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Effective Classroom Practice students don’t need a lot of mathematical background, giv-
ing them a little review of these topics may save them some
Chapter 2 is about using mathematics to find the location frustration. Build in a quick review of the mathematical
of individuals within a distribution and seeing the bigger skills needed before each lesson.
pattern of a distribution. Chapter 2 may seem a little more
“mathy” than Chapter 1, as it involves a little geometry
and algebra. The normal density curve is arguably the most Lesson-by-Lesson Content
important density curve and it’s introduced early in this
course. Here are some important classroom practices to Overview
keep in mind as you teach Chapter 2:
1. Insist on good pictures: Much of this chapter involves Lesson 2.1 Describing Location in
density curves – particularly normal distributions. Students a Distribution
may want to avoid drawing these curves but insist that they An important question in statistics is “how does this one
do. When drawing a normal distribution, students should value compare to the other values in a distribution?” There
label the number line, mark the mean, and mark the val- are two basic ways to measure the position of one value:
ues that are 1, 2, and 3 standard deviations from the mean. percentiles and standardized scores (z-scores). An individu-
Every. Time. Making a well-labeled picture will help pro- al’s percentile is the percent of values in a distribution that
mote success for your students. Pictures are indeed worth a are less than the individual’s value. A standardized score
thousand words! value − mean
(z-score) is defined as = . Values above the
z
2. Teach good eye work: In many sports, eye work and eye standard deviation
patterns are a key to success. The same is true of statis- mean will have positive z-scores, values below the mean
tics. From percentiles to z-scores to normal curves, stu- will have negative z-scores, and values equal to the mean
dents should be able to see whether their answers make will have a z-score of 0. A standardized score (z-score)
sense. Did a student get an answer of “5th percentile” is interpreted as the value’s number of standard devia-
for a value far out in the right tail of a distribution? Was tions greater (or less) than the mean. By comparing per-
a negative z-score calculated for a value larger than the centiles or standardized scores, we can compare two dif-
mean? Was an area of 0.12 found for a region larger than ferent values in the same distribution or across different
half the area under a density curve? In each of these cases, distributions.
students should rely on their eyes and the graphs and pictures
to make sure their answer is reasonable.
Lesson 2.2 Transforming Data
3. Pay attention to context: All data involve context. When When converting units or in other situations, the values in
students are finished, they should not only put units on a distribution are transformed using mathematical opera-
their answer (when appropriate), but also take a moment tions or functions. If a constant a is added to (or subtracted
to see if their answer is reasonable in the context of the from) all the values in a distribution, the measures of center
data they’re working with. For example, a box of cere- will increase (or decrease) by a, but the shape and measures
al that weighs 12 pounds is unreasonable, as is a stu- of variability will be unchanged. If all the values in a distri-
dent who is 630 inches tall. Context is what makes data bution are multiplied by (or divided by) a positive constant
interesting, and it can warn your students about potential b, the measures of center and the measures of variability
mistakes.
will be multiplied by (or divided by) b, but the shape will be
4. Review prior mathematical skills: Your students will unchanged. When multiple transformations are performed
encounter a few area formulas from geometry and need a in succession to a distribution, the changes to summary sta-
few equation-solving techniques from algebra. While your tistics will follow the same order as the transformations.

2-4 CHAPTER 2 • Modeling One-Variable Quantitative Data

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