Spread Out

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Objective

SWBAT rank data sets from least to most spread out based on their own reasoning. SWBAT examine and compare three different methods for ranking data. SWBAT understand mean absolute deviation.

Big Idea

How would you determine how spread out a data set is? Students put data sets in order from least to most spread out based on their own intuition and reasoning.

Opening

10 minutes

The two activities in today's lesson lay the foundation for students to understand standard deviation. The first activity is called "Data Spread" and can be found on page 323 of IMP's Year 1 textbook (2009).  The activity asks students to rank 4 sets of data that have the same mean and median from the least spread out to the most spread out and asks them explain their reasoning.  I can begin class by listing the 4 sets of data on the board and asking students to rank them on their own from least to most spread out. The data are named as groups from A to D.  I then ask students to share their rankings and their reasons why they ranked them in that order.  We will refer back to these lists later, so I save them on my Smartboard (or take a picture or record them in some other way).

Here is a screen capture from our rankings sharing and discussion this year. I tried to keep track of some of the ways students chose to rank their data from least to most spread out.

Investigation

30 minutes

Students now look at three different ways to arrange the data. These three different methods ask them to consider what spread out means in different ways.  

  • The first method is in the same activity and asks students to rank the data sets by range. The second and third methods are in the following activity "Kai and Mai Spread Data" on page 324-325 of IMP's Year 1 textbook (2009).  
  • The second method, Kai's method, is very close to mean absolute deviation. Kai's method takes the distance from the mean for each data item and adds those distances together to "score" the data sets.  
  • The third method asks students to remove the highest and lowest values and then find the data item that is farthest from the original mean. This number is then assigned as the score to that data set and the sets are again ranked.  

Students work to arrange the data from least to most spread out for each of the three methods.  Here is an example of some student work from this activity. We can see from this piece of work how the student "scored" each set of data according to the different methods that we assigned.

As students work, they may find that they like one of the methods better than the others. I try to get them to capture this thinking by writing it down and recording their thoughts.  I also want to encourage them to think about why this method is a better way of measuring spread.  These thoughts and comparisons can often lead to rich discussion.

Once they've sorted the data using the three different methods, students compare the new spreads with their original ranking of the data to see which method is closest to their own (it is likely that some students will have used some of the exact methods, especially using the range).  

Discussion + Closing

20 minutes

Next, I bring the class back together as a whole group and ask if anyone would like to change their original ranking.  I also ask students what method they liked best and why and how similar their own rankings were to the three methods. 

I bring up the concept of outliers in the context of the third method, though only one of the sets of data really has true outliers.  

We spend some time talking about the second method, which is very close to mean absolute deviation.  I bring up absolute value in the context of the data items.  I point out to students that sometimes they are subtracting the mean value from a data item that is larger than the mean and sometimes they are subtracting a smaller value than the mean from the mean. I ask them how they could standardize this process and try to elicit the idea of absolute value. I introduce the symbol "x bar" to students and ask them to generalize this difference from the mean.  

I also show students here that mean absolute deviation is just one more step from this point. Once they "score" the data set based on the sum of the differences from the mean, they only need to divide by the number of data items to get mean absolute deviation. I let them know that this is a simpler form of something called standard deviation that they will be learning about next.  Teaching mean absolute deviation here helps students to have a more intuitive understanding of standard deviation later so I try to make sure students have a solid understanding of the concept.

If  students are familiar with summation notation, I might show them how mean absolute deviation is written here. I find students like to be able to interpret what looks like complicated symbols and making sense of them gives them a sense of accomplishment.

To close class today, I give students the opportunity to reflect on these three different methods. I ask students to complete an exit ticket in response to the following prompt:

Which method did you like best to determine the ranking of least spread out to most spread out data sets? Why?  How was this method similar or different from your original way of sorting the sets?

Citations

  1. This material is adapted from the IMP Teacher’s Guide, © 2010 Interactive Mathematics Program. Some rights reserved.