• If your students do not have much experience with summation notation (Σ), you may need to explain the way this notation works when covering the standard deviation formula. Although covering the formula for standard deviation may not feel necessary (students will normally use technology to compute the standard deviation), understanding the underlying process behind the formula will help them develop a deeper conceptual understanding of what the standard deviation represents. If your students do not have much experience with summation notation (Σ), you may need to explain the way this notation works when covering the standard deviation formula. Although covering the formula for standard deviation may not feel necessary (students will normally use technology to compute the standard deviation), understanding the underlying process behind the formula will help them develop a deeper conceptual understanding of what the standard deviation represents.
• It may be helpful to call out the usefulness of IQR to identify how the middle 50% of data behaves. Doing so will help students to recall it as different from the 5-number summary they will see in lesson 1.6. Similarly, describing standard deviation as a way to measure how the data is dispersed away from the mean differentiates it from the range, which is simply the difference between the extreme data values.
• One nice connection to make with students is the relationship between percentiles and quartiles, where Q1 approximately corresponds to the 25th percentile and Q3 approximately corresponds to the 75th percentile.

First, download this lesson's slide deck and handout key to see the prompt and sample responses for the Lesson Starter. Then, check out the additional background notes below.

Instructional routine: Notice & Wonder. The lesson handout provides a Notice & Wonder T-frame for students to capture their notes and ideas. It is important that students recognize the difference between a noticing (observation) and a wondering (question that comes to mind). You can find more background on implementing a Notice & Wonder here.

Purpose & Background: The goal of this Lesson Starter is to compare and contrast data displays with the same summary statistics. Having recently considered the mean and the median of a data set in context, students may have developed their own mental picture of how these statistics visually appear in graphs. This Lesson Starter is designed to challenge that thinking and to show that measures of center can’t always tell the whole story. As datasets that are clearly different but with the same measures of center, students will be challenged to recognize what similarities the sets must have and what causes their graphs to look so different. Offered without context, this Notice & Wonder is designed to provide students with the opportunity to solidify their understanding of measures of center before looking more broadly at measures of spread.

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.

• When motivating the Discussion Question, some students may look at the situation as inherently unfair, but others may see it as a ‘change in the times’ with renting being the norm and finding new ways to build wealth through other types of investments. This latter stance offers opportunities for additional discussion of the implications of the change, such as people not being tied to the same roots that home ownership may require.
• Other topics that may inform the discussion are considerations of what else has changed. For example, in the 1970s, many people stayed with the same company for 20-30 years and often lived in the same place during that time. In 2024, the median tenure for employees at their current jobs was only 3.9 years. Changing employers or careers more frequently often means moving more frequently, which makes home ownership less attractive.
• Students may express concerns that it will be harder to build wealth and pass it on to their children. Home ownership (and the equity developed over time) sometimes gives parents the opportunity to give their children a ‘head start’ by helping them to pay for college or even get that first mortgage for a home of their own.
• Depending on the context in which you teach, you may find that your student population has a lopsided opinion on this Discussion Question. As a facilitator, it’s valuable to encourage students to voice the “other side,” so that students have a robust discussion that mirrors the broader civic discourse on this topic.

• Data sets of home values and incomes tend to have a right skew. Hence, the median is often a more useful measure of center for these variables than the mean. This may be a helpful point to reference during the next lesson in the course sequence, which specifically explores the median’s resistance to outliers and skew.
• The data explored by students in CODAP is a random sample from the famous Ames housing data set. The data set includes uniquely detailed data about housing, compiled from research of the housing market in Ames, Iowa. The city of Ames is home to Iowa State University. Like most college towns, the city of Ames has a diverse housing stock, including homes with a variety of features, sizes, and values. The full data set includes more than 20 variables for nearly 3,000 homes. The sampled data set students use to explore CODAP was curated to remove variables less recognizable or significant to students, and to make it manageable for students (300 cases) to analyze.

• To motivate each measure of spread, we continue to work with a small data set. This permits students to develop conceptual understanding of these measures by seeing their manual calculation. Then, we show students how to make use of technology to find summary measures of larger sets of data.
• Squaring differences is another key idea that occurs in the standard deviation formula. Also used in a variety of methods across statistics (variation, weighting errors, regression, coefficient of determination) students will see the strategy of squaring differences several times throughout the course.
• Students sometimes have these two common questions about the standard deviation formula. Some background is provided below. Note that each explanation is beyond the scope of our course, but can be helpful for curious students.
• Why do we square the differences to get rid of negatives? Why not just get the absolute value? There are two central reasons for this: i) Squaring emphasizes larger differences, which can be helpful for surfacing outliers. ii) Squares have nicer mathematical properties than absolute values. In particular, models that try to minimize the standard deviation often take the derivative of the standard deviation function. It’s much easier to find the derivative of a square than an absolute value. Check out a more thorough discussion of this topic here.
• Why do we divide by n - 1, instead of simply dividing by n? The reason is based on the concept of degrees of freedom. When calculating the sample standard deviation, we use the sample mean and every data value. However, with the sample mean in hand, we don’t actually need to be given every data to recreate the full sample. The final data value can be inferred from the sample mean and the remaining n - 1 data values. Hence, there are only n - 1 varying pieces of information in the standard deviation formula, or n - 1 degrees of freedom. This is why we divide by n - 1. A fuller discussion can be found here.