Lesson
3
Variability
Whenever two people observe an event, they are likely to see different things. If
two people measure something to the nearest millimeter, they may well get two
different measurements. If two people conduct the same experiment, they will get
slightly different results. In fact, there is variability in nearly everything; no two
leaves or snowflakes are exactly alike. Because variability is so common, it is
important that you begin to understand what causes variability and how it can be
measured and interpreted.
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Mean Height (in cm)
Mean Height (in cm)
Heights from Birth to 14
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Age of Boy
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Age of Girl
Think About This Situation
The data in the growth charts above come from a physician’s handbook. Use
the plots to answer the following questions.
a
Is it reasonable to call a 14-year-old boy “taller than average” if his height is
165 cm? Is it reasonable to call a 14-year-old boy “tall” if his height is
165 cm? What additional information about 14-year-old boys would you
need to know to be able to say that he is “tall”?
b At what height would you be willing to call a 14-year-old girl “tall”? Do
you have enough information to make this judgment?
c
During which year do children grow most rapidly?
LESSON
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VA R I A B I L I T Y
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INVESTIGATION 1
Measuring Variability: The Five-Number
Summary
If you are in the 40th percentile of height for your age, that means that 40% of
people your age are your height or shorter than you are and 60% are taller. Shown
below are physical growth charts for boys and girls, 2 to 18 years in age. The charts
were developed by the National Center for Health Statistics.
The curved lines for the height (top) and weight (bottom) tell a physician what
percentile a boy or girl is in. The percentiles are the small numbers 5, 10, 25, 50,
75, 90, and 95 towards the right ends of the curved lines. For example, suppose
John is a 17-year-old boy who weighs 60 kg or 132 pounds. John is in the 25th
percentile of weight for his age. Twenty-five percent of 17-year-old boys weigh
the same or less than John and 75% weigh more than John. If John’s height is
180 cm or almost 5'11", he is in the 75th percentile of height for his age.
1. Based on the information given about John, how would you describe John’s
general appearance?
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105
100
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75
in.
cm
Physical Growth Percentiles, Girls
2 to 20 Years
95
90
75
50
25
10
5
Weight
95
90
75
50
25
10
5
2
4
6
8
10
12
14
16
18
Age (in years)
Source: www.cdc.gov/growthcharts/
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20
kg
lb
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30
Height
Height
Physical Growth Percentiles, Boys
2 to 20 Years
74
72
70
68
66
64
62
60
58
56
54
52
50
48
46
44
42
40
38
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185
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175
170
165
160
155
150
145
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125
120
115
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105
100
95
90
85
80
75
in.
cm
95
90
75
50
25
10
5
Weight
95
90
75
50
25
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5
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Age (in years)
16
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20
kg
lb
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90
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2. With your group, spend some time learning to read the CDC growth charts.
They contain an amazing amount of information!
a. What is the approximate percentile for a 9-year-old girl who is 128 cm tall?
b. About how tall does a 12-year-old girl have to be so that she is as tall or taller
than 75% of the girls her age?
c. How tall would a 14-year-old boy have to be so that you would consider him
“tall” for his age? How did you make this decision?
d. How tall would a 14-year-old girl have to be so that you would consider her
“tall” for her age? How did you make this decision?
e. What is the 25th percentile of height for 4-year-old boys? The 50th percentile?
The 75th percentile?
f. How can you tell from the height and weight chart when children are growing the fastest? When is the increase in weight the greatest for girls? For
boys?
3. Some percentiles have special names.
a. What is another name for the 50th percentile?
b. The 25th percentile is sometimes called the lower quartile. Estimate the
lower quartile of height for 6-year-old girls.
c. The 75th percentile is sometimes called the upper quartile. Estimate the
upper quartile of height for 6-year-old girls.
The quartiles together with the median give some indication of the center and spread
of a set of data. A more complete picture of the distribution of a set of data is given
by the five-number summary: the minimum value, the lower quartile (Q1), the
median (Q2), the upper quartile (Q3), and the maximum value.
4. From the charts, estimate the five-number summary for 13-year-old girls’ heights
and for 13-year-old boys’ heights. Some estimates will be more difficult than
others.
The distance between the first and third quartiles is called the interquartile range
(IQR). The IQR is a measure of how spread out or variable the data are. The
distance between the minimum value and the maximum value is called the range.
The range is another, typically less useful, measure of how variable the data are.
5. Refer back to your estimates in Activity 4 and the CDC growth charts.
a. What is the interquartile range of the heights of 13-year-old girls? Of 13year-old boys?
b. What happens to the interquartile range of heights as children get older?
c. In general, do boys’ heights or girls’ heights have the larger interquartile range
or are they about the same?
d. What happens to the interquartile range of weights as children get older?
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6. Can you estimate the range of the heights of 18-year-old boys? Why is the
interquartile range more informative than the range?
For the children’s heights, you were able to estimate quartiles from the chart. Next
you will learn how to compute quartiles from sets of data.
When you explored the AAA ratings for cars in Lesson 2, Investigation 1, you
saw variability in the ratings of each car. Below are the ratings for the Volvo V40
Wagon and the Chevy Camaro and their corresponding histograms.
Volvo V40 Wagon Ratings
Chevy Camaro Ratings
8, 8, 7, 9, 7, 8, 8, 6, 8, 7,
8, 8, 8, 7, 8, 8, 10, 9, 9, 9
10, 7, 7, 9, 5, 8, 7, 4, 7, 5,
8, 8, 8, 4, 5, 6, 5, 8, 7, 7
V40 Wagon
Camaro
7. Which of the cars has greater variability in its ratings? Explain your reasoning.
8. Put the ratings for the Volvo V40 Wagon in an ordered list and find the median.
Mark the position of the median on your ordered list.
a. How many ratings are on each side of the median?
b. Once the data are ordered, you can divide them into quarters to find the
quartiles. Find the midpoint of each half of the ratings for the V40 Wagon.
Mark the positions of the quartiles on your ordered list of the ratings.
c. What fraction or percentage of the ratings is less than or equal to the lower
quartile?
d. What fraction or percentage of the ratings is greater than or equal to the upper quartile?
9. You can find the five-number summary quickly using technology.
a. Find the five-number summary for the Camaro’s ratings using your calculator or computer software.
b. What is the 25th percentile of the ratings for the Camaro? The 75th percentile?
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Checkpoint
A percentile gives the location of a value in a set of data while the range and
IQR are measures of how spread out the data are.
a
Will the range be changed if an outlier is added to a data set? Will the
interquartile range be changed?
b Why does the interquartile range tend to be a more useful measure of a
data set’s variability than the range?
c
If you get 75 points out of 100 on your next math test, can you tell what
your percentile is? Explain.
d
Give an example of when you would want to be in the 10th percentile
rather than in the 90th.
e
Give an example of when you would want to be in the 90th percentile
rather than in the 10th.
Be prepared to share your group’s thinking and examples
with the rest of the class.
On Your Own
Driving Position
Instruments
Controls
Visibility
Entry/Exit
Quietness
Cargo Space
Exterior
Interior
Value
10 7
Interior Space
6 7
Comfort
8
Fuel Economy
Driveability
5 6
Handling
Braking
8
Ride
Transmission
5
Steering
Acceleration
The ratings for the Toyota Prius are reproduced below.
7
8
8
8
8
8
9
5
8
8
6
a. Find the five-number summary for the Prius. What is the
interquartile range?
b. Refer to your results in Activity 8 and Activity 9 (page 50). Of the
ratings for the Prius, the Camaro, and the V40 Wagon, which has
the greatest interquartile range? What would this tell you as a possible buyer?
LESSON
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VA R I A B I L I T Y
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INVESTIGATION 2 Picturing Variability
Your heart continually pumps blood through your body. This pumping action can
be felt on the side of your neck or your wrist where an artery is close to the skin.
The small swelling of the artery as the heart pushes the blood is called your pulse.
1. Take your pulse for 60 seconds. Record the pulse rate (number of beats per
minute) for each member of your group.
a. Record your data and data from other groups on the Class Data Table you
started in Lesson 1.
b. Find the five-number summary for the pulse rates for your class.
The five-number summary can be displayed in a box plot. To make a box plot of
your pulse rates, first make a number line. Below this line draw a box from the
lower quartile to the upper quartile; then draw line segments connecting the box to
each extreme (the maximum and minimum values). Draw a vertical line in the box
to indicate the median. The segments at either end are often called whiskers, and
the plot is sometimes called a box-and-whiskers plot. Here are the results for one
class.
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Minimum
55
60
Lower
Quartile
(Q1)
65
70
75
80
Maximum
Median
(Q2) Upper
Quartile
(Q3)
2. Study the sample box plot above.
a. Is the distribution represented by the box plot skewed to the left or to the
right, or is it symmetric?
b. Draw a box plot of the pulse rates for your class.
c. Is the distribution of pulse rates for your class skewed to the left or to the
right, or is it symmetric?
d. What is the length of the box for your class data? What is the mathematical
term for this length?
3. Box plots are most useful when comparing two or more distributions.
a. How do you think box plots of class pulse rates before and after exercising
would compare?
b. Have some members of your group do some sort of mild exercise for a short
time. Take their pulse readings immediately afterwards. Record the data on
the Class Data Table.
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c. Combine your new data with data from other groups.
d. Construct a box plot of the new pulse rates. Place it below your first box plot.
e. How are the box plots different? Write a summary of your conclusions about
pulse rates of the class before and after exercise.
f. What other plot could be used to compare the two distributions? Make this
plot. Can you see anything interesting that you could not see from the box
plots?
You can produce box plots with technology by following a procedure similar to
that of making histograms. After entering the data and specifying the viewing window, select Box Plot as the type of graph desired.
4. Refer to the ratings for the V40 Wagon and Camaro given in Investigation 1 on
page 50.
a. Using your calculator or computer software, make a box plot of the ratings.
b. Use the trace feature to find the five-number summary for the V40 Wagon.
Compare the results with your computations in Activity 8 for Investigation 1
of this lesson.
c. Draw the box plot below a histogram of the V40 Wagon’s ratings. Use the
same scale for both.
d. How does the box plot compare to the histogram?
e. Produce a box plot of the ratings for the Camaro. Draw the box plot below a
histogram of the Camaro’s ratings. Use the same scale for both.
f. Compare the box plots for the V40 Wagon and the Camaro. What do the
different lengths of the boxes tell you about the variability in their AAA ratings? Is either distribution skewed? Symmetric?
LESSON
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VA R I A B I L I T Y
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Reproduced below are the ratings for the Concorde, Mustang, and Miata.
Concorde Ratings
Mustang Ratings
Miata Ratings
8, 9, 6, 7, 9, 8, 8, 6, 9, 9,
8, 7, 8, 7, 9, 9, 7, 7, 8, 9
9, 8, 6, 9, 7, 9, 8, 4, 8, 5,
9, 9, 9, 6, 4, 6, 5, 8, 8, 8
8, 9, 9, 9, 6, 9, 8, 6, 8, 6,
8, 8, 8, 6, 6, 6, 3, 8, 8, 9
5. Produce box plots for the Concorde, Mustang, and Miata ratings. Draw each
box plot below a corresponding histogram. Divide up the work among your
group. Including the V40 Wagon and the Camaro, your group now has five different box plots to compare.
a. Use the box plots to determine which of the five cars has the largest interquartile
range.
b. Which cars have almost the same interquartile range? Does this mean their
distributions are the same? Explain your thinking.
c. Why do the Concorde and Mustang have no whisker at the upper end?
d. Why is the lower whisker for the Miata so long?
e. Based on the box plots, which of the five cars seems to have the best ratings?
How did the plots help you make your decision?
Checkpoint
In this investigation, you have learned how to display the five-number summary on a box plot.
a How does a box plot convey how close together data are in a
distribution?
b What does a box plot tell you that a histogram does not?
c What does a histogram tell you that a box plot does not?
Be prepared to share your group’s thinking about the
usefulness of box plots.
Box plots can be used to compare several distributions. Some computer software and graphing calculators allow you to display several box plots on the
same screen. When making box plots, remember to include labels and a scale.
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On Your Own
Refer to the data on the amount of fat in the fast foods listed in Investigation 2 on
page 22. Produce a box plot of these data.
a. What is the five-number summary of these data?
b. What is the interquartile range for these data, and what information does it tell
you?
c. Where is the Burger King Double Whopper with Cheese located on the plot?
d. Choose a single item (such as your favorite, if you have one) and describe its
relation to the other foods in the list in terms of fat content.
LESSON
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VA R I A B I L I T Y
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