Some students may forget to sort the data when finding the median. Ask them, “What is a median? What does it tell you about the data?” Some students may not remember how to find the median when there is an even number of data values. Ask them, “What does the median tell you about the data? How could we find a middle number between these two values?”

The purpose of this activity is to get students to calculate the median and IQR, and to investigate how those values are affected by outliers.

The data sets in this lesson are small enough that finding summary statistics like measures of center or measures of variability are not necessary. The entire data set could be assessed fairly easily and does not need further analysis. The sets are small here for the purposes of practicing calculating and understanding the statistics. In reality, finding such statistics are much more useful when the data set is much larger. For example, some devices will find heart rate every 10 minutes, giving more than 1,000 values for a week, which might be analyzed to determine health information about a person. It would be difficult to understand all of the data by looking at a table or even a data display. Having some summary statistics, such as mean or median, would be useful to understand a person’s heart rate for the week.

In this activity, an optional graphic organizer is provided in the blackline master to help students compute the interquartile range. The boxes that are shaded are not to be used, and the position of the open boxes are meant to highlight the useful data for calculating the value for the row. For example, to compute the first quartiles, students use the average of the second and third values, so those boxes are left open to indicate that this data are useful.

Students may have difficulty calculating the median of a data set with an even number of data points. Ask them what the median represents for the data set and where that value might be. If they still struggle, remind them that the median is the average of the two middle numbers.

The purpose of this discussion is to discuss the method of calculating median and IQR as well as the interpretation of each. Here are some questions for discussion.

• “What would a box plot look like for the results of the vocabulary test?” (It would depend on the high and the low score, but the box would likely start at 5 and go up to 45 or it could start at 10 and go up to 50.)
• “What does the IQR tell you about a data set?” (The IQR is a measure of variability, which describes the range of the middle half of the data).
• “How do the median and quartiles divide the data?” (Into equal quarters so that the same number of values are in each quarter.)
• “How much of the data are between Q1 and Q3?” (The middle 50% of the data are between these values.)

The purpose of this activity is to get students to calculate and describe the mean absolute deviation. Students are given a data set and an organizer for calculating the MAD. Then they consider questions that are intended to get them thinking about MAD more conceptually as a measure of variability.

Monitor for students who have trouble finding the mean or who are using negative values for the distance from the mean. Remind them that the  symbol represents absolute value. If necessary, ask them to look at a number line and describe the distance between two values to remind them that distances should always be described as positive values.

Put pennies on a meter stick so that the centers of the pennies are at {20, 40, 40, 45, 45, 45, 45, 55, 55, 55, 55, 55, 55, 90}. Show how the stick balances when you put your finger at the 50 centimeter mark and how some are farther and some are closer than 10 centimeters away from the mean, but they’re spread out so that, on average, they’re 10 cm away.

Create a display that incorporates the measures of center (mean and median) and variability (interquartile range and mean absolute deviation) discussed so far. This display should be posted in the classroom for the remaining lessons within this unit. You will add to the display throughout the unit. The blackline master provides an example of what this display may look like after all items are added.

Here are some questions for discussion.

• “If you put pennies at 45, 35, and 70, where do you need to put a penny for the meter stick to balance at 50 cm? What is the MAD?” (50 cm, 10 cm)
• “If you put two pennies at 60, where do you need to put a penny to make the meter stick balance at 50 cm? What is the MAD?” (30 cm, cm)