Page 30 - 2021-bfw-SPA-4e-TE-sample.indd

LESSON 2.3 • Density Curves and the Normal Distribution 111

EXAMPLE TEACHING TIP

What does the left skew do? In this example, the peak of the
Mean versus median distribution is at point C. That makes C
the mode of the distribution. Students Lesson 2.3

PROBLEM: A density curve that models the distribution of a quantitative may have learned that the mode is

variable is shown. Identify the location of the mean and median by letter.
Justify your answers. a way to measure center. A better
SOLUTION: AB C understanding of mode is that it is a
=
=
MedianB,MeanA . B is the equal-areas point of the Even though C is directly under the peak of the peak—a point at which there is a high

distribution. The mean will be less than the median due to the curve, more than half of the area is to its left, so it (or the highest) frequency of values.
left-skewed shape.
cannot be the median.
Mean versus medianonly.
FOR PRACTICE TRY EXERCISE 11. AL TERNA TE EX AMPLE
Is it left skewed, too?
(C) 2021 BFW Publishers -- for review purposes
A density curve is an idealized model for the population distribution of a quanti-

tative variable. As a result, we label the mean of a density curve as µ and the standard
.
deviation of a density curve as σ We can roughly locate the mean of any density curve PROBLEM: A density curve that models
by eye, as its balance point. No easy way exists to estimate the standard deviation for
density curves in general. But there is one family of density curves for which we can the distribution of a quantitative variable
estimate the standard deviation by eye. is shown. Identify the location of the
mean and median by letter. Justify your
Normal Distributions
answers.
When we examine a distribution of quantitative data, how does it compare with

an idealized density curve? Figure 2.9(a) shows a histogram of the scores of all
seventh-grade students in Gary, Indiana, on the vocabulary part of the Iowa Test of
11
Basic Skills (ITBS). The scores are grade-level equivalents, so a score of 6.3 indicates

AB C
SOLUTION: This density curve is
normal skewed to the left. Median B,
=
density
=
curve MeanA. B is the equal-areas point
of the distribution. The mean (balance
point of the distribution) will be less
than the median due to the left-skewed
shape.
2 4 6 8 10 12 2 4 6 8 10 12
(a) ITBS vocabulary score (b) ITBS vocabulary score
FIGURE 2.9 (a) Histogram of the Iowa Test of Basic Skills (ITBS) vocabulary scores of all seventh-grade
students in Gary, Indiana. (b) The normal density curve shows the overall shape of the distribution. TEACHING TIP
The notion of population versus sample
is important and it was first alluded
to in Lesson 1.6. Populations are an
entire group of objects, events, or
03_StarnesSPA4e_24432_ch02_088_153.indd 111 07/09/20 1:55 PM
things. Samples are subsets of these
populations. When there are large
numbers of values in a distribution
(from large populations), we often see a
pattern and model it with a density curve.
In Figure 2.9(a), we see a population
distribution with a clear pattern. The
density curve in Figure 2.9(b) models that
pattern. The mean and standard deviation
of that density curve would be denoted µ
and σ, respectively.

LESSON 2.3 • Density Curves and the Normal Distribution 111

03_TysonTEspa4e_25177_ch02_088_153_4pp.indd 111 10/11/20 7:44 PM