• When your class arrives at the Key Question (How much money will you really make?), take a moment to surface students’ initial thoughts. In particular, prompt them to think about what questions they’d ask the employer about the statement: “The typical salary at our company is more than $65,000 per year.” In brainstorming questions, students will start to develop some intuition for how this figure could be misleading.
• Consider making the initial exploration of the effect of transformations on the simple data set (values 1 & 3) completely student-led, in small groups. As students perform addition and multiplication on the small data set and see how the mean and range transform, they can propose general rules for the effect of transformations on data sets.
• Our salary data set has one outlier, and it’s a high outlier. The lesson explores the effect of this high outlier on different summary statistics. For robustness, it’s also helpful to show them the effect that a low outlier would have on the different summary statistics (e.g. how a low outlier test score would lower the average test score).

First, download this lesson's Handout Key and read through its Discussion Question section. Then, check out our model discussion norms and the additional background notes below.

• As suggested in the Discussion Question’s hint, having students manually calculate the different summary statistics before and after the boss’s raise helps them build intuition for why some measures are resistant and other measures aren’t. The key point to emphasize is that the resistant measures (median, IQR) are based on central positions in the data set and, therefore, are not as affected by extreme values. The unresistant measures (mean, range, standard deviation) involve the values of the most extreme data points (the min and max) somewhere in their calculations. Hence, these measures are more sensitive to changes in these extreme values.
• An optional extension to the Discussion Question: What other contexts can you think of that would produce datasets with outliers or extreme skewness? How could different reported summary statistics be misleading in these contexts?

• The 13 company salaries are a simulated data set, created for pedagogical purposes. However, the high outlier and right skew shape are indicative of real income data, which tends to be right skew and contain high outliers. To surface this, ask students: “Do you believe the mean salary would paint a misleading picture at most companies, or just this one? Why?”
• The exchange rate between US Dollars and Canadian Dollars has hovered between 1.2 - 1.45 for several years. The value of 1.3 was used for the lesson because it’s within that range and makes calculations a bit simpler.

• Although it’s rarely done, data values can also be multiplied or divided by a negative scalar. In these cases, the measures of center get multiplied by the scalar, and the measures of spread get multiplied by the absolute value of the scalar.
• Boxplots are especially useful for larger data sets, in which a dotplot or histogram could be visually overwhelming. Because boxplots directly display certain measures of center (median) and spread (IQR), students should use those measures when describing a boxplot (rather than, for example, trying to visually estimate the mean or standard deviation).
• Although their visual simplicity is useful in many respects, boxplots lack visual information on two important aspects of quantitative distributions: sample size (the number of data values) and peaks (e.g. unimodal, bimodal, uniform). In particular, if peaks or modes are of interest, encourage students to use dotplots or histograms.
• In AP Statistics, outliers are visualized in boxplots with dots or asterisks. However, in other contexts, boxplots are graphed such that the whiskers extend all the way to the minimum and maximum – regardless of whether or not those values are outliers. It’s worth mentioning to students that both conventions for graphing boxplots are popular in the broader field.