Overview

• Students will explore how skew and outliers influence different measures of center (mean vs. median).
• Students will also interrogate misleading statements about means.
• This activity connects back to calculations of measures of center, and connects forward to boxplots and analyses of univariate data.
• C2, S2, C4, O2, C4, C3, C5, S4, V1 ← Link to EBTP descriptions 

Prerequisite assumptions

Students should be able to do each of the following after completing the What to Know assignment.

• Calculate and interpret the mean and median of a data set.
• Identify whether a data set is left-skewed, symmetric, or right-skewed.
• Make connections between the distribution of a dataset and how the mean and median relate.

Intended goals for this activity

After completing this activity, students should understand that medians are resistant to influence from skew and outliers, that means are not resistant to influence from skew and outliers, and that means, in certain circumstances, can be misleading. They should be able to identify misleading claims made using means and suggest the most appropriate
measure of center to use in different situations.

Synchronous Delivery and Activity Flow

The sample activity delivery below assumes a face-to-face class meeting but can be adapted to a fully online or hybrid delivery by using break-out rooms for pairs and small groups.

Frame the activity (3 minutes)

• Have students read and answer Question 1 independently. Then, ask students to discuss their answers in pairs before sharing with the full group. V1
• Transition to the in-class activity by briefly discussing the Objectives for the activity.

Activity Flow (18 minutes)

• Have students work on the dotplot and analysis in Questions 2 – 6 together.
  • For Question 5, emphasize that students do not need to get the exact percentages; it’s far more important to get the general percentages from reasoning than to get their counts and calculations exactly correct.
  • Questions 3 and 6 are good places for a whole class discussion after the groups have a chance to work on them together. C5
• Give students time to think about Question 7 independently and to write out their responses. Then,
  • First, ask for a student who thinks the mean is “typical” to share their reasoning.
  • Second, ask for students who think the mean is not “typical” to share their reasonings.
  • Third, return to the first sharer and ask if they have been convinced that the mean is not actually “typical.” S4
  •  Students should leave this discussion feeling the median is the best representation of the “typical” salary in this dataset. The next question will broaden this idea. C5
• For Questions 8 – 16, Have students answer this question in small groups, and then ask several groups to share their answers with the whole class. Ensure that students include not just specific examples in their explanations
  but also general patterns that can apply to multiple datasets (e.g., contains right skew, contains outliers, etc.)
• After students have discussed, show the simulation linked below on a projector. Illustrate the varying relationships between the mean and median after low and high outliers are added. Also, illustrate how similar the mean and median are when the distribution is symmetric. C3
• Simulation: https://dcmathpathways.shinyapps.io/MeanvsMedian/.

Wrap-up/transition (5 minutes)

• As a final formative assessment, consider drawing various distribution shapes (left skew, right skew, symmetric) on the board, and have students describe the predicted relationship between the median and mean. C4
• Have students refer back to the Objectives for the activity and identify ones they recognized completing. S2, C4, O2
• Assign the homework or Practice and any What to Know pages for the Forming Connections activities you plan to complete in the next class meeting. C2