# Blog Archives

## Day 61: The Challenges of Combining Random Variables

In AP Stats, we’ve begun our study of how random variables can be combined. I use a very simple partner activity to get students thinking about what we can and can’t do when combining random variables.

APS 6.2 – Day 2 Combining Random Variables Partner Activity

Through doing the first part of the activity, I found some students didn’t actually understand that the term “probability distribution” meant to create the table with the random variable, X, and their related probability values. They were just writing out what the probabilities were. Also, determining the standard deviation from a probability distribution was still elusive. So we spent some time reviewing the formula followed by how to use the calculator to make the calculations easier. (Sorry, forgot to take photos today ;{

I’m worried that my kiddos lost sight of the purpose of the activity in the process of completing the calculations. We did finish the questions for the combined random variable X + Y; yet I am very sure that the conceptual understanding is missing. So on block day Wednesday, we’ll revisit the reasons behind the calculation of the standard deviation using a simplified version of Dave Bock (of Cornell University)’s AP article entitled, *Why Variances Add–And Why It Matters*. We’ll talk about his coined phrase “Pythagorean Theorem of Statistics” as well as explore his three questions:

- Why do we
**add**the variances? - Why do we add even when working with the
**difference**of the random variables? - Why do the variables have to be
**independent**?

We’ll then look at one of the Math XL homework questions:

followed by having my students look at the work they did for the combined random variable X – Y on the Partner Activity. Wednesday will be a busy day, and luckily we have 90 minutes to fortify their understanding. Keeping my fingers crossed.

## Day 67: Speed Dating Random Variables

Today in AP Stats we tackled combining random variables. This has always been a “dry” topic, even when I’ve had the kids complete an activity….every year it seems the “big idea” gets lost on them in the minutia of collecting data by hand and crunching the numbers. So I used this example from our teacher resources, but used an idea I saw Josh Tabor use in his class notes of using a simulation to lay the foundation for developing the concepts and resultant formulas. It proved to be a great way to review simulation AND introduce the RandNorm( function on the Nspires.

First we talked through developing the simulation. Darn if the kids are still struggling with quantifying a situation…food for thought for later. But we finally set up the parameters (and set them up for two sample inference to boot! See the probability statements I stuck in)

Secondly, we used the power of the Nspires to collect 1000 random samples for the males and the females. I had the students verify that their set of numbers (at least the first few) were not the same as their groups. Then we used the probability statement to determine the value of the combined random variable M – F. We looked at the values in the table first, noticing that there were positive and negative differences. I asked if it looked about even between positive and negative values and they said they couldn’t tell…yay! So we graphed them and the collective “it looks like a normal curve” was exclaimed…yay again! We also used our probability statement, P(M-F>0) to draw in the vertical line and again estimate the probability. We sorted the “d” list and then counted the number of negative values…ugh, no fun…and then determined the ratio and then the complement. So much good stuff!

Thirdly, we calculated the means of the three random variables and looked for the (obvious) relationship, which they found in a heart beat. Asked them if this was the true population values and they chimed in “NO” only an approximation since this is a sample…helped that everyone had slightly different values but all close to the true values of μ_{M} = 68.5 and and μ_{F} = 64. They also were able to reason through how to find the value of μ_{M-F} = 4.5.

The standard deviation was a little different story. Again they saw that their values were close to the true values of σ_{M} = 4 and σ_{F} = 3, but σ_{M-f} = 5 not 1. So we had a chat about what happens to the variablility when we subtract variables; i.e. that subtracting does not reduce variability but actually increases it. Then a student blurted out “pythagorean theorem!” So we explored that idea and developed a generalization as shown below. All in all, a great lesson and I believe my “statisticians” turned “mathematicians” will remember this concept.

## Day 65: Random Variables and Gallery Walks

We are finishing up the introduction to Random Variables in AP Statistics today and I thought it was time for students to work together on some problems on discrete and continuous random variables including finding and interpreting the expected value and standard deviation of discrete RVs and using notation properly. Once again, I want the kids to be actively engaged in thinking about and doing the problems rather than passively watching me or another student work thorough a problem. Don’t get me wrong; there is a time and place for modeling good mathematical techniques, but we are past that today.

So I gave a 20 Minute Poster assignment. Groups were given 20 minutes to solve and write up a problem by applying their statistical knowledge and using good AP-level communication. Each member of the group chose one colored marker and could only write in their color on the poster. They signed their name in the same color so I could see the contributions made by each individual. Lots of good clarifying questions and discussion in making the poster: “Does a uniform distribution look like this (showing a normal curve)…then what does it look like?”, “why is P(X=3) equal zero in a continuous random variable distribution?”, “I don’t get why P(X>3) = P(X≥3) for a continuous random variable, but they aren’t equal for a discrete random variable,” and “how do we find the standard deviation if we don’t have the probability distribution given to us?”

After the posters were completed, I had students pair up using Clock Buddies…today it was their 7 o’clock buddies. Using a Gallery Walk protocol, the pairs had 5 minutes to read over and check the work on a poster that was not either of theirs, leaving sticky notes with “I notice…” and “I wonder…” comments regarding the correct application of statistical techniques and good communication. After 5 minutes, they moved to another problem and did the same. I will post the original problems and photos of their solutions on our website.

How do you use formative assessment techniques in your classes?