If students aren’t sure about the meaning of “variability,” tell them that it means how spread out the data are. For example, in a typical grade 9 class, the ages of students usually have low variability since they are mostly the same age, but the ages of people in the school parking lot after school has great variability since there are older adults as well as younger students and their siblings.

The goal of this activity is for students to recall how to calculate each measure and to practice the calculations. Discuss any complications or confusions that arose. Time permitting, discuss how to use the measures to interpret data. Here are sample questions to promote class discussion:

• “What kind of insights will the principal get from knowing these statistics about the tenth-grade students?” (The principal would know if tenth graders are generally qualified for the advanced literature class, and can use this to make decisions about who to place in the class. Since the median is 500, the principal knows that more than half of the tenth-grade students are qualified. On the other hand, the IQR of 205 means that there is a wide range of scores.)
• “What is one action you would take as principal if you had these statistics?” (Everyone with a score of 600 or higher would take the advanced course. I would pay extra attention to any students below the median of 500 by providing them with tutorials.)
• “What does the mean say about the tenth graders? Does the median say something different?” (The mean could tell us that the typical tenth grader is not qualified for the advanced class (). But the median tells us that more than half are qualified. So, some of the lower scores that are well below the level of qualification are pulling the mean score below 490.)
• “Why is it important to have multiple measures of center and measures of variability?” (Having multiple measures can give more information about the data than just using one measure. For example, when the mean is affected by really high or low scores, the median can give more insight about the data’s center. The mean uses every data value and gives a sense of the typical student score. The median divides the group in half.)

Consider asking students to explain why their solutions might be the same for each row.

Sample responses for each row:

1. Sharing a lot of the value from 100 to the other small numbers is similar to sharing a small amount of the value from the 25 to the other closer values.
2. When rearranged in order, both data sets have 6 in the middle. Since the actual values on either side do not matter for the median, the solution is the same.
3.  Each of the values in column A is 10 greater than the corresponding value in column B. Since IQR measures the range of the middle half of the values, it is the same for both sets of data.
4. The values in column B are each 1 greater than the values in column A. Since MAD measures variability and both data sets are spread out the same amount, the MADs are the same.

The goal of the lesson is to make sure students know the difference between each measure and how to calculate them. Discuss any misconceptions students encountered while working, and how each measure can be used. Here are sample questions to promote a class discussion:

• “In situations when you and your partner did not arrive at the same answer, how did you agree on an answer?” (We talked about what we did to get our answers, and checked our steps with a calculator. If we did not agree, we had to convince one another about our thinking.)
• “What is the difference between measures of center and measures of variability?” (Measures of center are median and mean, and they summarize the data set with a number that describes a typical value of the data set, which can also be thought of as a center of its distribution. Measures of variability here are IQR and MAD, and they tell us how spread out the data set is.)