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52 UNIT 1 Exploring One-Variable Data
Learn Statistics by Doing Statistics
PROPERTIES OF THE MEAN
The preceding example illustrates an important weakness of the mean as a mea-
sure of center: The mean is not resistant to extreme values, such as outliers. The
bag of Fritos, with only 19% air, decreases the mean by 1.81 percentage points.
DEFINITION Resistant
SECTION 1F Normal Distributions
A statistical measure is resistant if it is not affected much by extreme data values. 111
6.84 and standard deviation σ =
mean µ =
1.55. How unusual is it for a Gary
The median is a resistant measure of center. In the preceding example, the
seventh-grader to get an ITBS score less than 3.74? The figure shows the normal
median percent air in all 14 bags of chips is 45.5. If we remove the possible outlier
Experience statistical principles with simulations and technology tools
curve for this distribution with the area of interest shaded. Note that the boundary
bag of Fritos, the median percent air in the remaining 13 bags is nearly the same (46).
© 2024 BFW Publishers PAGES NOT FINAL - For Review Purposes Only - Do Not Copy
value, 3.74, is exactly 2 standard deviations below the mean.
Why is the mean so sensitive to extreme values? The following activity pro-
vides some insight.
ACTIVITY Interpreting the mean
In this activity, you will investigate a physical interpretation of the
mean of a distribution.
ACTIVITIES Every unit includes 1. Stack 5 pennies on top of the 6-inch mark on a 12-inch ruler.
5.29
2.19
11.49
8.39
3.74
6.84
9.94
ITBS vocabulary score
a hands-on activity in the first Place a pencil under the ruler to make a “seesaw” on a desk
or table. Move the pencil until the ruler balances. What is the
few pages to introduce the How can we estimate the shaded area, which represents the proportion of all
relationship between the location of the pencil and the mean of
the five data values 6, 6, 6, 6, and 6?
content, with other activities Gary, Indiana, seventh-graders with ITBS vocabulary scores less than 3.74? The
following activity reveals one way to do it.
2. Move one penny off the stack to the 8-inch mark on your ruler.
appearing later in the unit. Ann Heath Now move one other penny so that the ruler balances again with-
out moving the pencil. Where did you put the other penny? What
Many of the 37 activities use ACTIVITY What’s so special about normal distributions?
is the mean of the five data values represented by the pennies now?
dynamic applets that help 3. Move one more penny off the stack to the 2-inch mark on your ruler. Now
In this activity, you will use an applet to discover an interesting property of nor-
you experience the process of move both remaining pennies from the 6-inch mark so that the ruler still bal-
ances with the pencil in the same location. Is the mean of the data values still 6?
mal distributions.
collecting data and drawing 4. Discuss with your classmates: Why is the mean called the “balance point”
1. Go to www.stapplet.com and launch the Normal
Normal Distributions of a distribution?
conclusions from those data. Operation: Calculate an area under the Normal curve Distributions applet.
All of the applets are available Mean = 6.84 SD = 1.55 Plot distribution 2. Choose “Calculate an area under the Normal curve”
from the Operation menu at the top. Enter 6.84 for
the mean and 1.55 for the standard deviation. Then
to you whether you do the a distribution. For the data on percent air in each of 14 brands of chips, the dot-
The activity gives a physical interpretation of the mean as the balance point of
click the “Plot distribution” button. (These are the
70 activity in class or on your own. plot balances at x = 42.57%. values for the distribution of ITBS vocabulary scores
UNIT 1 Exploring One-Variable Data
of seventh-graders in Gary, Indiana.) A figure like
this should appear.
60
50
40
30
10 20 3. Use the applet to help you answer the following
questions about the distribution of ITBS scores.
test to their classes and grade the test together. Mr. Starnes’s students earned an Percent air
8.39
6.84
9.94
11.49
5.29
3.74
2.19
average score that was 8 points higher than the average for Ms. McGrail’s class. (a) About what proportion of Gary, Indiana,
Calculate the area between two values
COMPARING THE MEAN AND MEDIAN
Ms. McGrail wonders whether Mr. Starnes might have “adjusted” the class seventh-graders have ITBS vocabulary scores
Right boundary:
Left boundary:
rosters from the computer scheduling program. In other words, she thinks he between 5.29 and 8.39? That is, what percentage
Which measure — the mean or the median — should we report as the center of
might have “stacked” his class. He denies this, of course. of the area under the normal curve lies within
Calculate area
a distribution? That depends on both the shape of the distribution and whether
there are any outliers.
To help resolve the dispute, Mr. Starnes provides data on the cumulative 1 standard deviation of the mean?
(b) About what
grade point averages of the students in both classes from his computer. The proportion of Gary, Indiana, seventh-graders have ITBS
following table displays the data. vocabulary scores between 3.74 and 9.94? That is, what percentage of
the area under the normal curve lies within 2 standard deviations of the
mean?
2.900 3.300 3.980 2.900 3.200 3.500 2.800 2.900 3.950
McGrail TECHNOLOGY Use technology as a tool for discovery and analysis. The 30 Tech Corners,
2.900
3.200
3.100 2.850 2.900 3.245 3.000 3.000 2.800 (c) About what proportion of Gary, Indiana, seventh-graders have ITBS
vocabulary scores between 2.19 and 11.49? That is, what percentage of
3.600
2.600
3.200
3.750
2.700
3.085
2.860
3.100
2.900 placed strategically throughout the book at the optimal point of use, give step-by-step 09/10/23 11:35 AM
02_StarnesTPS7e_40934_un01_p1_001_086.indd 52
Starnes the area under the normal curve lies within 3 standard deviations of the
3.100
3.800
3.200
3.338
3.560
3.400 instructions for using the TI-83/84 calculator. Instructions for additional calculators are
mean?
available on the book’s website.
Based on these data, did Mr. Starnes stack his class? Give appropriate graph-
ical and numerical evidence to support your conclusion.
Supporting Tech Corner Videos are available to walk you through the keystrokes needed to
perform each analysis.
You can use technology to make boxplots, as the following Tech Corner illustrates.
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3. Tech Corner MAKING BOXPLOTS
TI-Nspire and other technology instructions are on the book’s website at bfwpub.com/tps7e.
The TI-83/84 can plot up to three boxplots in the same viewing window. Let’s use the calculator to make
parallel boxplots of the overall rating data for Apple and Samsung tablets.
1. Enter the ratings for the Apple tablets in list L1 and those for the Samsung tablets in list L2.
2. Set up two statistics plots: Plot1 to show a boxplot of the Apple data NORMAL FLOAT AUTO REAL RADIAN MP
and Plot2 to show a boxplot of the Samsung data. The setup for Plot1 Plot2 Plot3
Plot1 is shown. When you define Plot2, be sure to change L1 to L2. On Off
Note: The calculator offers two types of boxplots: one that shows Type:
outliers and one that doesn’t. We’ll always use the type that identifies Xlist:L1
Freq :1
outliers. Mark :
Color: BLUE
3. Press ZOOM and select ZoomStat to display the parallel boxplots. NORMAL FLOAT AUTO REAL RADIAN MP
xxii Then press TRACE and use the arrow keys to view the five-number
Plot1: L1
summary.
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