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TRANSCRIPT
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
2 - 1
Describing DataDescribing DataDescribing DataDescribing Datarequency requency DistributionsDistributionsf
Graphic PresentationsGraphic Presentations
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2 - 2
Organize raw data into frequency distribution
Produce a histogram, a frequency polygon, and a cumulative frequency polygon from quantitative data
Develop and interpret a stem-and-leaf display
When you have completed this chapter, you will be able to:
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2 - 3
Present qualitative data using such graphical techniques such as a clustered bar chart, a stacked bar chart, and a pie chart
Detect graphic deceptions and use a graph to present data with clarity, precision, and efficiency
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2 - 4
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2 - 5
A Frequency Distribution is a grouping of data into
non-overlapping classes (mutually exclusive)…
showing the number of observations
in each category or
class.
The range of categories includes all values in the data set (collectively exhaustive classes).
The range of categories includes all values in the data set (collectively exhaustive classes).
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2 - 6
Class Midpoint or Class Mark:A point that divides a class into two equal parts, i.e. the
average of the upper and lower class limits.
12.5
Class frequency:The number of observations in each class.
Class interval:The class interval is obtained by subtracting the lower limit of
a class from the lower limit of the next class, e.g.
517.522.527.532.5
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2 - 7
Dr. Tillman is Dean of the School of Business. He wishes to prepare a report showing the number of hours per week students spend studying.
He selects a random sample of 30 students and determines the number of hours
each student studied last week.
15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.
Organize the data into a frequency distribution.
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2 - 8
Decide how many classes you wish to use.
Frequency Distributions
by hand
Frequency Distributions
by hand
Determine the class width.
There are five steps that can be used to
Construct a Frequency Distribution:
Set up the individual class limits.
Tally the items into the classes.
Count the number of items in each class.
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2 - 9Decide how many classes you wish to use
Use the 2 to the K rule.
Choose k so that 2 raised to the power of k is greater than the number of data points (n) or 30.
Rule of Thumb:
For most data sets, you would want
between 3 and 12 classes!
Rule of Thumb:
For most data sets, you would want
between 3 and 12 classes!
2k = 30 students25 = 32, so use k = about 5 classes
In this case…
In this case…
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2 - 10Determine the class width
Generally, the class width should be the same size for all classes.
Class width >= Max - Min K
(33.8 – 10.3)/ 5 = 4.7
Therefore, use class size of 5 hours
Therefore, use class size of 5 hours
15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.
15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.
10.3,33.8,
Max Min
K=5K=5
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2 - 11
Minimum Value is 10.3, therefore,
classes should start at 10 hours
Minimum Value is 10.3, therefore,
classes should start at 10 hours
10.0 – 14.915.0 – 19.920.0 – 24.925.0 – 29.930.0 – 34.9
10.0 – 14.915.0 – 19.920.0 – 24.925.0 – 29.930.0 – 34.9
Lower class limits
will be: 10, 15, 20, etc.
Lower class limits
will be: 10, 15, 20, etc.
Classes oror10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35
10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35
Classes
Set up the individual class limits
15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.
15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.
10.3,33.8,
Class Width 5 hoursClass Width 5 hours
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2 - 12
15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.
15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.
Tally the items into the classes
10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35
10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35
Classes TallyTally
…and so on with the remaining
hours
10.3,
13.514.213.7
14.0
12.9
12.9FindFind
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2 - 13Count the number of items in each class
10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35
10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35
Hours Studying x Frequency f
7
12
7
3
1
30
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2 - 14
Using different limitsUsing different limits
7.5 to under 12.512.5 to under 17.517.5 to under 22.522.5 to under 27.527.5 to under 32.532.5 to under 37.5
7.5 to under 12.512.5 to under 17.517.5 to under 22.522.5 to under 27.527.5 to under 32.532.5 to under 37.5
Hours Studying x Frequency f
11210
11
305
…will give you a different distribution, e.g.
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2 - 15Construct a Frequency Distribution
Using Excel
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2 - 16
Click on MegaStat
SeeSee
Click on Frequency
Distributions
Click on Frequency
Distributions
See…See…
Using
Click on Quantitative
Click on Quantitative
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2 - 17Using
SeeSee
INPUT NEEDS INPUT NEEDS
$A:$A
5
10
See…See…
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2 - 18
SeeSee
Using
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2 - 19
Relative Frequency Distribution
Relative Frequency Distribution
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2 - 20
RRelativeelative FFrequency requency DDistributionistributionRRelativeelative FFrequency requency DDistributionistribution
…shows the percent of observations in each class!
Hours Studying x f7
12
7
3
1
Relative f
3030
10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35
10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35
TotalTotal
7/30 = 0.2333
12/30 = 0.40
7/30 = 0.2333
3/30 = 0.10
1/30 = 0.0333
30/30 =130/30 =1
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2 - 21
Using different limitsUsing different limits
7.5 to under 12.512.5 to under 17.517.5 to under 22.522.5 to under 27.527.5 to under 32.532.5 to under 37.5
7.5 to under 12.512.5 to under 17.517.5 to under 22.522.5 to under 27.527.5 to under 32.532.5 to under 37.5
Hours Studying x f Relative f
3030TotalTotal
1/30 = 0.0333 12/30 = 0.40 10/30 = 0.3333
1/30 = 0.0333 1/30 = 0.0333
30/30 =130/30 =1
11210
11
5 5/30 = 0.1666
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2 - 22
Stem-and-leaf DisplaysStem-and-leaf DisplaysStem-and-leaf DisplaysStem-and-leaf Displays
Each numerical value is divided into two parts:
1. the leading digits become the stem and
2. the trailing digits become the leaf.
…an advantage of the stem-and-leaf display over a frequency distribution is
that we retain the value of each observation!
…an advantage of the stem-and-leaf display over a frequency distribution is
that we retain the value of each observation!
A statistical technique for displaying a set of data.
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2 - 23
A student achieved the following scores on the twelve accounting
quizzes this semester: 86, 79, 92, 84, 69, 88, 91,
83, 96, 78, 82, 85.
Construct a stem-and-leaf chart to illustrate the results.
Stem-and-leaf DisplaysStem-and-leaf DisplaysStem-and-leaf DisplaysStem-and-leaf Displays
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2 - 24
Stem-and-leaf DisplaysStem-and-leaf DisplaysStem-and-leaf DisplaysStem-and-leaf Displays
First, find the lowest score
86, 79, 92, 84, 69, 88, 91, 83, 96, 78, 82, 85.
Now list the next scores with the highest leading digits.
You should now have the following STEMS:
69, 78, 82, 916 7 8 9
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2 - 25
Stem
69
78
82
91
7
8
9
6
Split Leaf
6 9
7 8
8 2
9 1
Now, list the remaining ‘leaf’ scores!
Now, list the remaining ‘leaf’ scores!
9
3 4 5 8
2 6
6
All 12 Scores
Stem-and-leaf DisplaysStem-and-leaf DisplaysStem-and-leaf DisplaysStem-and-leaf Displays
86, 79, 92, 84, 69, 88,
91, 83, 96, 78, 82, 85.
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2 - 26
The grades on a statistics exam for a sample of 40 students are as
follows:
The grades on a statistics exam for a sample of 40 students are as
follows:Stem Leaf
3 6 8
4 1 2 7 8
5 0 1 2 5 5 8 9
6 0 1 1 1 2 5 7 8 8 8 9
7 0 0 2 5 6 6 7
8 4 6 8 8 9
9 0 2 4 6
Stem Leaf
3 6 8
4 1 2 7 8
5 0 1 2 5 5 8 9
6 0 1 1 1 2 5 7 8 8 8 9
7 0 0 2 5 6 6 7
8 4 6 8 8 9
9 0 2 4 6
How many students
earned an A on this test?
How many students
earned an A on this test?
55
What is the most common
letter grade earned?
What is the most common
letter grade earned?
FF
A+ = 90%-100%
A = 80%-89%
B+ = 75%-79%
B = 70%-74%
C+ = 65%-69%
C = 60%-64%
D = 55%-59%
F = 0%-54%
A+ = 90%-100%
A = 80%-89%
B+ = 75%-79%
B = 70%-74%
C+ = 65%-69%
C = 60%-64%
D = 55%-59%
F = 0%-54%
Alpha-NumericGrading
Alpha-NumericGrading
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2 - 27
Graphic Graphic Presentation of a Presentation of a
Frequency Frequency DistributionDistribution
Graphic Graphic Presentation of a Presentation of a
Frequency Frequency DistributionDistribution
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2 - 28Graphic Presentation of a Graphic Presentation of a Frequency DistributionFrequency Distribution
Graphic Presentation of a Graphic Presentation of a Frequency DistributionFrequency Distribution
The three commonly used graphic forms are:
Histograms
Frequency Polygons or Line Charts
Cumulative Frequency Distributions
The three commonly used graphic forms are:
Histograms
Frequency Polygons or Line Charts
Cumulative Frequency Distributions
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2 - 29
The class frequencies are represented by the heights of the bars and
the bars are drawn adjacent to each other.
A Histogram is a graph in which the classes are marked on the horizontal axis
and the class frequencies on the
vertical axis
Fre
quen
cyClass
Graphic Presentation of a Graphic Presentation of a Frequency DistributionFrequency Distribution
Graphic Presentation of a Graphic Presentation of a Frequency DistributionFrequency Distribution
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2 - 30
10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35
10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35
Hours Studying x f
712731
Graphic Presentation of a Graphic Presentation of a Frequency DistributionFrequency Distribution
Graphic Presentation of a Graphic Presentation of a Frequency DistributionFrequency Distribution
0 10 15 20 25 30 35 Hours spent studying
14
12
10
8
6
4
2
Fre
quen
cy
HistogramHistogram
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2 - 31
A frequency polygon consists of
line segments connecting the points formed by the class midpoint and
the class frequency.
A frequency polygon consists of
line segments connecting the points formed by the class midpoint and
the class frequency.0
2
4
6
8
10
12
14
7.5 12.5 17.5 22.5 27.5
0
5
10
15
20
25
30
35
10 15 20 25 30 35
A cumulative frequency distribution is used to determine
how many or what proportion of the data
values are below or above a certain
value.
A cumulative frequency distribution is used to determine
how many or what proportion of the data
values are below or above a certain
value.
Graphic Presentation of a Graphic Presentation of a Frequency DistributionFrequency Distribution
Graphic Presentation of a Graphic Presentation of a Frequency DistributionFrequency Distribution
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2 - 32
Making a Making a Histogram Histogram
in Excel in Excel
Making a Making a Histogram Histogram
in Excel in Excel
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2 - 33Using
Click on DATA ANALYSIS
SeeSee
Click on HISTOGRAM
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2 - 34
The upper limits of the classes you have determined
The upper limits of the classes you have determined
Using
Complete INPUTTING of DATAComplete INPUTTING of DATA
must now be entered from Column B (Excel calls these “bins”)
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2 - 35
To remove the Legend on the right side… Right mouse click and Click on Clear
Using
To remove the spaces between the bars… Right mouse click on one of the bars and
Click on Format Data Series
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2 - 36Using
Now, Click on the Options tab;
To reduce/remove the spaces between the barsAdjust the Gap width down to 0 and Click on OK.
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2 - 37Using
Edit the size of the histogram, titles, etc
as appropriate.
Note that the upper limit values are included in each class – this explains the difference between this
Excel Frequency Distribution
and the one we did by hand.
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2 - 38
0 10 15 20 25 30 35 Hours spent studying
14
12
10
8
6
4
2F
requ
ency
Frequency Polygon Frequency Polygon or or Line Chart Line Chart for Hours Spent Studyingfor Hours Spent Studying
Frequency Polygon Frequency Polygon or or Line Chart Line Chart for Hours Spent Studyingfor Hours Spent Studying
0 10 15 20 25 30 35 Hours spent studying
14
12
10
8
6
4
2F
requ
ency10.0 to under 15
15.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35
10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35
Hours Studying x f
712731
Notice that the class midpoints (the plotted points) aren’t as “user friendly” in this distribution choice.
Notice that the class midpoints (the plotted points) aren’t as “user friendly” in this distribution choice.
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2 - 39
10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35
10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35
Hours Studying x f
712731
Cumulative Frequency Distribution Cumulative Frequency Distribution For Hours StudyingFor Hours Studying
Cumulative Frequency Distribution Cumulative Frequency Distribution For Hours StudyingFor Hours Studying
under 15under 20under 25under 30under 35
under 15under 20under 25under 30under 35
Hours StudyingCumulative
f
19262930
7
Graph…..Graph…..
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2 - 40Cumulative Frequency Distribution Cumulative Frequency Distribution For Hours StudyingFor Hours Studying
Cumulative Frequency Distribution Cumulative Frequency Distribution For Hours StudyingFor Hours Studying
0 10 15 20 25 30 35 Hours spent studying
35
30
25
20
15
10
5
Fre
quen
cy
under 15under 20under 25under 30under 35
under 15under 20under 25under 30under 35
19262930
7
Hours StudyingHours Studying
Cumulative f
Cumulative f
Notice that the limits are the plotted points.Notice that the limits are the plotted points.
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2 - 41
Pie Bar Line
Pie Bar Line… used primarily for Qualitative Data
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2 - 42
…is useful for displaying a
Relative Frequency Distribution
PiePie
A circle is divided proportionally to the
relative frequency and portions of the circle are allocated for the
different groups.
A circle is divided proportionally to the
relative frequency and portions of the circle are allocated for the
different groups.
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2 - 43
PiePie
200 runners were asked to indicate their favourite type of running
shoe. TypeType
Nike 92Adidas 49Reebok 37Asics 13Other 9
# of runners selecting:# of runners selecting:
Draw a pie chart based on this information.Draw a pie chart based on this information.
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2 - 44
Nike 92Adidas 49Reebok 37Asics 13Other 9
TypeType ##
200200
%46.024.518.56.54.5
100
Adidas 24.5% Nike46.0%
Reebok
18.5%
Asics6.5%
Other
4.5%
Relative Frequency Distribution for the running shoes
PiePie
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2 - 45
Nike 92Adidas 49Reebok 37Asics 13Other 9
TypeType ##
200200
%46.024.518.56.54.5
100
Using Excel, follow the steps in the Chart Wizard to construct a Pie Chart!
Using Excel, follow the steps in the Chart Wizard to construct a Pie Chart!
PiePie
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2 - 46BarBar
…can be used to depict any of the levels of measurement (nominal, ordinal, interval, or ratio).
(also known as a ‘column chart’)
Examples of…
3-D
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2 - 47BarBar
Use bar charts also when the order
in which qualitative data are
presented is meaningful.
Use bar charts also when the order
in which qualitative data are
presented is meaningful.
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2 - 48
How could we chart this data?
BarBar
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2 - 49BarBar
Using Excel we can produce this…
Using Excel we can produce this…
Other formats…Other formats…
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2 - 50
Employment Rate
Canadian City
Victoria 57.7
Halifax 60.5
Montreal 60.4
Sherbrooke 59.2
Quebec 59.7
Toronto 65.1
Hamilton 63.2
Kitchener 66.0
London 63.3
Thunder Bay 61.0
Regina 67.4
Saskatoon 63.7
Edmonton 67.1
Vancouver 61.4
Winnipeg 66.7
BarBar
Halifax
Montreal
Sherbro
oke
Quebec
Toronto
Hamilt
on
Kitchener
London
Thunder Bay
Regina
Saskatoon
Win
nipeg
Edmonto
n
Vancouver
Victoria
% e
mp
loym
ent
52
54
56
58
60
62
64
66
68
70
Employment Rate in Canadian Cities
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2 - 51
Employment Rate
Canadian City
Victoria 57.7
Halifax 60.5
Montreal 60.4
Sherbrooke 59.2
Quebec 59.7
Toronto 65.1
Hamilton 63.2
Kitchener 66.0
London 63.3
Thunder Bay 61.0
Regina 67.4
Saskatoon 63.7
Edmonton 67.1
Vancouver 61.4
Winnipeg 66.7
BarBar
Employment Rate in Canadian Cities
% e
mp
loym
ent
52
54
56
58
60
62
64
66
68
70
Halifax
Montreal
Sherbro
oke
Quebec
Toronto
Hamilt
on
Kitchener
London
Thunder Bay
Regina
Saskatoon
Win
nipeg
Edmonto
n
Vancouver
Victoria
- by Province
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2 - 52BarBar
Did any of the previous Bar Charts adequately display
all the information that was provided?
Did any of the previous Bar Charts adequately display
all the information that was provided?
The following has been modified from that data found by Statistics Canada.
Does it do an effective job of displaying the StatCan data?
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2 - 53
Real estate and rental and leasingProfessional, scientific and technical servicesManagement of companies and enterprisesEducational services (private sector)
Health care and social assistance (private sector)Administration and support, waste management and
remediation servicesArts, entertainment and recreationAccommodation and food services
All private sector
Information and cultural industriesFinance and insurance
ManufacturingWholesale trade
Retail trade
0
20
40
60
80
100
% o
f e
nte
rpri
ses
Clustered Bar
Comparison of Internet Use in 2000 and 2001Comparison of Internet Use in 2000 and 2001
% of enterprises thatuse the Internet 2000
% of enterprises thatuse the Internet 2001
% of enterprises with aWeb site 2000
% of enterprises with aWeb site 2001
Data Source: Statistics Canada
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2 - 54
Full-Time University Faculty By Gender,Canada and Jurisdictions, 1987-88 and 1997-98
Stacked Bar
Canadian Full Time University Faculty
020406080
100120
1987-88 1997-98
% o
f T
ota
l
% males
% females
Data Source: Statistics Canada
Total
34,651 33,925 12,829 13,910 12,650 12,095 9,172 7,817
Full Professor Associate Professor
1987-88 1997-98 1987-88 1997-98 1987-88 1997-98
Other
1987-88 1997-98
% Male% Female 17
832575
793
1387
1783
2872
3268
4456
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2 - 55
Make sure that your charts are
not overly cluttered
Make sure that your charts are
not overly cluttered
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2 - 56
There are four typical shape characteristics
Shapesof
Histograms Modal
Class
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2 - 57
…a balanced effect!
Both ‘balanced’ or ‘have symmetry’
Both ‘balanced’ or ‘have symmetry’
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2 - 58
… occurs when the observations are graphed as being skewed or tilted more to one side of the centre
of the observations than the other.
… occurs when the observations are graphed as being skewed or tilted more to one side of the centre
of the observations than the other.
The skewness, if on the right side is said to be
‘positive’.
The skewness, if on the left side is said to be
‘negative’.
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2 - 59
ClassModal
A modal class is the one with the largest number of observations
A modal class is the one with the largest number of observations
This is a uniModal HistogrambiModal
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2 - 60
ClassModal
biModalbiModal
This is a biModal Histogram
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2 - 61
Population distributions are often bell shaped. Drawing a histogram helps verify the shape
of the population in question.
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2 - 62LineLine
Line charts are particularly useful when the trend over time is to be
emphasizedExamples …
3-D In combination
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
2 - 63
Time PlotTime PlotTime PlotTime Plot
LineLine
OSAJJMAMFJDNOSAJJMAMFJDNOSAJJMAMFJ
8.5
7.5
6.5
5.5Mo n th
M o n th ly S te e l P ro d u c tio n
Mil
lion
s of
Ton
s
2000 2001 2002
M o n t h l y S t e e l P r o d u c t i o n
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
2 - 64
Employment Rate in Canadian Cities
52
54
56
58
60
62
64
66
68
70%
em
plo
ymen
t
Halifax
Montreal
Sherbro
oke
Quebec
Toronto
Hamilt
on
Kitchener
London
Thunder Bay
Regina
Saskatoon
Win
nipeg
Edmonto
n
Vancouver
Victoria
LineLine
Preparing a Line Chart for this type of data is not overly useful!Preparing a Line Chart for this type of data is not overly useful!
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
2 - 65
Employment Rate in Canadian Cities
52
54
56
58
60
62
64
66
68
70%
em
plo
ymen
t
Halifax
Montreal
Sherbro
oke
Quebec
Toronto
Hamilt
on
Kitchener
London
Thunder Bay
Regina
Saskatoon
Win
nipeg
Edmonto
n
Vancouver
Victoria
LineLine
Is this combination any better for displaying the data?Is this combination any better for displaying the data?
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
2 - 66
ffrequency requency PolygonPolygon and and OgiveOgiveffrequency requency PolygonPolygon and and OgiveOgive
frequency Polygon
50403020100
0.3
0.2
0.1
0.0
Re
lati
ve
Fre
qu
enc
y
Sales
Ogive
50403020100
1.0
0.5
0.0
Cu
mu
lati
ve R
ela
tiv
e F
req
ue
ncy
Sales
LineLine
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
2 - 67
Test your learning…Test your learning…
www.mcgrawhill.ca/college/lindClick on…Click on…
Online Learning Centrefor quizzes
extra contentdata setssearchable glossaryaccess to Statistics Canada’s E-Stat data…and much more!
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2 - 68
This completes Chapter 2