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108 CHAPTER 2 • Modeling One-Variable Quantitative Data
FYI
Density curves are one example of
mathematical modeling. The great
statistician George Box once said
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“Essentially, all models are wrong, but
some are useful.” That quote admits a
fundamental truth: models are never
a perfect fit for real data, but they
sometimes fit well enough to provide
useful insights and information.
COMMON ERROR
The term “curve” is sometimes strange
for students, particularly when the
first density curve they see is “straight.”
Remind them that in mathematics, curve
is a general term for any graph. Later
in this lesson, they’ll see some density
curves that are . . . curved.
TEACHING TIP
If you have students who took or are
taking a calculus course, they will
immediately recognize the importance
of areas under curves. However, your
other students can rest easy: this course
requires only a little algebra.
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108 CHAPTER 2 • Modeling One-Variable Quantitative Data
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Density Curves
Selena works at a bookstore in the Denver International Airport. She takes the airport
train from the main terminal to get to work each day. The airport just opened a new
walkway that would allow Selena to get from the main terminal to the bookstore in
4 minutes. She wonders if it will be faster to walk or take the train to work.
Figure 2.5(a) shows a dotplot of the amount of time it has taken Selena to get to
the bookstore by train each day for the last 1000 days she worked. To estimate the
percent of days on which it would be quicker for her to take the train, we could find
the percent of dots (marked in red) that represent journey times less than 4 minutes.
Surely there’s a simpler way than counting all those dots!
Another approach is to model the dotplot with a density curve, like the one in
Figure 2.5(b). We can use the red shaded area under the density curve to approximate
the proportion of red dots in Figure 2.5(a).
Density curve
1
Height = 3
2.0 2.5 3.0 3.5 4.0 4.5 5.0 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Journey time (min) Journey time (min)
(a) (b)
FIGURE 2.5 (a) Dotplot of Selena’s travel time via train from the Denver airport main terminal to the book-
store where she works for each of 1000 days. The red dots indicate times when it took her less than 4 minutes
to get to work. (b) Density curve modeling the dotplot in part (a). The red shaded area estimates the propor-
tion of times that it took Selena less than 4 minutes to get to work via airport train.
You might wonder why the density curve is drawn at a height of 1/3. That’s so the
rectangular area under the density curve between 2 minutes and 5 minutes is equal to
baseheight = ×1/31.00 = 100%
3
×
=
representing 100% of the observations in the distribution shown in Figure 2.5(a).
DEFINITION Density curve
A density curve models the distribution of a quantitative variable with a curve that:
• is always on or above the horizontal axis.
• has an area of exactly 1 underneath it.
The area under the curve and above any interval of values on the horizontal axis esti-
mates the proportion of all observations that fall in that interval.
The red shaded area under the density curve in Figure 2.5(b) gives a good approx-
imation for the proportion of times that it took Selena less than 4 minutes to get to
work via the airport train:
×
×
area = base height = 21/3 = 2/3 = 0.667 = 66.7%
So we estimate that it would be quicker for Selena to take the train to work on about
66.7% of days. In fact, on 669 of the 1000 days, Selena’s journey from the terminal to
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