Students compare the mean of a set of data with the mean after an outlier is introduced, and notice that the outlier has a significant effect on the mean.

Making statistical technology available gives students an opportunity to choose appropriate tools strategically (MP5) to quickly compute the new means with technology instead of computing by hand each time.

The goal of this discussion is to ensure students understand how an outlier affects the mean.

Ask students what they notice and wonder about the mean values related to whether Andre’s grandfather goes on the hike with him or not. (Students may notice that the mean significantly increases when Andre’s grandfather goes on the hike, and that their estimate was likely inexact. They may wonder why the mean changed so drastically with only 1 new data point, and, if their estimate was wrong why the mean changes in a way that’s different from what they thought would happen.)

Point out that the mean changes so much because 130 minutes is significantly greater than all the other values. When the seventh value is closer to the rest of the data (59 minutes), the mean is not very different from the original.

• “As Andre trains for a mountain hiking race, he wants to track how long it takes him to climb the mountain. Should he include the data point of 130 minutes in the mean time it takes him to hike the mountain? Explain your reasoning.” (He should not include the time in his calculation of the mean since he will probably not be running the race with his grandfather.)
• “Even if Andre’s grandfather did not come with him on the hike, there are reasons it might take Andre 130 minutes to complete the hike. What are some circumstances that might increase Andre’s hiking time to 130 minutes and would make it reasonable to include in his mean time for the race?” (If the weather was bad or if Andre was injured, his time might be much longer and could also affect his race time, so he might consider including the value in his calculation of mean.)

Highlight that a value should be left in the analysis if it was collected accurately and in the right conditions for the situation at hand or if we are unsure of why it is different. If we are sure it is a typo, or the value is under significantly different conditions that don’t fit the situation at hand, we might leave it out. The most important thing, though, is that students are not under the impression that data should be thrown out just because it’s different and doesn’t match what we want. The default should always be to include the data and remove the data only if we are certain that it was an error or is measuring something very different than we intended.

Students are presented with several data sets and have to use a formula to decide whether they have outliers or not. Students then make guesses about possible situations in which the outlier is included in the data set. Since students are only guessing, it is important they understand that none of the outlier values should be removed. Students should explain their reasoning for deciding whether to include the outlier. Students reason abstractly and quantitatively (MP2) when they decide if a data point is an outlier and consider the effects of an outlier. Students look for and express regularity in repeated reasoning (MP8) when using the formula to interpret data.

The purpose of this discussion is to clarify how to calculate outliers and when outliers should be included in data analysis. Here are sample questions to promote a class discussion:

• “If a value is accidentally recorded twice, should both data points be included in the data set?” (No, if it was an accident, that means that the second appearance of the data point did not actually happen.)
• “Could you find an outlier without using the formula?” (No, the formula ensures that finding an outlier is not left to chance. It needs to be supported mathematically, or else people would find outliers on the basis of their opinions, which is not appropriate when making statistical decisions.)

Tell students that in some cases, what can be considered an outlier is clear because it is so atypical compared to the rest of the data often revealed by visual representations. At other times, whether a point should be considered an outlier is harder to decide just by looking at the data. In those cases, the rule that students are encouraged to use in this class is the one given in this activity developed by John Tukey to use  and .