High School Statistics

Chi-Squared Tests: Rock, Paper, Scissors.

At the beginning of the school year, I shared a post about a fun Rock, Paper, Scissors applet on the New York Times website.  Back then, my class used the applet to collect data for 2-way tables, and considered appropriate methods for displaying the data set.

Fast-forward 6 months: my AP Statistics class is knee-deep into hypothesis testing, and we’re now up to Chi-Squared tests.  These are some of my favorite tests, as the data is often richer than what we find in tests for means or proportions.  Here’s how we used the Rock-Paper-Scissors applet to produce data:

  • Teams played the game in veteran mode.
  • In round 1, teams were given 3 minutes to play the game normally, which I’ve labeled the “guts” method.
  • In round 2, teams were given 3 minutes to play the game randomly, using “randint” on their calculator to generate a digit from 1 to 3, which corresponds to a move.

We then considered an appropriate test for assessing the data.  This comes on the heels of Chi-Squared Goodness of Fit tests.  But here we have two samples, and we want to determine if the proportions are similar in both samples: this was our first test for homogeneity, and it was easy to move through the mechanics of the test.


Doug Page also shared a worksheet he has developed for using the Rock, Paper, Scissors applet.  I do not have contact info for Doug, but I hope he provides some details on his success in the comments.

This activity will now become a yearly staple in my AP Stats arsenal!  Enjoy.

High School Middle School

Some of My Students Failed Today! Woo Hoo!

A new semester has just begin here at my high school, and one of my classes is a co-taught course we call Prob/Stat.  The Prob/Stat course is one we offer to our 9th graders, as a follow up for Algebra 1.  It includes concepts in probability and statistics, along with algebraic concepts like systems, polynomial operations, and matrices.  The students in this academic class will take the Pennsylvania Keystone Exam in May, a graduation requirement, so this course is quite important for them.

My math department colleague and I, along with both co-teachers, agreed that we did not want this to “feel” like math courses they had taken up until now.  We wanted our students to become more reflective in their approach, think about their strengths and weaknesses, and devlelop their own learning paths.  We have embraced Standards-Based Grading and a policy of re-dos and retakes to help meet our ideals for this course.

On the first day of the class, I wanted to set the tone that communication and discussion would be valued in my classroom.  I asked the students to arrange their desks in a circle, which brought many questioning looks and rolling eyes.  But once we established our circle (actually, it was more like an oblong), I passed around small slips to paper to every student.  I asked the group to list any factors which had caused them to not perform well in their past math classes.  Many students were willing to share their stories: “I don’t do homework”, “Teacher X didn’t like me.”, “I don’t like to ask for help.”…the list was rich.  Placing a trash can in the center of the floor, I instructed students to ball up their slips of paper, and toss them into the bucket…they are in the past! Chum BucketI stole this idea from my time at the Siemens STEM Academy, where we started the week by catapulting our educations hold-backs into the chum bucket (it was Shark Week at Discovery Ed).  You can read more about the chum bucket activity on the Siemens STEM Institute blog.

Next, I asked the students to write something they could do, moving forward, to improve their math outlook.  What an awesome conversation!  One student shared her fear of reading problems in math, but a desire to work through it and seek help.  Many students confessed their need to complete assignments.  Others communciated the need to start self-advocating, asking more questions.


For many students in my class, this is their first experience with Standards-Based Grading.  Before the course began, I took all course concepts and arranged them into 4 anchors, mimicking the anchor language of the PA Keystone Algebra 1 content.  Each anchor contains 5-7 standards, written as “I can” statements.  The document also contains room for multiple attempts on the same standard.  As students complete notes or assignments, I instruct them to write the standard we are working on clearly at the top of the page.

View this document on Scribd

In this course, we start off with the probability sections, so we actually led off with 4.5 “I can find the probability of a simple event”.  Probability is a topic which haunts students of all ages, sizes, and ability levels.  And while many students did just fine on their first quiz, a number of students struggled.  Under normal circumstances, this would cause deep sighs from me, and steamrolling on. But, to be honest: I HAVE NEVER FELT MORE ENERGIZED ABOUT STUDENTS STRUGGLING IN MY CLASS!

All students in the class have their own binder, which houses the Standards Tracker, and all assessments. During the next few class meetings, my co-teacher and I will develop groups for small group instruction to discuss mis-conceptions, and work towards the re-do on their 4.5 quiz.  At the same time, we have moved forward into 4.6, multi-stage events.  We are striving to set-aside time each Friday to be reflection and redo time, in order to establish regularity with these new grading concepts.  I find myself looking forward to students dicussing their needs, and working with them to do better next time.  It’s early in the semester, but already things feel different.

Check out some of my earlier blog posts on Redos, Retakes, and Standards-Based Grading:

Rick Wormeli – Redos and Retakes

Standards-Based Grading, twitter chat recap

Quality Assignments, #sbgchat

High School Statistics

Matched Pairs with Hallway Bowling

The experimental design unit in AP Statistics is a fun one, with lots of opportunities to design activities, discuss possibilities and collect data.  For a few years, a “Hallway Bowling” activity I created has been one of my favorites for discussing matched-pairs experiments.  This year, I added a new wrinkle to this activity day, in order to economize class time.  As students entered the class, they drew a playing card, each having one of three suits which determined their group assignment for the day.  Each group had 7 or 8 students.  Groups then rotated through 3 stations, with 15 minutes on the clock, and with each activity designed to review a different aspect of the chapter.

In Station 1, students met with me in a small group, where we discussed experimental design, writing ideas and experiment trees on desks.  This was a departure from whole-class discussions, and more students had the chance to share their ideas on experiments dealing with clothes washing temperatures and drug trials.  Experimental design vocabulary like blocking and matched-pairs were clarified, and the small-group discussions were rich.  At the end of the day, the students shared how much they liked being able to share in a more intimate setting.

In Station 2, the group completed an actual AP item dealing with experimental design.  Papers were collected as a group, and I will randomly choose 2 paper from the group to grade.  Students knew this going into the activity, and this procedure holds all students accountable for the group grade/

In Station 3, the group went out of the room to play and collect data with “Hallway Bowling”.  15 minutes was enough time for students to practice, play, and collect data.

You can down loading the rules here:  Hallway Bowling

Here’s how Bowling works.

  • 2 markers are placed 5 meters apart (I had pre-taped blue X’s on the floor)
  • players stand behind one marker, and roll a golf ball as close to the other marker as possible.
  • During the data recording, players will roll 4 times; alternating hands and measuring the disatance to the marker.

BowlingAfter the activity, a whole-class discussion is held to talk about Hallway Bowling as an experiment.  What are we trying to prove?  How does our activity provide data for the experiment?  Where is the randomization?  What could be done to improve the design?  Here, we are looking to encourage “matched-pairs thinking”; where all subjects are exposed to both treatments (rolling with dominant and non-dominant hands), and we are interested in those differences.  We can also consider blocking here if we feel that males and females may be effected diffferently by the treatments.  We can also revisit the data later when we look at hypothesis testing procedures.

And about that data we collected?  My kids entered their data into a Google form.  There are some great comparisons to consider: right hand vs left hand, boys vs girls.  But how did the distances come out for dominant hands vs non-dominant hands?

Graph 1

Note the difference in medians here.  But can we directly compare individual player performances?  To do this, we can subtract dominant and non-dominant hand scores, and observe the differences:

Graph 2

If players are truly better with their cominant hands, we should see many negative differences here.  We see over 50% negative, but is there enough evidence to prove a mean difference for ALL players?  Time to start linking to inference.

So have fun with hallway bowling, and try some classroom stations!