Monthly Archives: January 2015

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!

Thoughts on Teacher Preparation

Just before winter break, I received an e-mail from a former student who was now teaching in a private school in Maryland. This is his first year teaching AP Statistics, and he was looking for some nuggets of wisdom and advice from someone he trusted.  Drew was one of my favorite students, mainly because he can keep up with my warped sense of humor, and it’s thrilling to add another former student to my teaching “tree”.  The conversation began with a comment I suppose is universal to new teachers. (note – many quotes here paraphrased by my weak memory).

Nobody told me it was going to be this much work.

Yep. Drink up, Drew! Anything else?

I don’t know how I find time to plan. Between getting tests ready and grading the homework I collect every night, there’s just not enough time.

Wait…you collect and grade homework every night?  Why the hell are you doing that?  Let’s have a chat about the intent and value of homework.

How did college prepare me for any of this?

They didn’t. You need another beer Drew?  The lack of classroom realism education colleges provide seems to be a common tale, and for my money is one of the big problems we need to tackle if we seek to improve the profession. So here comes the big reveal I’d like to share with all new teachers and/or those considering the profession.  Ready?  This is free advice, though putting a buck in the tip jar is always welcome.

The day-to-day profession of teaching is nothing like the tired stereotypes, nor does it resemble the vision you have of teaching when you think back to your own schooling. It’s likely that for the first few years of your profession you will adhere to stereotype: neat rows, textbook pages, planned lectures. The path of your career, your effectiveness as a teacher, and your own happiness all rest on how quickly you challenge your methods. Find colleagues doing great things, learn from them, tweak them, and make them your own.

PodcastThis past week I caught up on past episodes on the Freakonomics Podcast while working out. One episode from November was of particular interest, titled “Is America’s Education Problem Just a Teacher Problem?“. When I talk about the teaching profession with non-educators, my prescription for “fixing” the system often comes down to 3 easy-to-understand steps:

  1. Hire the best people
  2. Train them well
  3. Keep the best and let them lead

Isn’t that how all professions should operate? Where are we faling short? And while this podcast started off slow, with a narrow focus on where teachers tend to fall in their graduating classes, it was a later conversarion regarding teacher training which moved me. The quotes below come from David Levin, co-founder of KIPP schools, which you can find on the podcast transcript.

The way we train teachers is fundamentally broken in this country.

Yep….My bachelors degree is in mathematics. Only after graduation did to I go back to Drexel University to obtain teaching certification. I clearly recall a conversation with my advisor back then, who touted the Drexel program with “we believe anyone can teach”, and assurance that I could teach with an intern certificate after taking just 1 class.  Somehow I made it through, but I can’t recall much from my training which was helpful to me as a young educator. I’ve also discovered that not everyone with a bachelor’s degree can teach…not surprising.

It’s disproportionately theory-based. …I’m not sure of what good the theory of math instruction is if you don’t actually know how to deliver a lesson on math as well.

I would go even further here. It’s not just about planning and delivering a lesson – it’s about having the expectation that lesson study is a critical element in becoming an effective educator. How do we know if a lesson is effective? How do we adjust for student needs? What do we do to improve even our best lessons?

Sometimes the best math teachers weren’t necessarily the best math students, because you know you often teach better what you weren’t so good at, because you actually had to work to learn it.

Whoa….so, should we be hiring all of the top students to teach, or no? I immediately had a flashback to college calculus. I fully admit to not being a top math student, mainly because my work habits were supect. But I could learn new things quickly on my own if I had the inclination. I have to say that I didn’t really understand calculus until I started tutoring it. Having to explain the chain rule to a peer caused me to think about how it worked, to prepare for snags, to consider how ideas fit within the big picture. There’s an assumption that math teachers simply transmit information, with little regard for how we facilitate learning. This was quite a refreshing statement.

Back to my conversation with Drew, where it was also a great time to confess many of my teaching sins from over the past years, and measure the reaction. So, over some beverages, I had to confess something to Drew about the AP Stats class he attended, which was in my 3rd year of teaching the course.

Drew, I didn’t know what the hell I was doing back then.

It’s true. Sorry. I mean, everyone passed the AP exam and everything, but I look back on some of my old techniques and can confidently place them in the “stuff I would never advise people to do” file.  Drew gave me look on par with the wide-eyed glare a 9-year-old gives you when you finally tell him Santa isn’t real, that awkward, incredulous stare which is followed by knowing head nods.

So what changed? I used the same textbook, provided many of the same materials, revised my tests a bit. Tweaking problems and getting better at identifying the tricky part of problem 45 on page 312 just isn’t enough.  My development as a teacher comes from finding and utilizing a support system to help me improve. This goes beyond conversations with building colleagues, though having an enthusiastic department here at my home school has been valuable. Local groups like the Philadelphia Area Stats Teacher Association and ATMOPAV have helped me develop a local network of master teachers with whom I can discuss lessons. Recently, reaching out through twitter and the AP Stats message boards grew my learning community, and help me continually improve.

I wonder how many teacher prep programs are helping teachers find and build their own professional networks? In this age of connectivity, it’s simply silly to leave out this crucial piece.