Construct and Draw Inferences Circle Graphs
Copyright 2005 Kendall/Hunt Publishing Company statistics
17. Which of these data sets below has the greatest negative correlation? a.
c.
$ 35 30 25 20 15 10 5 0 1965
$ 40 35 30 25 20 15 10 5 0 1965
• •
• • •• • • •
1970
1975
91
•
1980
1985
•
• • •
1990
•
1995
2000
b.
•
•
1970
•
• • • • • • • •
1975
1980
1985
1990
•
• • 1995
2000
d. $
$ 40 35 30 25 20 15 10 5 0 1965
• • • • • • • • • • • • • •
1970
1975
1980
1985
1990
1995
2000
40 35 30 25 20 15 10 5 0 1965
• • • •
1970
1975
• • •
1980
• • • • • • • 1985
1990
1995
2000
Construct and Draw Inferences Constructing and drawing inferences are essential to critical thinking and problem solving. When faced with statements, problems and puzzles, we do more than use common sense. We use problem solving skills to try to find patterns and infer statements that follow logically from the statements given. We determine what is reasonable and what is not. We determine what should logically follow and what should not in order to make good decisions.
Circle Graphs Taken directly from newspaper headlines: Should a juvenile be tried as an adult? To address this issue, we should ask ourselves many questions and look at this crucial problem from many perspectives. For many of us, the first question we ask may be “Do juveniles who murder pose a chronic problem in this country?” Well, what is chronic? If a large percentage of all murders were done by juveniles over a period of years, this could be called chronic. 12.5% We return to the Crime Index as defined by the FBI or 45º from 2001. Let us ask the question, “Does there exist 25% or 90º a correlation between age and those who commit murder in this country?” As long as we have the information grouped by category, which in this case is by age, we can recognize large numbers displayed in data as a percent of the whole in a pie chart or circle graph. First, let us see how a circle graph or pie chart is made. 50% or 180º We tend to subdivide a circle into sectors represented by their central angle in either degrees (out of 360 degrees) or the percent of the circle that is to be shaded (out of 100%).
AILM-c03-083-124.indd 91
6.25% or 22.5º 3.125%—or 11.25º
7/21/05 5:41:27 PM
Copyright 2005 Kendall/Hunt Publishing Company 92
statistics
So, for our question: “Is there a correlation between age and those who commit murder in this country?”, we examine the data taken from the Crime Index. Of the 10,113 known murderers in the country in 2001, their age distribution was given as follows: Age, in Years
Number
1 to 4
0
5 to 8
0
9 to 12
14
13 to 16
454
17 to 19
1,695
20 to 24
2,767
25 to 29
1,571
30 to 34
992
35 to 39
855
40 to 44
645
45 to 49
455
50 to 54
272
55 to 59
158
60 to 64
85
65 to 69
59
70 to 74
37
75 and over
54
Total
10,113
Since the data is already organized, let us find the density of each age group. This means we will reconstruct the table and find the percent of murderers for each category, 1 to 4, 5 to 8, 9 to 12, 13 to 16 and so on. Note, not all categories are partitioned into equal time intervals. Age, in Years
Number
Relative Frequency
1 to 4
0
0
5 to 8
0
0
14
14 = 0.0013 _____
9 to 12
AILM-c03-083-124.indd 92
10,113
454 _____ 10,113
= 0.0449
Central Angle
0.0013 × 360° = 0.0468° 0.0449 × 360° = 16.2°
13 to 16
454
17 to 19
1,695
1,695 _____ = 0.1676
0.1676 × 360° = 60.34°, or 1/6 of the circle
20 to 24
2,767
2,767 _____ = 0.2736
0.2736 × 360° = 98.5°
25 to 29
1,571
0.115
41.4 degrees
30 to 34
992
0.098
35.3 degrees
35 to 39
855
0.085
30.4 degrees
40 to 44
645
0.064
23 degrees
45 to 49
455
0.045
16.2 degrees
50 to 54
272
0.027
9.7 degrees
10,113 10,113
7/21/05 5:41:28 PM
Copyright 2005 Kendall/Hunt Publishing Company statistics
55 to 59
158
0.016
5.62 degrees
60 to 64
85
0.008
3 degrees
65 to 69
59
0.005
2.1 degrees
70 to 74
37
0.004
1.3 degrees
75 and over
54
0.005
2 degrees
Total
10,113
1
360 degrees, a whole circle
93
The pie chart below is illuminating. Very quickly, by glancing at the chart, we can tell that 20 to 24 year olds commit the most murders, but a close second seems to be 17 to 19 year olds, as well as 25 to 29 year olds. If a juvenile is defined to be under 18 years of age, then this appears to be a chronic problem because the second most dense population of murderers occurs in the age group 17 to 19 year olds. Now when we factor in the 13 to 16 year olds (454), the problem of juvenile murder seems to be more acute. For teenager murderers alone, we have within the 13 to 19 year old age group accounted for 454 + 1,695 or 2,149 murders committed by teenagers. This comes to 2,149/10,113 or just a little over 20 percent, and this does not include the children who are 12 or under. Murder Offenders by Age, 2001 1 to 4 5 to 8 9 to 12 13 to 16 a
a
17 to 19 b b
20 to 24 c 25 to 29 d 30 to 34 35 to 39 40 to 44
c d
45 to 49 50 to 54 55 to 59 60 to 64 65 to 69 70 to 74 75 and over
AILM-c03-083-124.indd 93
7/21/05 5:41:28 PM
Copyright 2005 Kendall/Hunt Publishing Company 94
statistics
Now, let us continue to address this problem. Numbers never lie. But rearranged, could they deceive? Could the very same numbers be used by the opposing side of the argument to make the opposing view more viable? As said, first, we rearrange the numbers. 1 to 19 14 + 454 + 1,695 = 2,163 20 to 39 2,767 + 1,571 + 992 + 855 = 6,185 40 to 59 645 + 455 + 272 + 158 = 1,530 60 and over 85 + 59 + 37 + 54 = 235 We then construct a pie chart from these new subdivisions. Again, keep in mind we are only considering the cases where we know the age of the murderer. There were 10,113 of these murders. Murderers by Age, 2001
1 to 19 20 to 39 40 to 59 60 and over
But we are trying to represent the opposing point of view and we are trying to show murder by juveniles is not a “chronic problem.” So, in 2001, there were an additional 5,375 murders where the age of the perpetrator was unknown. Regrouping, our table looks like: 1 to 19 2,163 20 to 39 6,185 40 to 59 1,530 60 and over 235 Unknown 5,375 Let us examine the new pie chart. Notice how much smaller the piece of the pie for the 1 to 19 year old segment now is compared to the whole. This is a significant difference from the previous pie charts where we did not factor in the murders committed by people of unknown ages. Murders by Age of Offender, 2001
1 to 19 20 to 39 40 to 59 60 and over unknown
AILM-c03-083-124.indd 94
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Copyright 2005 Kendall/Hunt Publishing Company statistics
95
To further enhance our Murders by Age of Offender, 2001 argument, we may construct the slices of the pie with a 3dimensional representation. 1 to 19 We then shift the angle of the 20 to 39 segment of the pie we are try40 to 59 ing to ostensibly hide so that 60 and over it is less prominent. Our point that juvenile crime is not a unknown chronic problem seems more justified to the viewer’s eye. To add a final touch in enhancing our argument, let us re-categorize and change two groupings: 1 to 19 and 20 to 29 to 1 to 16 and 17 to 39. If we keep the category of unknown murderers in the groupings, let us compare the original pie chart with the final one. To the naked eye, a quick glance reveals the juveniles’ slice to be a mere sliver on the left chart compared to nearly a quarter of the pie on the right. Murders by Age of Offender, 2001
Murders by Age of Offender, 2001
1 to 19
1 to 19
20 to 39
20 to 39
40 to 59
40 to 59
60 and over
60 and over unknown
Statistics do not lie, they can be re-arranged though to show whatever is on one’s agenda.
Example Two The graph below is shown and a TV anchorman states, “There was a sharp dramatic increase in drunk driving convictions between the year 1999 and the year 2000.” Consider the statement and reply to its accuracy. 732 730 728 726 724 722 720 1999
2000
Drunk Driving Convictions
AILM-c03-083-124.indd 95
7/21/05 5:41:29 PM
17. Which of these data sets below has the greatest negative correlation? a.
c.
$ 35 30 25 20 15 10 5 0 1965
$ 40 35 30 25 20 15 10 5 0 1965
• •
• • •• • • •
1970
1975
91
•
1980
1985
•
• • •
1990
•
1995
2000
b.
•
•
1970
•
• • • • • • • •
1975
1980
1985
1990
•
• • 1995
2000
d. $
$ 40 35 30 25 20 15 10 5 0 1965
• • • • • • • • • • • • • •
1970
1975
1980
1985
1990
1995
2000
40 35 30 25 20 15 10 5 0 1965
• • • •
1970
1975
• • •
1980
• • • • • • • 1985
1990
1995
2000
Construct and Draw Inferences Constructing and drawing inferences are essential to critical thinking and problem solving. When faced with statements, problems and puzzles, we do more than use common sense. We use problem solving skills to try to find patterns and infer statements that follow logically from the statements given. We determine what is reasonable and what is not. We determine what should logically follow and what should not in order to make good decisions.
Circle Graphs Taken directly from newspaper headlines: Should a juvenile be tried as an adult? To address this issue, we should ask ourselves many questions and look at this crucial problem from many perspectives. For many of us, the first question we ask may be “Do juveniles who murder pose a chronic problem in this country?” Well, what is chronic? If a large percentage of all murders were done by juveniles over a period of years, this could be called chronic. 12.5% We return to the Crime Index as defined by the FBI or 45º from 2001. Let us ask the question, “Does there exist 25% or 90º a correlation between age and those who commit murder in this country?” As long as we have the information grouped by category, which in this case is by age, we can recognize large numbers displayed in data as a percent of the whole in a pie chart or circle graph. First, let us see how a circle graph or pie chart is made. 50% or 180º We tend to subdivide a circle into sectors represented by their central angle in either degrees (out of 360 degrees) or the percent of the circle that is to be shaded (out of 100%).
AILM-c03-083-124.indd 91
6.25% or 22.5º 3.125%—or 11.25º
7/21/05 5:41:27 PM
Copyright 2005 Kendall/Hunt Publishing Company 92
statistics
So, for our question: “Is there a correlation between age and those who commit murder in this country?”, we examine the data taken from the Crime Index. Of the 10,113 known murderers in the country in 2001, their age distribution was given as follows: Age, in Years
Number
1 to 4
0
5 to 8
0
9 to 12
14
13 to 16
454
17 to 19
1,695
20 to 24
2,767
25 to 29
1,571
30 to 34
992
35 to 39
855
40 to 44
645
45 to 49
455
50 to 54
272
55 to 59
158
60 to 64
85
65 to 69
59
70 to 74
37
75 and over
54
Total
10,113
Since the data is already organized, let us find the density of each age group. This means we will reconstruct the table and find the percent of murderers for each category, 1 to 4, 5 to 8, 9 to 12, 13 to 16 and so on. Note, not all categories are partitioned into equal time intervals. Age, in Years
Number
Relative Frequency
1 to 4
0
0
5 to 8
0
0
14
14 = 0.0013 _____
9 to 12
AILM-c03-083-124.indd 92
10,113
454 _____ 10,113
= 0.0449
Central Angle
0.0013 × 360° = 0.0468° 0.0449 × 360° = 16.2°
13 to 16
454
17 to 19
1,695
1,695 _____ = 0.1676
0.1676 × 360° = 60.34°, or 1/6 of the circle
20 to 24
2,767
2,767 _____ = 0.2736
0.2736 × 360° = 98.5°
25 to 29
1,571
0.115
41.4 degrees
30 to 34
992
0.098
35.3 degrees
35 to 39
855
0.085
30.4 degrees
40 to 44
645
0.064
23 degrees
45 to 49
455
0.045
16.2 degrees
50 to 54
272
0.027
9.7 degrees
10,113 10,113
7/21/05 5:41:28 PM
Copyright 2005 Kendall/Hunt Publishing Company statistics
55 to 59
158
0.016
5.62 degrees
60 to 64
85
0.008
3 degrees
65 to 69
59
0.005
2.1 degrees
70 to 74
37
0.004
1.3 degrees
75 and over
54
0.005
2 degrees
Total
10,113
1
360 degrees, a whole circle
93
The pie chart below is illuminating. Very quickly, by glancing at the chart, we can tell that 20 to 24 year olds commit the most murders, but a close second seems to be 17 to 19 year olds, as well as 25 to 29 year olds. If a juvenile is defined to be under 18 years of age, then this appears to be a chronic problem because the second most dense population of murderers occurs in the age group 17 to 19 year olds. Now when we factor in the 13 to 16 year olds (454), the problem of juvenile murder seems to be more acute. For teenager murderers alone, we have within the 13 to 19 year old age group accounted for 454 + 1,695 or 2,149 murders committed by teenagers. This comes to 2,149/10,113 or just a little over 20 percent, and this does not include the children who are 12 or under. Murder Offenders by Age, 2001 1 to 4 5 to 8 9 to 12 13 to 16 a
a
17 to 19 b b
20 to 24 c 25 to 29 d 30 to 34 35 to 39 40 to 44
c d
45 to 49 50 to 54 55 to 59 60 to 64 65 to 69 70 to 74 75 and over
AILM-c03-083-124.indd 93
7/21/05 5:41:28 PM
Copyright 2005 Kendall/Hunt Publishing Company 94
statistics
Now, let us continue to address this problem. Numbers never lie. But rearranged, could they deceive? Could the very same numbers be used by the opposing side of the argument to make the opposing view more viable? As said, first, we rearrange the numbers. 1 to 19 14 + 454 + 1,695 = 2,163 20 to 39 2,767 + 1,571 + 992 + 855 = 6,185 40 to 59 645 + 455 + 272 + 158 = 1,530 60 and over 85 + 59 + 37 + 54 = 235 We then construct a pie chart from these new subdivisions. Again, keep in mind we are only considering the cases where we know the age of the murderer. There were 10,113 of these murders. Murderers by Age, 2001
1 to 19 20 to 39 40 to 59 60 and over
But we are trying to represent the opposing point of view and we are trying to show murder by juveniles is not a “chronic problem.” So, in 2001, there were an additional 5,375 murders where the age of the perpetrator was unknown. Regrouping, our table looks like: 1 to 19 2,163 20 to 39 6,185 40 to 59 1,530 60 and over 235 Unknown 5,375 Let us examine the new pie chart. Notice how much smaller the piece of the pie for the 1 to 19 year old segment now is compared to the whole. This is a significant difference from the previous pie charts where we did not factor in the murders committed by people of unknown ages. Murders by Age of Offender, 2001
1 to 19 20 to 39 40 to 59 60 and over unknown
AILM-c03-083-124.indd 94
7/21/05 5:41:28 PM
Copyright 2005 Kendall/Hunt Publishing Company statistics
95
To further enhance our Murders by Age of Offender, 2001 argument, we may construct the slices of the pie with a 3dimensional representation. 1 to 19 We then shift the angle of the 20 to 39 segment of the pie we are try40 to 59 ing to ostensibly hide so that 60 and over it is less prominent. Our point that juvenile crime is not a unknown chronic problem seems more justified to the viewer’s eye. To add a final touch in enhancing our argument, let us re-categorize and change two groupings: 1 to 19 and 20 to 29 to 1 to 16 and 17 to 39. If we keep the category of unknown murderers in the groupings, let us compare the original pie chart with the final one. To the naked eye, a quick glance reveals the juveniles’ slice to be a mere sliver on the left chart compared to nearly a quarter of the pie on the right. Murders by Age of Offender, 2001
Murders by Age of Offender, 2001
1 to 19
1 to 19
20 to 39
20 to 39
40 to 59
40 to 59
60 and over
60 and over unknown
Statistics do not lie, they can be re-arranged though to show whatever is on one’s agenda.
Example Two The graph below is shown and a TV anchorman states, “There was a sharp dramatic increase in drunk driving convictions between the year 1999 and the year 2000.” Consider the statement and reply to its accuracy. 732 730 728 726 724 722 720 1999
2000
Drunk Driving Convictions
AILM-c03-083-124.indd 95
7/21/05 5:41:29 PM
