Measures of Dispersion
Lesson 8 of 11
Objective: SWBAT use standard deviation to analyze data sets.
To open today's class, I post the set of student's scores on Linear Practice #1, which students completed during the previous class. (The data set in today's Prezi is from one of my Fall 2012 classes, and as I describe below, the nature of a particular data set may influence the moves I make later in the lesson.) The task is for students to sketch a histogram of this data set. My two primary goals are to engage students with the data and to give myself the chance to check on their abilities to construct histograms.
I also want each student to have an example in their notebook that they'll be able to reference when we start getting into what the normal distribution looks like. I'm not sure exactly what this histogram will look like in each class - it depends on both the data that our trial produced and on the bin sizes chosen by each student - but that's part of the fun of it. This may start to resemble a bell-shaped curve, it might not.
If students ask to draw a box plot or dot plot instead, I say they can, but that they should also try to make a histogram in their notes. If they're not sure what bin size to choose, I try to get them to to discuss it at their tables, and I suggest that each group member tries a different bucket size so they can compare the different options. If everyone is really stuck, I tell them that I'd like a minimum of 4 bins on these graphs, before asking what bin size it would take to have 4 or 5 or 6 or more.
For this semester's Statistics class, I choose not to take students through the work of calculating standard deviation by hand. There is definitely value in doing this in some contexts - for both theoretical understanding and procedural fluency - but in my experience, it slows down the work we're doing here without adding too much value. You may certainly find that it's worth your time to take students through a more thorough treatment of how standard deviation is calculated. Instead, I take my classes directly to the TI-83 calculator to show them how to compute standard deviation, then we use its resulting value in some calculations.
I tell students that I'm going to show them some calculator tricks. I prepare only a simple introductory slide (see #3 in today's Prezi) for this part of class, because I use the chalkboard as my "notebook" as I demonstrate for students what I would write as I took instructions on how to do this for the first time. I proceed by explaining which buttons to push on the TI-83, and I record these steps on the board as I go. I walk around the room asking students to show me what they've got on their calculator screens and in their notebooks. I also try to identify student experts who will be able to help troubleshoot as the class continues.
Teacher's Note: I'll leave it to you to make sure to study up on how the calculator works, because there are far better resources already available than I could produce here. Just as an outline, here are the key things I want students to be able to do:
- Clear the memory. ("Second, plus, 7, 1, 2" becomes a mantra of sorts in my classes.)
- Enter a data set as a list.
- Analyze that data set by running 1-Var Stats.
Students are immediately excited - then a bit confused - by what they see on their calculator screens after learning these steps, and they are eager to know what they're looking at. When I tell them that "x hat" is the mean, and that by scrolling down they can see all five components of a box plot, they're sold on the idea that this is indeed a worthwhile trick to know. I ask them again if their notes are sufficient: "Are you all sure you'd be able to do this again without any help? What if I erased the notes on the board - would you still know what you're doing?" This sets off a last minute scramble among a few students to make sure they've got the notes. Whether I like it or not, I'm teaching habits of scholarship here: a number of my students are poor note-takers, and I've found that the main reason for this is that no one ever helped them see the value in that skill. I've found that calculator procedures are a great place to start establishing the need for great notes.
The Greek letter sigma is new to most of my students: both the capital letter for summation, and the lowercase letter for population standard deviation. I take a few moments to name each of these values for students, briefly explaining the idea of a summation and naming both sample standard deviation and population standard deviation. I continue recording these notes on the board and reminding students that this is what my notebook would look like.
Later in the class, when students work on Delta Math, I circulate and make sure their notes are solid - organize, clear, and useful for when needed.
Teacher Talking: Spread vs. Dispersion
In statistics, The terms "measures of spread" and "measures of dispersion" are often treated synonymously, and I tell my students this. I also tell them that I like to make a distinction between the two. "When I hear the word spread," I say, "I think of how far apart two elements are. On the other hand, when I think of dispersion, I think of elements moving away - dispersing - from some common starting place." (Teachers: as you're reading this, consider the hand gestures that you'd use as you make these comments.) I go on to explain that range and IQR simply give us the distance between the low and high values in a data set - showing how spread out the data is - and that the measures of center may or may not be placed symmetrically within the data. In the previous class, we looked at differences between the mean and each element in a list, seeing how dispersed each value was from the mean. "Today we're studying the measure of dispersion called standard deviation," I conclude. "Of course, there is a relationship between range, IQR and standard deviation, and if you'd like to call all of them measures of spread or measures of dispersion, that's fine. I just wanted you to think about the difference between these words."
How to Use Standard Deviation (An Introduction)
Again we review learning target 1.4: I can use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages.
I tell students that over the next two classes, we're going to look at what it means to "estimate population percentages" using standard deviation. In order to do that, I show them what it means in the context of our real data set, explaining that we can ask, "How many student scores were within one standard deviation of the mean?" I tell students that they'll hear this question again for other data sets, and that right now I'm going to show them how to answer it.
In the example I've shared, 50% of the students are within 1 standard deviation of the mean. There have been results better matched to the normal distribution. This is a place for improvisation in this sequence of lessons: if your class results happen to come close to matching the normal distribution, you can talk about that. If they don't, then this is a chance to talk about what kind of distribution you have and what might happen if you were to have a much bigger population. In this particular case, I have to be careful to explain that we have a special case here: usually, we won't have 50% of the data within one standard deviation of the mean
One interesting extension is to combine the stats from multiple class sections or even multiple years, and look at those results, and to see how bigger data sets will come closer to forming a bell-shaped curve.
Side Note: Sample Standard Deviation vs. Population Standard Deviation
Each year, the extent to which I differentiate between sample standard deviation and population standard deviation is based on how attentative to detail I find each of my classes to be. In this case, population standard deviation is the right choice to use, because we're not trying to extend the results of Linear Practice #1 to any larger groups of students.
There are two parts to the current Delta Math assignment. I want students to practice solving linear equations. I want them to get faster and more effective than they are, so we exercise on that. I also want them to practice finding measures of dispersion.
Linear Equation Practice
We're recapping the Mastermind Project here, by reviewing its central question: If we practice a skill, will everyone get better at it? Again, I'm using linear equations, but this review exercise will work for the skill of your choice. This is an example of something I consider to be a very important teaching move: I try to figure out ways to naturally fold remediation/exercise into the flow of the class. It's never a punishment, not drudgery, and certainly not the main point of the lesson, but I know my students need to practice linear equations, and that they know they've studied it before. So I acknowledge that this is something they've seen, and that they're going to practice. We spent 10 minutes on this yesterday, they're spending a few today, and they'll run another Linear Practice trial tomorrow. All in all, it's a small portion of our time (which actually helps to build some urgency), and hopefully the results will show that it was worth it.
In contrast to the Mastermind Project, this time we're going to see what group improvement looks like in terms of mean and standard deviation instead of box plots with range and IQR.
Measures of Dispersion
Two modules dedicated to measures of dispersion and standard deviation give students a chance to apply what I introduced today. They will definitely need my help, so I circulate with the expectation that I'll help them flesh out their notes, polish up on their use of the TI-83, and maybe learn another "calculator trick" or two, like how to use the 1-var stats function on a frequency table.
They will also see the word variance for the first time here. Again, I choose to gloss over the derivation of standard deviation, so this is a word students have not yet seen. For now, I give them the additional note that variation is simply standard deviation squared. My students and I will dig deeper into this later, but for now, this familiarized them with the term.
I provide students with frequent updates about how much time they have left as they work on Delta Math. With a few minutes left in class, I tell them it's time to return the laptops, and I use the Smartboard to take a public look at the current results of today's assignment. I make supportive comments to both groups and individual students, and I ask everyone what they're going to work on tonight. I also remind students that there's another Linear Practice trial next class, and that I expect everyone to improve.