The mathematical purpose of this activity is for students to investigate the effect of outliers on measures of center and variability, and to make decisions about whether or not to include outliers in a data set.

Students may incorrectly compute the expression for outliers. Remind them to use the correct order of operations, using the Math Talk Warm-up in a previous lesson.

The goal is to make sure that students understand that outliers can significantly affect measures of center and variability. Discuss the effect of the outlier on the median, mean, and standard deviation and the student responses to “Do you think that 9.8923 should be eliminated from the data set? Why or why not?”

If time permits, discuss questions such as:

• “Which measure of center is more greatly affected by the inclusion of extreme values, the mean or median? Explain your reasoning.” (The mean because it uses the actual numerical value rather than the position of the values like the median does.)
• “Which measure of variability is more greatly affected by the inclusion of extreme values, the standard deviation or the interquartile range? Explain your reasoning.” (The standard deviation because it uses the mean as well as the numerical value of each number in the data set, whereas the IQR uses only the position of the middle half of the data.)

The purpose of this discussion is to highlight different reasons that outliers appear in data. For example, they could be data-entry or data collection errors, or they could be representative of the sample. The goal is to make sure that students understand that the inclusion of outliers in a data set needs to be evaluated in the context of the data. For the number cube rolls, it is clear that the data should not be used since it is impossible to achieve in the right circumstances. For the other two scenarios, students should understand that a deeper investigation should be done to determine whether the outlier should be included and be able to state circumstances for including or excluding the outlier in each context.

Here are some questions for discussion:

• “Why is it important to analyze the source of outliers?” (To determine if all of the data fit conditions for the question being asked so that the analysis is valid. It can also give insights into different situations that may not be common.)
• “What are reasons to keep an outlier in a data set?” (Just because a value is extremely large or small does not discount its reality. To be honest in the analysis, all valid data should be included.)
• “What are reasons to remove an outlier from a data set?” (If there is an error in data collection or recording, the data may be faulty, and it would not be honest to include data that do not fit the question asked.)
• “What could be done about the 6 outliers for the electric vehicle data to account for county size as the source of the outliers?” (The question could be changed to examine the number of electric vehicles in counties with a population  below some level. It may make better sense to look at the percentage of cars that are electric or something similar that takes into account the county size rather than just the number of electric cars.)
• “How do you know that a value is an outlier?” (If it is greater than the third quartile by more than one and a half times the interquartile range or less than the first quartile by the same amount.)