this fun, you are doing something wrong. Second, it often feels like the first time in the course we are doing some heavy lifting. Until now, we have discussed ideas like sampling, scatterplots and describing distributions – nothing really “new”, though we are certainly taking a much deeper dive.

The section on combining distributions contains a number of “major league” ideas; non-negotiable concepts which help build the engine for hypothesis testing later.  The activity I’m sharing today will focus on these facts:

  • The variance of the sum of independent, random variables is the sum of their variances.
  • The variance of the difference of independent, random variables is the sum of their variances.
  • The sum of normal variables is also normal.

First, we need to have student “buy in” that variances add. Then we have the strange second fact: how can it be that we ADD variances, when we are subtracting random variables? In this activity, we’ll look at large samples, and what happens when we add and subtract these samples. Since many students taking AP Stats have the SAT on their brain, and there is a natural need to add and subtract these variables, we have a meaningful context for exploration.


The printable classroom instructions for this activity are given at the bottom of this post.

photoTo begin, students use their graphing calculator to generate 200 simulated SAT math scores, using the “randnorm” feature on their TI calculators, and using the fact that section scores have an approximately normal distribution with mean 500 and standard deviation 110. Note – some older, non-silver edition TI-84’s won’t be happy with this, and a few students had to downgrade and use a sample of size 50 instead. There are a few issues with realism here: SAT section scores are always multiples of 10, which randnorm doesn’t “know”, and occasionally we will get a score below 200 or above 800, which are outside the possible range of scores. Also, there is a clear dependence on SAT section scores (higher math scores are associated with higher verbal scores, and vice versa), and here we are treating them independently.  But since our intent is to observe behavior of distributions, and not reach conclusions about actual SAT scores, we can live with this. In my class, no student questioned this as problematic.

Repeat the simulation in another column to simulate verbal scores. Then, for both columns, compute and record the sample mean and standard deviation. For my simulated data, we have the following:

It’s time to pause and make sure all students are clear on what we are simulating. We now have 200 students with paired data – the math and verbal score for each. Like most students, our simulated students would like to know their overall score, so adding math and verbal scores is natural. I help students write this command in a new column, then let them loose with the remaining instructions on both sides of the paper.

Students had little trouble finding the sum of the math and verbal scores, and computing the summary statistics. For my sample data, we have:

As students work through this, I want to make sure they are making connections to the notes they have already taken on combining distributions. I visited each student group (my students sit in groups of 4) to discuss their findings. Most groups could quickly identify that the means add, but what about those standard deviations? By now, if my students have taken good notes, they know that standard deviations don’t add, and that variances should. I leave groups with the task of verifying that the variances add.

Here’s the beautiful thing: students who immediately tell me that they “checked” the variances and verified the addition get the evil eye from me. In this simuation, students should find that the variances are “close” to adding, but not quite.  At the end of the acitivity, I ask students to conjecture why the addition is a “not quite” – even after I have beat into them that variances add.  There are two main reasons for this, and I was happy that a number of students sniffed these out.

  1. We are dealing with samples, not populations. There is inherent variability in the samples which causes the sample variances to not behave nicely.
  2. Variances add – but only if distributions are independent. Here, even though we created large random samples, there is still some small dependence. And while we don’t specifically cover the formula for dependent distributions in AP Stats, it’s worth discussing.

Next, it’s time to look at the differences.  Here’s students are asked to subtract math and verbal scores, compute the summary statistics, and compare the sum and differences. This was a nice way to go back and re-visit center, shape and spread.


CENTER: Sums are centered around 1000, while differences are centered around zero.

SHAPE: Both distributions appear approximately normal.

SPREAD: The sum and difference distributions appear to have similar variability.

And this idea that the spread, and standard deviation, will be similar for both the sum and difference, can be also be explained by looking at the range of each population distribution.

  • For the sums, the max score is 1600 (800 M and 800 V), with a min of 400 (200 each)
  • For the differences, the max score is +600 (800M and 200 V), with a min of -600 (200M and 800 V).

Here, we can see that both distributiuons has the same range.

From start to finish, this exploration took about 30-40 minutes, and was worthwhile for verifying and developing understanding of the facts for combining distributions.  The student instructions and video notes students take beforehand are given below.  Enjoy!

Class Opener – Day 66 – Surprising Coin Patterns

A short post today, as I am out the door for a meeting with our NCTM local group, ATMOPAV. Please check out our website, where we have information on local awards, and house our award-winning newsletter!

I enjoy giving problems with solutions which go against our instinct. In statistics, there are many opporunities for this, and today’s opener in my AP class seemed innocent enough:

Which will more likely occur first in a string of coin tosses: HTH or HTT?

After a few moments of debate, there was universal agreement that the two patterns are equally likely, and therefore we should have an equal expectation of seeing them occur first in a string.  But the correct answer goes against this intuitive notion.

Peter Donnelly’s TED Talk – “How Stats Fool Juries” is easily digestible for the high school crowd. I show it over 2 days, first to present the coin-tossing problem. Then in our next class meeting I will show the second half, where conditional probabilities and the multiplication rule make appearances in courtroom trials.  In the video below, fast-forward to about the 5:30 mark if you want to learn about the coin-tossing problem, or watch from the beginning for some statistics humor.

Class Opener – Day 64 – Can My Students be Random?

Today begins out probability chapter in AP Statistics, which is often deceptively tricky for students. Until now, probability has meant simple experiments – drawing cards, flipping coins or picking marbles from urns (why are we probability folks always so fixated on urns, anyway?). Thinking about probability as a long-term proportion of success is a foreign concept, and separating short-term “bad luck” from a suspected effect requires much deeper understanding. Here is one of my favorite openers to start conversation about short-term probability, which is adapted from an activity done in a college statistics course.


pic3Students are separated into teams of 2 (or 3).  5 minutes are on the projected clock, and each team is given a grid with 50 squares, along with instructions, face down. Students are told that I will leave the room for exactly 5 minutes, during which time they are to complete the instructions.  At the end of 5 minutes, I will return to the room (after enjoying my coffee) and class will commence.  All instructions are expected to be followed precisely, and without talking to other teams.

In the room, there are two sets of instuctions, which I have passed out without knowing who received which.  The instructions are mostly similar, but with an important difference:



So, one group tosses “real” coins, while another group tries their best to act randomly. When I came back to the room, 8 sheets were hanging on the board.  Without comment, I write “RandInt” and “Guts” on the board to indicate the two methods, and now the challenge is on: can I successfully separate the “real” from the “fake”? As I examine the papers, and slide them into their groups, students begin to sense what I am up to as I hear groans , cheers and grunts…but I don’t want to know who is who yet.  I am sometimes quite good at separating these, but some days I over-think things or believe that a group or two may have sabotaged the experiment.  So, what I am looking for here?

  • Runs of heads or tails: the probability of a run of 5 heads (or tails) in 50 coins is 55%, verified with Wolfram. Usually, non-random students will not let a run go beyond 3, maybe 4.
  • Alternating starting behavior: my hypothesis is that if a group is developing a pseudo-random string, they will alternate often at the start.

For the class, this becomes a rich discussion about short-term versus long-term behavior.  That while we can expect a group of 50 tosses to settle to about 50/50 heads and tails, the short term can yield surprises. And how long of a string should cause us to begin to suspect something is amiss, versus a natural occurrance?

So, how did I do this time?  Unfortunately, not so hot. But that’s OK.


Asking kids what they thought I was looking for led to many of the big ideas of the section I was looking for, and we were off and running into our probability chapter!

pic2There was one paper I really struggled over, as it started with 5 consecutive tails. While my guidelines should have clearly placed this into the “randint” group, my suspicion was student sabotage. But it was RandInt all the way….even short-term events can fool the instructor.

One student in my colleague’s class then summarizes the entire activity quite nicely, and also provided a needed Friday dose of comic relief:

The calculator gives you runs

Well, if it’s a TI, maybe…


Class Opener – Day 60 – Herding the Cats

After a wonderful Thanksgiving break (made more wonderful by the Eagles win over the Cowboys!), it’s the 3-week sprint to the holidays, followed by 2 full weeks before final exams.  There’s a lot of stopping and starting going on, which doesn’t help continuity when thinking about class content.  In my AP Stats class, we are deep into our unit on experimental design, which is filled with ideas, terms and arguments must different than a traditional math class. Groups are working through their “Old Wives’ Tales” project, and after grading some student responses this weekend I need an opener which brings the whole class back into the Stats circle. My friend Glenn Waddell has some awesome resources for statistics on his website, which provided inspiration for today’s opener – a short video from ABC News featuring the placebo effect.
More ABC US news | ABC World News

For today’s opener, I asked students to design an experiment which could prove (or disprove) the efficacy of the WYFFT “energy drink”.  This gave groups much to talk about, and a thorough discussion of elements of experimental design, including:

  • Treatments: WYFFT is not a “real” drink, it’s just soda. Students conjectured that the labeling and associated signage were the actual treatment. We can compare this vs a plain bottle, or against no drink at all.
  • Matched-Pairs: could subjects plausibly participate in both treatments? Is this reasionable?
  • Blocking: could the implied reaction be different in men than in women? Perhaps we should have two different experiments?
  • Response: what exactly are we measuring? What would be a suitable activity to measure a change in energy?
  • Randomization: how will subects be selected for the treatments?

And we are off and running after a long turkey-induced rest!