Chapter 16 l Distributions of Discrete Data 711
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16.1 Choosing CirclesSampling Methods and Bias | p. 713
16.2 Surveys and Studies and Experiments, Oh My!Surveys, Studies, and
Experiments | p. 727
16.3 Do It YourselfDesigning and Collecting Data Using a
Survey, Study, or Experiment | p. 737
16.4 Numbers, Graphs, and ConclusionsInterpreting Results and Drawing
Conclusions from a Survey, Study, or
Experiment | p. 745
Distributions of Discrete Data16
CHAPTER
16
Text messaging while driving increases the likelihood of an accident or near miss by 2,300%,
according to a 2009 study of professional drivers. Texting drivers take their eyes off the road for
an average of 4.6 out of every six seconds. You will analyze a survey of teenage texters.
712 Chapter 16 l Distributions of Discrete Data
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Lesson 16.1 l Sampling Methods and Bias 713
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Problem 1 You Pick! When analyzing data to make conclusions, more data generally leads to more
accurate results. However, collecting large amounts of data can be both time
consuming and expensive. So, only a portion of the data may be collected with the
hope that this collected data is representative of the total population.
The last two pages of this lesson show 100 circles and a table. The table lists an
identification number, a diameter, and an area for each circle. The average area of
the 100 circles is
∑ n�0
99
Area of circle n
__________________ 100
� 0.58�.
A subjective sample is a sample in which an individual makes a judgment about
which data items to select.
1. Select a subjective sample by choosing 5 circles that you think best
represent the entire set of circles. List the identification numbers of the
5 circles you chose.
16.1 Choosing CirclesSampling Methods and Bias
ObjectivesIn this lesson you will:
l Use different sampling techniques.
l Identify possible bias due to sampling
methods or wording of questions.
Key Termsl subjective sample
l simple random sample
l random digit table
l stratified random sample
l cluster sample
l systematic sample
l convenience sample
l volunteer sample
l biased sample
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2. Explain why you chose those particular circles.
3. Calculate the area of each of the five circles you chose.
4. Calculate the average area of the five circles you chose.
5. List the average areas calculated by the other students in your class.
6. Create a histogram of the average areas calculated by all students in your class.
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7. What do you notice about the average areas calculated by the students in
your class? How close is each student’s average to the average area of all
100 circles?
Research has shown that asking people to choose a subjective sample based
on what a person thinks is accurate is not a good sampling technique. Even for
relatively small data sets, the average of the sample often varies significantly with
the actual average of the entire data set.
Fortunately, there are other sampling techniques that often yield results that are
more representative of the data.
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Problem 2 Random SamplesSimple random sampling involves selecting a sample in such a way that every
member of the population has the same chance of being selected.
Just as trying to select a subjective sample can be difficult, most people are unable
to select data that are truly random. So, technology is often used to generate the
random numbers.
A random digit table is a table of single digits, 0 through 9, that have been
randomly generated by a computer so that every digit has the same chance of
being chosen each time. The following is an example of a random digit table.
Random Digit TableLine 1 65285 97198 12138 53010 94601 15838 16805 61004 43516 17020
Line 2 17264 57327 38224 29301 31381 38109 34976 65692 98566 29550
Line 3 95639 99754 31199 92558 68368 04985 51092 37780 40261 14479
Line 4 61555 76404 86210 11808 12841 45147 97438 60022 12645 62000
Line 5 78137 98768 04689 87130 79225 08153 84967 64539 79493 74917
Line 6 62490 99215 84987 28759 19177 14733 24550 28067 68894 38490
Line 7 24216 63444 21283 07044 92729 37284 13211 37485 10415 36457
Line 8 16975 95428 33226 55903 31605 43817 22250 03918 46999 98501
Line 9 59138 39542 71168 57609 91510 77904 74244 50940 31553 62562
Line 10 29478 59652 50414 31966 87912 87154 12944 49862 96566 48825
Line 11 96155 95009 27429 72918 08457 78134 48407 26061 58754 05326
Line 12 29621 66583 62966 12468 20245 14015 04014 35713 03980 03024
Line 13 12639 75291 71020 17265 41598 64074 64629 63293 53307 48766
Line 14 14544 37134 54714 02401 63228 26831 19386 15457 17999 18306
Line 15 83403 88827 09834 11333 68431 31706 26652 04711 34593 22561
Line 16 67642 05204 30697 44806 96989 68403 85621 45556 35434 09532
Line 17 64041 99011 14610 40273 09482 62864 01573 82274 81446 32477
Line 18 17048 94523 97444 59904 16936 39384 97551 09620 63932 03091
Line 19 93039 89416 52795 10631 09728 68202 20963 02477 55494 39563
Line 20 82244 34392 96607 17220 51984 10753 76272 50985 97593 34320
The random digit table shown is divided into groups of 5 digits to make the table
easier to read. To use the table, begin at any digit and follow the numbers in a
systematic way, such as moving across a row until it ends and then moving to the
beginning of the next row.
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1. Explain how you can use the random digit table shown to choose a sample of
5 circles.
2. Select a simple random sample of 5 circles by starting with the first digit of
line 7.
3. Select a simple random sample of 5 circles by starting with the first digit of
line 9.
4. Calculate the average area of the circles included in the random samples from
Questions 2 and 3.
5. Suppose a student in your class selected the same circles that appear in
Question 3 when picking a subjective sample in Problem 1. Did that student
select a simple random sample?
6. Select a simple random sample of 5 circles by starting with any digit in the
random digit table. Be sure to explain where you started.
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7. Calculate the average area of the circles included in the random samples from
Question 6.
8. Create a histogram of the average areas of the random samples of all
students in your class.
9. What do you notice about the average areas of the students in your class?
How close is each student’s average to the average area of all 100 circles?
10. Compare the histogram in Question 8 to the histogram in Problem 1, Question 6.
What do you notice?
Most graphing calculators can also be used to generate random numbers. To
generate random numbers using a graphing calculator, perform the following steps.
l Press the MATH button.
l Select PRB and then go down to 5 which reads randInt (press Enter).
l Enter the range of numbers separated by a comma. For example, to
generate numbers between 0 and 99, enter randInt(0, 99).
l Press Enter to generate another random number.
11. Select a simple random sample of 5 circles using a graphing calculator.
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12. Calculate the average area of the circles included in the random sample from
Question 11.
Problem 3 Other Sampling TechniquesA stratified random sample is a sample obtained by dividing the population into
different groups, or strata according to a characteristic and randomly selecting data
from each group.
For example, the circles can be divided into groups based on their relative size as
follows.
Small Circles (46) Medium Circles (39) Large Circles (15)
1, 4, 6, 13, 14, 16, 17, 19, 22,
24, 26, 28, 30, 33, 34, 37, 39,
42, 45, 46, 47, 51, 53, 56, 57,
58, 59, 62, 63, 67, 68, 72, 74,
78, 79, 82, 85, 87, 88, 89, 93,
94, 95, 97, 98, 99
0, 2, 3, 8, 9, 10, 11, 12, 21,
23, 25, 29, 31, 35, 36, 40, 41,
43, 49, 50, 52, 61, 64, 65, 66,
69, 71, 73, 75, 76, 77, 80, 81,
83, 84, 86, 90, 91, 96
5, 7, 15, 18, 20, 27,
32, 38, 44, 48, 54,
55, 60, 70, 92
1. Select a stratified random sample by randomly selecting 2 small circles,
2 medium circles, and 1 large circle.
2. Calculate the average area of the circles included in the stratified random
sample from Question 1.
A cluster sample is a sample obtained by creating clusters with each cluster
containing the characteristics of the population and randomly selecting a cluster.
3. On the page at the end of this section that contains the circles, draw
4 horizontal lines and 2 vertical lines so that the page is divided into
12 equal rectangles. Each rectangle represents a cluster of circles.
Number each cluster from 1 to 12.
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4. Select a cluster sample by randomly selecting one of the clusters.
5. Calculate the average area of the circles included in the cluster sample from
Question 4.
A systematic sample is a sample obtained by selecting every nth data in the
population.
6. Select a systematic sample by choosing every 20th circle. First randomly
choose a number from 0 to 20 to start at and then choose every 20th circle
after that.
7. Calculate the average area of the circles included in the systematic sample
from Question 6.
Not all sampling techniques are random. For example, at the beginning of this
lesson, you chose a subjective sample by individually making a judgment about
which circles to choose. Other sampling techniques that are not random include:
l A convenience sample is a sample whose data is based on what is
convenient for the person choosing the sample.
l A volunteer sample is a sample whose data consists of those who
volunteer to be part of the sample.
8. Determine the type of sampling technique being used in each situation: simple
random sampling, cluster sampling, stratified sampling, systematic sampling,
subjective sampling, convenience sampling, or volunteer sampling.
a. A principal personally chooses 10 students she believes are representative
of the junior class to participate in a survey about school spirit.
b. A principal randomly chooses the number ‘15’ and chooses the 15th
student and every 10th student after that to participate in a survey about
school spirit.
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c. A principal selects the first 15 students to walk past her office to
participate in a survey about school spirit.
d. A principal gives each student a number and randomly chooses
15 numbers to be a sample of students to take a survey about school spirit.
e. A principal gives students in each grade, 9–12, a number and randomly
selects 5 students from each grade to participate in a survey about school
spirit.
Problem 4 BiasSamples are chosen to learn something about a population. However, not all
samples are representative of the population. A biased sample is a sample of data
that does not accurately represent all of the population.
1. Identify how each sampling technique could be biased.
a. Stratified random sample
b. Cluster sample
c. Systematic sample
d. Convenience sample
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2. In the famous election poll of 1936, the Literary Digest predicted Alf Landon
would be the winner over Franklin Delano Roosevelt by a huge margin. The
poll was based on a sample of over 2 million voters who were selected based
on automobile and telephone ownership. When results of the election were
totaled, Roosevelt was the winner. Can you think of a reason why the pollsters
were so far off in their prediction?
Often, those who respond to surveys do so because they have strong opinions.
Bias can also occur when those selected to be in a sample are unable or refuse to
participate in the sample or if the wording of the question being asked is confusing.
3. Identify and explain any possible sources of bias in the following samples:
a. A cell phone company wants to know how many text messages teenagers
typically send in one month. They post an online survey on the Internet
asking for teenagers to respond to the question, “How many minutes do
you spend sending text messages per month?”
b. A cell phone company wants to know how many text messages teenagers
typically send in one month. They ask the first 100 teens who stop by
their store this question, “How many minutes do you spend sending text
messages per month?”
c. A cell phone company wants to know how many text messages teenagers
typically send in one month. They send out surveys to 2000 teens asking,
“How many minutes do you spend sending text messages per month?”
Only 35 teens respond.
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d. A cell phone company wants to know how many text messages teenagers
typically send in one month. They ask a random sample of teens,
“To the nearest half minute or 30 seconds, how many minutes do you
usually spend sending text messages on your cell phone during an average
month when you regularly use your cell phone?”
0
1 2
4
5
3
6
7
89
10
11
12
1314
15
1617
19 21
18
34
3331
3738
35
36
4547 46
32
43
5759
58
60
44
48
61 62
6364
6571
72
73
74 75 76
94
77
78
79
96
9593
6667
68
69
81
8283
84 85
63
86
70
87
97 9888 89 90
91
92
99
2022
23
24
2526
2728
2930
404239
41
55
56
5150
49
52
53
54 80
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Circle Number
Diameter(in.)
Area (in.2)
Circle Number
Diameter(in.)
Area (in.2)
CircleNumber
Diameter(in.)
Area (in.2)
0 1 __ 2
1 ___
16 � 25 1 __
2 1 __
4 � 50 1 __
2
1 ___
16 �
1 1 __ 4 1 ___
64 � 26 1 __
4 1 ___
64 � 51 1 __
4 1 ___
64 �
2 1 __ 2
1 ___
16 � 27 1 1 __
2 9 ___
16 � 52 1 __
2 1 ___
64 �
3 1 1 __ 4 � 28 1 __
4 1 ___
64 � 53 1 __
4 1 ___
64 �
4 1 __ 4 1 ___
64 � 29 1 1 __
4 � 54 1 1 __
2 9 ___
16 �
5 1 1 __ 2 9 ___
16 � 30 1 __
4 1 ___
64 � 55 2 �
6 1 __ 4 1 ___
64 � 31 1 1 __
4 � 56 1 __
4 1 ___
64 �
7 2 � 32 2 � 57 1 __ 4 1 ___
64 �
8 1 __ 2
1 ___
16 � 33 1 __
4 1 ___
64 � 58 1 __
4 1 ___
64 �
9 1 1 __ 4 � 34 1 __
4 1 ___
64 � 59 1 __
4 1 ___
64 �
10 1 1 __ 4 � 35 1 __
2
1 ___
16 � 60 2 �
11 1 __ 2
1 ___
16 � 36 1 1 __
4 � 61 1 __
2
1 ___
16 �
12 1 __ 2 1 ___
16 � 37 1 __
4 1 ___
64 � 62 1 __
4 1 ___
64 �
13 1 __ 4 1 ___
64 � 38 2 � 63 1 __
4 1 ___
64 �
14 1 __ 4 1 ___
64 � 39 1 __
4 1 ___
64 � 64 1 __
2
1 ___
16 �
15 2 � 40 1 __ 2 1 ___
16 � 65 1 1 __
4 �
16 1 __ 4 1 ___
64 � 41 1 1 __
4 � 66 1 __
2
1 ___
16 �
17 1 __ 4 1 ___
64 � 42 1 __
4 1 ___
64 � 67 1 __
4 1 ___
64 �
18 2 � 43 1 1 __ 4 � 68 1 __
4 1 ___
64 �
19 1 __ 4 1 ___
64 � 44 2 � 69 1 __
2
1 ___
16 �
20 1 1 __ 2 9 ___
16 � 45 1 __
4 1 ___
64 � 70 2 �
21 1 __ 2
1 ___
16 � 46 1 __
4 1 ___
64 � 71 1 __
2
1 ___
16 �
22 1 __ 4 1 ___
64 � 47 1 __
4 1 ___
64 � 72 1 __
4 1 ___
64 �
23 1 1 __ 4 � 48 2 � 73 1 __
2
1 ___
16 �
24 1 __ 4 1 ___
64 � 49 1 __
2
1 ___
16 � 74 1 __
4 1 ___
64 �
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Circle Number
Diameter(in.)
Area (in.2)
Circle Number
Diameter(in.)
Area (in.2)
CircleNumber
Diameter(in.)
Area (in.2)
75 1 __ 2 1 ___
16 � 84 1 __
2 1 ___
16 � 93 1 __
4 1 ___
64 �
76 1 __ 2 1 ___
16 � 85 1 __
4 1 ___
64 � 94 1 __
4 1 ___
64 �
77 1 1 __ 4 � 86 1 1 __
4 � 95 1 __
4 1 ___
64 �
78 1 __ 4 1 ___
64 � 87 1 __
4 1 ___
64 � 96 1 __
2 1 ___
16 �
79 1 __ 4 1 ___
64 � 88 1 __
4 1 ___
64 � 97 1 __
4 1 ___
64 �
80 1 1 __ 4 � 89 1 __
4 1 ___
64 � 98 1 __
4 1 ___
64 �
81 1 __ 2 1 ___
16 � 90 1 __
2 1 ___
16 � 99 1 __
4 1 ___
64 �
82 1 __ 4 1 ___
64 � 91 1 __
2 1 ___
16 �
83 1 __ 2 1 ___
16 � 92 2 �
Be prepared to share your methods and solutions.
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Lesson 16.2 l Surveys, Studies, and Experiments 727
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Problem 1 Surveys, Studies, and Experiments
Data are collected and analyzed to help answer questions about the world that we
live in. Answering different questions requires different methods for collecting data.
A sample survey poses a question of interest to a random sample of the targeted
population. The question concerns something we wish to know about the
characteristic of the population. Political polls, TV ratings, and questionnaires are all
types of surveys. The following is one example of a survey.
A recent survey of nearly 1200 young people from across the U.S. shows that
40% of 16 to 20 year olds who have a driver’s license admit to texting while
they are driving on a regular basis.
1. Identify the population, the sample, and the characteristic that was of interest
in the sample survey.
16.2 Surveys and Studies and Experiments, Oh My!Surveys, Studies, and Experiments
ObjectivesIn this lesson you will:
l Differentiate between sample surveys,
observational studies, and experiments.
l Identify characteristics of sample
surveys, observational studies, and
experiments.
Key Termsl sample survey
l observational study
l experiment
l treatment
l experimental unit
l confounding
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An observational study gathers data about a characteristic of the population
without trying to influence the data. Many medical studies are observational studies
in which data from several studies of the same population are compiled in order to
draw a conclusion. The following is one example of an observational study.
New research funded by a pediatric agency found that nearly 70% of in-house day
care centers show as much as 2½ hours of television to the children in the center
per day. The study examined 132 day care programs in 2 midwestern states.
2. Identify the population, the sample, and the characteristic that was of interest
in the observational study.
An experiment gathers data on the effect of one or more treatments, or
experimental conditions, on the characteristic of interest. Members of the sample,
sometimes called experimental units, are randomly assigned to a treatment group.
The following is one example of an experiment.
A sample of 200 asthma patients participated in the clinical trial for a new
asthma drug. One hundred of the patients received a placebo treatment along
with an inhaler while the remaining 100 patients received the new drug along
with an inhaler. Monthly blood and breathing tests were performed on all
200 patients to determine if the new drug was effective.
3. Identify the population, the sample, and the characteristic that was of interest
in the experiment.
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4. Classify each situation as a sample survey, an observational study, or an
experiment. Then identify the population, the sample, and the characteristic
of interest.
a. To determine if there is a link between high voltage power lines and cancer
in children, researchers examined the rate of cancer for 100 children living
within 1 __
4 mile of power lines and the rate of cancer for 100 children living
more than 1 __
4 mile from power lines.
b. 70 calculus students are randomly divided into 2 classes, one in which
a graphing calculator is used at all times and one in which a graphing
calculator is never used. The department chair wants to determine if
students’ calculus grades are higher if they can always use a calculator.
c. A medical researcher wants to determine if children who watch more than
3 hours of TV per day are more obese than children who watch 3 hours
or less. She gathers data from 15 different pediatricians’ offices where
doctors are tracking children’s weight and TV habits.
d. A researcher wants to know if children who watch more than 3 hours of TV
per day are more obese than children who watch 3 hours or less. He sends
out a survey to 500 children in the city asking them how much TV they
watch and how much they weigh.
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Problem 2 Designing a SurveyWhen designing a survey, ask yourself the following questions:
l What is the population of interest?
l How will the sample be randomly selected so that it is representative of
the population?
l Will the question being asked on the survey answer the question
accurately?
1. Consider the survey example from Problem 1, Question 1 about texting while
driving.
a. Augie identified the population as all young people. Sandy identified the
population as all young people who have a driver’s license. Who is correct?
Explain.
b. What types of different young people should be represented in the sample
to avoid bias?
c. In a survey, it is important not to leave part of the question for the survey
taker to determine. What phrasing in the example might need more clarity?
2. A politician conducts a survey to determine what percent of city residents
support a tax increase to provide additional funds for public schools.
a. What is the population of interest?
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b. How could a simple random sample be selected?
c. What is the characteristic of interest?
d. List one survey question that might be biased.
e. List one survey question that might be unbiased.
3. A parents’ group organizes a survey to learn how much money per month
teenagers receive as an allowance.
a. What is the population of interest?
b. Who should the sample include in order to avoid bias?
c. What is the characteristic of interest?
d. List one survey question that might be biased.
e. List one survey question that might be unbiased.
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Problem 3 Designing an Observational Study
When seeking to gather information or data, even for an observational study, there
may be factors and situations that might be overlooked which distort the final
results. This is known as confounding. Confounding occurs when there are other
possible reasons for the results to have occurred that were not identified prior to
the study.
When designing an observational study, ask yourself the following questions:
l What is the population of interest?
l How will the sample be identified and is it representative of the population?
l How will confounding be avoided or addressed?
1. Consider the observational study example from Problem 1, Question 2 about
the amount of TV shown in day care programs.
a. Lezlee identified the population as young children. Dave identified the
population as directors of day care centers. Who is correct? Explain.
b. Does the sample represent the population?
c. Under what circumstances could confounding have occurred? Explain.
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2. To determine if there is a link between high voltage power lines and cancer
in children, researchers examined the rate of cancer for 100 children living
within 1 __
4 mile of power lines and the rate of cancer for 100 children living more
than 1 __
4 mile from power lines.
a. What is the population of interest?
b. What is the sample?
c. Under what circumstances could confounding have occurred? Explain.
Problem 4 Designing an ExperimentWhen designing an experiment, ask yourself the following questions:
l What is the population of interest?
l What are the treatments in the experiment?
l How will the experimental units be identified and how will they be
randomly assigned to a treatment?
l How will the differences in treatments be analyzed and interpreted in order
to draw a conclusion?
1. Consider the experiment example from Problem 1, Question 3 about the new
asthma drug.
a. Dawson identified the population as people with asthma. Matt identified
the population as people with asthma who took the new drug. Who is
correct? Explain.
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b. What are the treatments in the experiment?
c. How will the experimental units be identified and how will they be randomly
assigned to a treatment?
d. How will the differences in treatments be analyzed and interpreted in order
to draw a conclusion?
2. A teachers’ union conducts an experiment to determine if using a graphing
calculator regularly in an algebra class results in higher grades.
a. What is the question being addressed in the experiment?
b. What is the population of interest?
c. What are the treatments in the experiment?
d. How will the experimental units be identified and how will they be randomly
assigned to a treatment?
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3. How will the differences in treatments be analyzed and interpreted in order to
draw a conclusion?
Be prepared to share your methods and solutions.
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Lesson 16.3 l Designing and Collecting Data Using a Survey, Study, or Experiment 737
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Problem 1 Designing a Survey, Study, or Experiment
In this lesson, you will use what you have learned about collecting data to design
and implement a survey, study, or experiment to answer a question that you are
curious about. For example, some questions might include:
l Do females spend more time on homework than males?
l Do teachers call on students in the front of the classroom more than
students in the back of the classroom?
l What effect does playing soft music have on students’ test scores?
In the next lesson, you will analyze and interpret your results and draw conclusions
about the question you asked.
A scoring guide for designing and implementing your survey, study, or experiment
is included on the last page of this lesson.
1. What questions would you like your survey, study, or experiment to answer?
Be as specific as possible.
16.3 Do It YourselfDesigning and Collecting Data Using a Survey, Study, or Experiment
ObjectivesIn this lesson you will:
l Design a sample survey, observational study, or experiment to answer a question.
l Identify potential sources of bias in a sample survey, observational study,
or experiment.
l Randomly select a sample for a sample survey, observational study,
or experiment.
l Collect data using a sample survey, observational study, or experiment.
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2. Is your question best answered by a survey, study, or experiment? Explain
why the method that you identified is most appropriate.
a. a sample survey
b. an observational study
c. an experiment
3. What is your population of interest?
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4. How will you randomly select your sample?
5. What is your characteristic of interest?
6. Are there any potential sources of bias?
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7. Describe the design of your survey, study, or experiment. Remember to
consider the key questions discussed in the previous lesson.
For a sample survey:
l What is the population of interest?
l How will the sample be randomly selected so that it is representative of
the population?
l Will the question being asked on the survey answer the question
accurately?
For an observational study:
l What is the population of interest?
l How will the sample be identified and is it representative of the population?
l How will confounding be avoided or addressed?
For an experiment:
l What is the population of interest?
l What are the treatments in the experiment?
l How will the experimental units be identified and how will they be
randomly assigned to a treatment?
l How will the differences in treatments be analyzed and interpreted in order
to draw a conclusion?
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Problem 2 Conducting a Survey, Study, or Experiment
1. Collect the data for the survey, study, or experiment you designed in
Problem 1, Question 7.
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2. Identify any problems or issues that you encountered when collecting the
data. For example:
l For a survey, did any of the questions on your survey cause problems
for people to answer? If so, you will need to report this as possible
response bias.
l For a study, was there any possible confounding or other issues
encountered in your study? If so, you will need to report this.
l For an experiment, did you encounter any issues with your treatments or
random assignment in your experiment? If so, you will need to report this.
3. Were you able to collect data from the random sample you identified in
Problem 1? If not, what problems arose with your sample? How many
declined or were unable to participate?
4. Was the wording of the question you asked confusing or capable of multiple
interpretations? If so, what caused confusion?
Be prepared to share your methods and solutions.
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Scoring Guide
I. Designing a survey, study, or experiment14
points
l Selects a question of interest that can be answered from a
survey, study, or experiment.2
l Selects a question that can be answered by collecting
quantitative data.2
l Identifi es population of interest. 2
l Identifi es characteristic being studied. 2
l Describes method for choosing random sample. 2
l Addresses potential sources of bias. 2
l Addresses design principles for a survey, study, or experiment. 2
II. Conducting a survey or experiment20
points
l Collects data using the survey, study, or experiment. 10
l Collects data from the random sample using the method described. 10
III. Summarizing the results of the survey, study, or experiment40
points
Numerically analyzing the results of the survey or experiment—25 points
l Correctly calculates measures of central tendency—mean
and median.10
l Correctly calculates measures of variation—standard deviation,
quartiles, and IQR.15
Graphically displaying the results of the survey or experiment—15 points
l Correctly displays the results in either a dot plot, a histogram,
or stem-and-leaf plot (includes correct scales and appropriate labels).5
l Correctly displays the results in a box-and-whisker plot
(includes correct scales and appropriate labels).5
l Identifi es the shape of the graphs as symmetric, skewed left, or
skewed right.5
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IV. Interpreting the results of the survey, study, or experiment20
points
l Interprets the mean and median in the context of the question that
was asked. 5
l Describes which measure of central tendency would be a better
measure to describe the center of the data and justifi es reason(s) why.5
l Interprets the standard deviation, quartiles, and IQR in the context
of the question that was asked. 5
l Describes which measure of dispersion would be a better measure to
describe the spread of the data and justifi es reason(s) why.5
V. Drawing conclusions about the results of the survey, study, or experiment
6 points
l Writes a conclusion about the question that was studied based on
the results of the survey, study, or experiment and uses information
from the graphical displays and numerical analysis as evidence for
the conclusion.
6
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Problem 1 Analyzing Data GraphicallyIn the previous lesson, you designed a survey, study, or experiment in order
to answer a question of interest to you. In this lesson, you will numerically and
graphically analyze and interpret the data collected from your survey, study, or
experiment to draw conclusions about your question.
The scoring guide for designing, implementing, and analyzing your survey, study, or
experiment is included on the last page of the previous lesson.
Graphs provide a way to visually analyze data and determine how it is distributed.
l Dot plots (or line plots) are used to display discrete data and provide a
method of organizing data from the smallest value to the largest value.
Ages of students in a college English class
18 19 20 21 22 23 24 25
16.4 Numbers, Graphs, and ConclusionsInterpreting Results and Drawing Conclusions from a Survey, Study, or Experiment
ObjectivesIn this lesson you will:
l Analyze the results of a sample survey, observational study, or
experiment graphically.
l Analyze the results of a sample survey, observational study, or
experiment numerically.
l Interpret the results of a sample survey, observational study, or experiment.
l Draw conclusions from the results of a sample survey, observational study,
or experiment.
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l Histograms are used to display data grouped in intervals. A histogram
does not display each data point but does show how many data points
are in each interval.
2
4
6
8
10
12
14
0 – 4.9 5 – 9.9 10 – 14.9 15 – 19.9Number of Hours
Hours spent on homework weekly
Num
ber
of
Peo
ple
l Stem-and-leaf plots are used to organize quantitative data and can be
used with discrete or continuous data. Stem-and-leaf plots are helpful to
determine the median and quartiles.
0
1
2
3
2
0
2
1
4
4
2
2
4 = 24 days2
5
5
3
4
8
4
4
9
7
Average Number of Days of Snow
l Box-and-whisker plots are used to divide a set of data into 4 equal groups
by identifying the median and the first and third quartiles so that the
spread, or variation, in our data can be seen.
0 10 20Average Number of Days of Snow
30 40
1. Create a visual display of your data using each of the following graphs.
a. Dot plot
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b. Histogram
c. Stem-and-leaf plot
d. Box-and-whisker plot
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2. Describe the shape of the distribution of your data.
Problem 2 Analyzing Data NumericallyData are often analyzed using measures of central tendency and measures of
dispersion or spread.
A measure of central tendency is a single number that best represents
a set of data.
l The mean, or average, is the sum of the data values in a set divided by
the number of data values in the set.
l The median is the middle number in a set of data that is listed in numerical
order.
l The mode is the data value that occurs most frequently.
1. Determine the mean, median, and mode for your data set.
2. Which measure of central tendency best describes your data set? Explain.
The measures of central tendency are useful to describe the symmetry of data.
l When the mean is equal to the median, the data is symmetric.
meanmedian
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l When the mean is less than the median, the data is skewed to the left.
mean median
l When the mean is greater than the median, the data is skewed to the right.
meanmedian
3. Describe the symmetry of your data set.
A measure of dispersion is a number that describes how the data are spread about
its center.
l The standard deviation is a number that describes how the data are
spread about the mean.
l Quartiles are numbers that divide the data into 4 groups of equal size.
4. Determine the standard deviation of your data set.
5. Determine the quartiles of your data set.
6. Which measure of dispersion best describes your data set? Explain.
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Problem 3 Interpreting the Results and Drawing Conclusions
1. How do the measures of central tendency help answer the question in your
survey, study, or experiment?
2. How do the measures of dispersion help answer the question in your survey,
study, or experiment?
3. How do the graphs help answer the question in your survey, study, or
experiment?
4. Write a paragraph describing any conclusions that you were able to draw
about the question in your survey, study, or experiment. Include references to
your numerical and graphical analysis.
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5. If you had the opportunity to perform your survey, study, or experiment again,
what would you have done differently? Explain.
Be prepared to share your methods and solutions.
752 Chapter 16 l Distributions of Discrete Data
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