10.2 Displaying Data Part I - Dr. Travers Page of Math
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§ 10.2 Displaying Data Part I
Types of Variables
1 Categorical
2 QuantitativeTypes of Variables
1 Categorical
2 Quantitative
Definition
A categorical variable is a variable that can take on one of a limited,
and usually fixed, number of possible values.Types of Variables
1 Categorical
2 Quantitative
Definition
A categorical variable is a variable that can take on one of a limited,
and usually fixed, number of possible values.
Definition
A quantitative variable is naturally measured as a number for which
meaningful arithmetic operations make sense.Pictographs Definition A pictograph is a form of a bar graph where the bars are graphics that each represent a predetermined quantity. Generally these are graphs used to display categorical variables.
Pictographs
What’s Good
Gives good visual and can easily be adapted to multiple graphicsPictographs
What’s Good
Gives good visual and can easily be adapted to multiple graphics
Easy for children to understand since it includes graphicsPictographs
What’s Good
Gives good visual and can easily be adapted to multiple graphics
Easy for children to understand since it includes graphics
What’s Not Good
Tedious to drawPictographs
What’s Good
Gives good visual and can easily be adapted to multiple graphics
Easy for children to understand since it includes graphics
What’s Not Good
Tedious to draw
Fractions of the scale amount can be difficult to determine
exactly which amount was intendedDot Plots Definition A dot plot is the representation of a set of data over a number line. The number of dots over a number represents the relative quantity of the value.
Dot Plots
Definition
A dot plot is the representation of a set of data over a number line.
The number of dots over a number represents the relative quantity of
the value.
Example
The following are the test scores for a particular high school student
in their math class over the course of an academic year.
64 73 85 74 83
71 56 83 76 85
83 87 92 84 95
92 95 92 91Dot Plots
Grades of a High School Student
3 • •
Frequency
2 •• ••
1 • • •••• •••• •••
50 60 70 80 90 100
GradesDot Plots
What’s Good
Gives a good idea of distributionDot Plots
What’s Good
Gives a good idea of distribution
Preserves all of the data pointsDot Plots
What’s Good
Gives a good idea of distribution
Preserves all of the data points
What’s Not Good
Tedious to plotDot Plots
What’s Good
Gives a good idea of distribution
Preserves all of the data points
What’s Not Good
Tedious to plot
Can be hard to readDot Plots
What’s Good
Gives a good idea of distribution
Preserves all of the data points
What’s Not Good
Tedious to plot
Can be hard to read
Not practical for large data setsDistributions Definition A distribution is a representation of data vs. frequency. It shows all possible values and how often they occur.
Distributions
Definition
A distribution is a representation of data vs. frequency. It shows all
possible values and how often they occur.
Range: Highest value - lowest value
Here, the range would be 95 − 56 = 39.Distributions
Definition
A distribution is a representation of data vs. frequency. It shows all
possible values and how often they occur.
Range: Highest value - lowest value
Here, the range would be 95 − 56 = 39.
Center: The central value(s) is the center. It could be a value or a
class, depending on the type of graph.
Here, the center is the 10th value, since there are 19 data points in
the set. The value we seek is 84.Distributions
Definition
A distribution is a representation of data vs. frequency. It shows all
possible values and how often they occur.
Range: Highest value - lowest value
Here, the range would be 95 − 56 = 39.
Center: The central value(s) is the center. It could be a value or a
class, depending on the type of graph.
Here, the center is the 10th value, since there are 19 data points in
the set. The value we seek is 84.
Shape: How many peaks are there? Is it roughly in the middle or
to one side?
Here we have one peak, so we would say the distribution is
unimodal. That peak is to the right, so the tail stretches out to the
left. We would say this graph is left skewed.Distributions
Definition
A distribution is a representation of data vs. frequency. It shows all
possible values and how often they occur.
Range: Highest value - lowest value
Here, the range would be 95 − 56 = 39.
Center: The central value(s) is the center. It could be a value or a
class, depending on the type of graph.
Here, the center is the 10th value, since there are 19 data points in
the set. The value we seek is 84.
Shape: How many peaks are there? Is it roughly in the middle or
to one side?
Here we have one peak, so we would say the distribution is
unimodal. That peak is to the right, so the tail stretches out to the
left. We would say this graph is left skewed.
We must ALWAYS have axes labeled and have a title for our graphs.Stem-and-Leaf Plots
Similarities to Dot Plots
Gives idea of distributionStem-and-Leaf Plots
Similarities to Dot Plots
Gives idea of distribution
Preserves dataStem-and-Leaf Plots
Similarities to Dot Plots
Gives idea of distribution
Preserves data
Not practical for large data setsStem-and-Leaf Plots
Similarities to Dot Plots
Gives idea of distribution
Preserves data
Not practical for large data sets
Differences from Dot Plots
Used for quantitative variablesStem-and-Leaf Plots
Similarities to Dot Plots
Gives idea of distribution
Preserves data
Not practical for large data sets
Differences from Dot Plots
Used for quantitative variables
Easier to read actual data elementsStem-and-Leaf Plots
Similarities to Dot Plots
Gives idea of distribution
Preserves data
Not practical for large data sets
Differences from Dot Plots
Used for quantitative variables
Easier to read actual data elements
Can be used for comparisons of two data setsStem-and-Leaf Plots Example Using the same data set as we did for the dot plot, construct a stem-and-leaf plot.
Stem-and-Leaf Plots
Example
Using the same data set as we did for the dot plot, construct a
stem-and-leaf plot.
First thing we need to do is order the data elements.
56 64 71 73 74
76 82 83 83 83
84 85 85 87 91
92 92 95 95Stem-and-Leaf Plots
Example
Using the same data set as we did for the dot plot, construct a
stem-and-leaf plot.
First thing we need to do is order the data elements.
56 64 71 73 74
76 82 83 83 83
84 85 85 87 91
92 92 95 95
Grades for a High School Student
5
6
7
8
9Stem-and-Leaf Plots
This is the spine with the stemsStem-and-Leaf Plots
This is the spine with the stems
The leaves are the last digitStem-and-Leaf Plots
This is the spine with the stems
The leaves are the last digit
List in increasing order from spineStem-and-Leaf Plots
This is the spine with the stems
The leaves are the last digit
List in increasing order from spine
No commasStem-and-Leaf Plots
This is the spine with the stems
The leaves are the last digit
List in increasing order from spine
No commas
Repetition is important
Grades for a High School Student
5 6
6 4
7 1 3 4 6
8 2 3 3 3 4 5 5 7
9 1 2 2 5 5Analysis of Stem-and-Leaf Plot
Grades for a High School Student
5 6
6 4
7 1 3 4 6
8 2 3 3 3 4 5 5 7
9 1 2 2 5 5
Same range as dot plotAnalysis of Stem-and-Leaf Plot
Grades for a High School Student
5 6
6 4
7 1 3 4 6
8 2 3 3 3 4 5 5 7
9 1 2 2 5 5
Same range as dot plot
Same center as dot plot although we would only give classAnalysis of Stem-and-Leaf Plot
Grades for a High School Student
5 6
6 4
7 1 3 4 6
8 2 3 3 3 4 5 5 7
9 1 2 2 5 5
Same range as dot plot
Same center as dot plot although we would only give class
Shape is unimodal and skewed leftAnalysis of Stem-and-Leaf Plot
Grades for a High School Student
5 6
6 4
7 1 3 4 6
8 2 3 3 3 4 5 5 7
9 1 2 2 5 5
Same range as dot plot
Same center as dot plot although we would only give class
Shape is unimodal and skewed left
Notice that the values on the right are essentially in columns - this is
what allows us to quickly see which classes have more elements.Stem-and-Leaf Plots
What if we had a 3 digit number? Suppose the student got a 100 on
the next exam?
Grades for a High School Student
5 6
6 4
7 1 3 4 6
8 2 3 3 3 4 5 5 7
9 1 2 2 5 5
10 0Back-to-Back Stem-and-Leaf Plots Example Suppose we wanted to compare the careers of Babe Ruth and Mark McGwire in terms of their yearly home run totals to determine which player was the more consistent long ball hitter. Make a back-to-back stem-and-leaf plot to make the is determination. Ruth: 54, 59, 35, 41, 46, 25, 47, 60, 54, 46, 49, 46, 41, 34, 22 McGwire: 49, 32, 33, 39, 22, 42, 9, 9, 39, 52, 58, 70, 65, 32, 29
Back-to-Back Stem-and-Leaf Plots
Ruth v. McGwire
0
1
2
3
4
5
6
7
We set up the graph with one set of data increasing out to the right
and the other increasing out to the left. This way we have a
side-by-side comparison of the data sets.Back-to-Back Stem-and-Leaf Plots
Ruth v. McGwire
0 9 9
1
5 2 2 2 9
5 4 3 2 2 3 9 9
9 7 6 6 6 1 1 4 2 9
9 4 4 5 2 8
0 6 5
7 0
Who is more consistent and why?Histograms
Used for quantitative variablesHistograms
Used for quantitative variables
Tracks frequency and shows distributionHistograms
Used for quantitative variables
Tracks frequency and shows distribution
Does not preserve individual valuesHistograms
Used for quantitative variables
Tracks frequency and shows distribution
Does not preserve individual values
Good for a large number of valuesHistograms
Used for quantitative variables
Tracks frequency and shows distribution
Does not preserve individual values
Good for a large number of values
Bars must be vertical and must touchHistograms
Example
For our test scores example, construct a histogram and analyze the
distribution.
It is easier if the values are in order as we will be grouping them into
classes.
56 64 71 73 74
76 82 83 83 83
84 85 85 87 91
92 92 95 95Histograms We first want to create a frequency table. This is a collection of non-overlapping classes and the frequency of observation in each of those classes. We need to determine the following in this order:
Histograms
We first want to create a frequency table. This is a collection of
non-overlapping classes and the frequency of observation in each of
those classes. We need to determine the following in this order:
Number of classes
The rule of thumb with the number of classes is to use the square root
of the number of observations in the data set.
√
19 ≈ 4.36
So, we can use 4 or 5 classes. I tend to go up to the next integer to be
sure I have enough classes. So we will use 5 for our graph.Histograms
Size of each class
We want them to be the same width so that the taller classes will be
known to have the most elements. If not then we have to find the area
of each rectangle to determine relative size.
To find the size, we divide the ‘range’ by the number of classes.
95 − 56 + 1 40
size = = =8
5 5
So we will use 8 for the class size.Histograms
Size of each class
We want them to be the same width so that the taller classes will be
known to have the most elements. If not then we have to find the area
of each rectangle to determine relative size.
To find the size, we divide the ‘range’ by the number of classes.
95 − 56 + 1 40
size = = =8
5 5
So we will use 8 for the class size.
If this came out to be a decimal, we can use the decimal value for the
class size or we can round here too. Remember, this is an
approximation tool, so as long as we are using values close to those
we calculate, we will have an accurate representation to analyze.Histograms
Endpoints of each class
We start the smallest class with a left endpoint of 56, since that was
our minimum. Then, to find the next left endpoint, add 8 to 56.
Continue in this manner until we have 5 classes.
Grade Range Frequency
56-
64-
72-
80-
88-Histograms
Then, we subtract 1 from each left endpoint to find the right endpoint
of the previous class.
Grade Range Frequency
56-63
64-71
72-79
80-87
88-95Histograms
Then, we subtract 1 from each left endpoint to find the right endpoint
of the previous class.
Grade Range Frequency
56-63
64-71
72-79
80-87
88-95
Finally, we count how many elements go in each class.
Grade Range Frequency
56-63 1
64-71 2
72-79 3
80-87 8
88-95 5Histograms
Grades of a High School Student
8
Frequency 6
4
2 56-63
64-71
72-79
80-87
88-95
Grades
We see the same range and shape. Here, we’d have no choice but to
give the class only for the center as we would lose the ability to see
individual values.Histograms
Example
The EPA lists most sports cars in its “two-seater” category. The table
below gives the city mileage in miles per gallon. Make and analyze a
histogram for the the city mileage.
Model Mileage Model Mileage
Acura NSX 17 Insight 57
Audi Quattro 20 S2000 20
Audi Roadster 22 Lamborghini 9
BMW M Coupe 17 Mazda 22
BMW Z3 Coupe 19 SL500 16
BMW Z3 Roadster 20 SL600 13
BMW Z8 13 SLK230 23
Corvette 18 SLK 320 20
Prowler 18 911 15
Ferrari 360 11 Boxster 19
Thunderbird 17 MR2 25Histograms
√
There are 22 cars, so we would use 4 < 22 < 5 classes, so here I
will choose 5. The size of each class would be
57 − 9 + 1 49
= = 9.8
5 5
So we will use 10.Histograms
√
There are 22 cars, so we would use 4 < 22 < 5 classes, so here I
will choose 5. The size of each class would be
57 − 9 + 1 49
= = 9.8
5 5
So we will use 10.
Mileage Frequency
9 -18 11
19-28 10
29-38 0
39-48 0
49-58 1Histograms
MPG for Sports Cars
12
Frequency 9
6
3
19-28
29-38
39-48
49-58
9-18
MPG
Center:Histograms
MPG for Sports Cars
12
Frequency 9
6
3
19-28
29-38
39-48
49-58
9-18
MPG
Center: Boundary between the first two classes
Range:Histograms
MPG for Sports Cars
12
Frequency 9
6
3
19-28
29-38
39-48
49-58
9-18
MPG
Center: Boundary between the first two classes
Range: 58 − 9 = 49
ShapeHistograms
MPG for Sports Cars
12
Frequency 9
6
3
19-28
29-38
39-48
49-58
9-18
MPG
Center: Boundary between the first two classes
Range: 58 − 9 = 49
Shape Unimodal, skewed rightBar Graphs
These look similar to histograms but there are a few differences.
Generally used for categorical variablesBar Graphs
These look similar to histograms but there are a few differences.
Generally used for categorical variables
Bars can be vertical or horizontalBar Graphs
These look similar to histograms but there are a few differences.
Generally used for categorical variables
Bars can be vertical or horizontal
Cannot analyze distribution like histogram because the order of
the classes is not necessarily in numerical orderBar Graphs
These look similar to histograms but there are a few differences.
Generally used for categorical variables
Bars can be vertical or horizontal
Cannot analyze distribution like histogram because the order of
the classes is not necessarily in numerical order
Can be used for comparisonsBar Graphs
Example
The growth of the US population age 65 and over is given in the table.
Create a bar graph to represent this data.
1900 4.1 1970 9.8
1910 4.3 1980 11.3
1920 4.7 1990 12.5
1930 5.5 2000 12.4
1940 6.9 2010 13.2
1950 8.1 2020 16.5
1960 9.2 2030 20.0Percent
5
10
15
20
Bar Graphs
1900
1910
1920
1930
1940
1950
Year
1960
1970
1980
Age of Seniors by Decade
1990
2000
2010
2020
2030Bar Graphs for Comparisons
Example
Create a bar graph for the given causes of death and analyze the
results. Values given are the number per 100,000 people.
Cause of Death 1970 1980 1990 2000
Cardiovascular 640 509 387 318
Cancer 199 208 216 201
Accidents 62 46 36 34Bar Graphs for Comparisons
Causes of Death
600
Number of Deaths
Legend
(per 100,000)
450
Cardiovascular
300 Cancer
Accidents
150
1970
1980
1990
2000
YearBar Graphs for Comparisons
Causes of Death
600
Number of Deaths
Legend
(per 100,000)
450
Cardiovascular
300 Cancer
Accidents
150
1970
1980
1990
2000
Year
Cancer and accidents are roughly the same in each decadeBar Graphs for Comparisons
Causes of Death
600
Number of Deaths
Legend
(per 100,000)
450
Cardiovascular
300 Cancer
Accidents
150
1970
1980
1990
2000
Year
Cancer and accidents are roughly the same in each decade
Cardiovascular disease decreases each decade and is
approaching level of cancer deathsPie Charts Pie charts are used to compare different categories in relation to each other. It is only good for one type of variable; we cannot do comparisons between different types of observations with one pie chart.
Pie Charts
Pie charts are used to compare different categories in relation to each
other. It is only good for one type of variable; we cannot do
comparisons between different types of observations with one pie
chart.
Example
You sit on an overpass and record the color of the first 100 cars you
see. The results are as follows:
color frequency
red 15
blue 21
green 18
white 22
black 19
other 5
Construct a pie chart to illustrate the relationship between the colors
of these cars.Pie Charts We have to make sure that the size of each slice is correct in relation to the other slices. To do this, we make sure the central angle is the correct size. Since we have 100 observations, the number of observations is the percent of the circle we need for that wedge. For example, for the red cars, we saw 15 of them, so we would need a central angle of .15(360◦ ) = 54◦ .
Pie Charts
Blue
21% Red
Green
15%
18%
5%
Other
22% 19%
White BlackPie Charts
Example
The following is a breakdown of the solid waste that made up
America’s garbage in 2000. Values given represent millions of tons.
Material Weight
Food 25.9
Glass 12.8
Metal 18.0
Paper 86.7
Plastics 24.7
Rubber 15.8
Wood 12.7
Yard Trimmings 27.7
Other 7.5
Create a pie chart to represent this data.Pie Charts
We can’t make a pie chart with this data; at least not yet. What do we
need?
Material Weight Relative Frequency
Food 25.9 11.2 %
Glass 12.8 5.5%
Metal 18.0 7.8%
Paper 86.7 37.4%
Plastics 24.7 10.7%
Rubber 15.8 6.8%
Wood 12.7 5.5%
Yard Trimmings 27.7 11.9%
Other 7.5 3.2%
231.9Pie Charts
Now we can find the central angles and create our pie chart.
Material Weight Relative Frequency Central Angle
Food 25.9 11.2% 40.3◦
Glass 12.8 5.5% 19.8◦
Metal 18.0 7.8% 28.1◦
Paper 86.7 37.4% 134.6◦
Plastics 24.7 10.7% 38.5◦
Rubber 15.8 6.8% 24.5◦
Wood 12.7 5.5% 19.8◦
Yard Trimmings 27.7 11.9% 42.8◦
Other 7.5 3.2% 11.5◦Pie Charts
Metal
Glass
7%
Paper 6%
Food
37% 11%
3%
Other
12%
11% Trimmings
7% 6%
Plastics Wood
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