Lesson 14

The purpose of this warm-up is to elicit the idea that outliers are often present in data, which will be useful when students investigate the source of outliers and what to do with them in a later activity. 

As students work, monitor for students who

• estimate the IQR from values in the box plot
• use a measurement tool to determine the IQR from the box plot

Students are given the formulas for outliers: a value is considered an outlier for a data set if it is greater than Q3 + 1.5 \(\boldcdot\) IQR or less than Q1 - 1.5 \(\boldcdot\) IQR. To find extreme values, we are comparing very large or small values to the bulk of the data. This means using the quartiles and interquartile range to compare the value to typical distances to the center of the data.

Display the histogram and box plot for all to see. Tell students to think of one thing they notice and one thing they wonder about the images. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner. Listen for students who notice that there is a value that seems greatly different from the rest of the data. Select a few students to share things they notice and wonder making sure to select the identified student that notice an extreme value.

Select previously identified students in the order listed in the lesson narrative to share their method for creating this visualization of outliers in the box plot.

Tell students:

• Values in a data set that are greatly different from the rest of the data are called outliers. The precise meaning of greatly different will be different for different situations. For example, a possible $4,000 difference in this graph does seem like a lot, but if the data represented the entire budgets of these countries in the billions or trillions of dollars (rather than spending on each member of the population for healthcare), it would not be a great difference.
• Using the IQR to determine outliers helps to adjust the difference to the variability of the bulk of the middle data. Using 1.5 times the IQR allows for some variability on the ends of the distribution to be considered usual.
• It is also possible for there to be values that are unusually low compared to the rest of the data set. Consider this box plot that displays \(\text{Q1} - 1.5 \boldcdot \text{IQR}\). The minimum value for this data set should be considered an outlier.

  

  

• For the purposes of this unit, a value will be considered an outlier for a data set if it is greater than Q3 + 1.5 \(\boldcdot\) IQR or less than Q1 - 1.5 \(\boldcdot\) IQR. These formulas compare extreme values to the middle half of the data to determine if the value should be considered an outlier.

Create a display for the outlier formulas so that can be referenced throughout the rest of the unit.

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

Arrange students in groups of 2. Provide access to devices that can run GeoGebra or other statistical technology.

Display the data showing Per Capita Health Spending by Country in 2016 for all to see. Orient students to the data and explain that the distribution of this data set represents the histogram used in the warm-up. Ensure that students understand what per capita means. “Per capita health spending” means the average health spending per person. For example, the United States spends approximately \$9,892 on healthcare for each person in the population.

Remind students that we will classify a value in a data set as an outlier if it is greater than Q3 + 1.5 \(\boldcdot\) IQR or less than Q1 - 1.5 \(\boldcdot\) IQR.

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 impact measures of center and variability. Discuss the impact 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 since 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 since it uses the mean as well as the numerical value of each number in the data set whereas the IQR only uses the position of the middle half of the data.)

The mathematical purpose of this activity is to get students thinking about the source of outliers and whether or not it is appropriate to include them when analyzing data. It is important to stress that data should not be removed simply because it is an outlier. If there is any doubt about the reason for the outlier, the data should be included in any analysis done on the data set.

Arrange students in groups of 2. Give students quiet think time to answer the first question. Ask partners to compare answers. Follow with a whole-class discussion. Ask partners to answer and discuss the remaining questions.

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 there is a reason to exclude the data from an analysis.)
• “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 does not fit the question asked.)
• “What could be done about the 3 outliers for the college crime data to account for school size as the source of the outliers?” (The question could be changed to examine crime at schools with enrollment below some level. It may make better sense to look at “crime rate” like a measure of crimes per 100 students or something similar that takes into account the school size rather than just the number of crimes.)
• “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.)