Measures of center
• Recall how to calculate mean and median.
• Understand mean and median as a “typical value” that can answer a statistical question.
• Know that mean and median are equal for a symmetrical data distribution (S-ID.A.2).
• Explain why mean and median are unequal for a skewed data distribution (S-ID.A.2).
• Select mean as the better measure for symmetrical distributions, and median as the better measure for skewed distributions (S-ID.A.2).
• Make generalization what kinds of distributions have means larger than medians, and what kinds have medians larger than means (S-ID.A.2).
• Recognize outliers when they exist, and know to investigate their source—that data point is way out there, why is that? Is there something weird about it that means we should disregard it (S-ID.A.3)?
• Know that outliers affect the mean, but not the median of a data set (S-ID.A.3).
Students deepen their understanding of mean and median as measures of center, gaining a better understanding which to use in summarizing a given data distribution. They work with data sets where mean and median are equal and where they are different, and explain, using the context of the data, why this occurs. Teachers may continue to use the class’s available technology to calculate summary statistics and make plots.
Tasks
1
Haircut Costs
WHAT: In this task, students are given the minimum, maximum, quartiles, median, and mean for two data sets and are asked to sketch side-by-side box plots in order to compare the two distributions.
WHY: This task allows students to use summary statistics to compare two data sets rather than requiring computation. By asking which measure, mean or median, is more appropriate, an opportunity is presented for students to describe an effect of outliers, e.g.., some extreme high values “pull” the mean to the right.
2
Identifying Outliers
WHAT: In this task, students analyze given data about distances traveled to school. Students examine what happens when one of the data points changes, and its effect on the mean and median. The task gives a definition of “outlier” and a data point that satisfies this definition. Students learn how to identify an outlier and examine how an extreme value affects a measure of center.
WHY: This task is intended to help students develop their understanding of extreme data points (outliers) and how they affect the measures of center of a data set. It is also an opportunity for students to develop their skills in constructing arguments and critiquing the reasoning of others because students are asked to use the given data (MP3), and to use technology to create box plots and calculate measures of center (MP5).
3
Describing Data Sets with Outliers
WHAT: In this task, students examine means and medians in different real-world situations (distances between celestial bodies, real-estate prices, annual income) and how outliers in each situation may affect (or not affect) mean and median.
WHY: Students have an opportunity to gain experience with different measures of center in real-world contexts and see examples where mean is equal to or different from median. In order to support their answers, students use their understanding of mean and median, and how their values are affected by outliers in the given data sets.