© 2012 McGraw-Hill
Ryerson Limited
© 2009 McGraw-Hill Ryerson Limited
1
Lind
Marchal
Wathen
Waite
© 2012 McGraw-Hill Ryerson Limited
2
Learning Objectives
LO 1 Organize qualitative data into a frequency table.
LO 2 Present a data frequency table as a bar chart or a pie
chart.
LO 3 Organize qualitative data into a frequency distribution.
LO 4 Present a frequency distribution for quantitative data
using histograms, frequency polygons, and
cumulative frequency polygons.
LO 5 Develop and interpret a stem-and-leaf display.
© 2012 McGraw-Hill Ryerson Limited
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LO
1
CONSTRUCTING A FREQUENCY
DISTRIBUTION
© 2012 McGraw-Hill Ryerson Limited
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Frequency Table
The Frequency Table is a grouping of qualitative data into
mutually exclusive classes showing the number of
observations in each class.
The number of observations in each class is called the
class frequency.
Example:
Dwelling
Type
Number of
Listings
Apartment
58
House
26
Townhouse
14
Total
98
TABLE 2–1 Frequency Table for Halifax Real Estate Listings by Type
LO
1
© 2012 McGraw-Hill Ryerson Limited
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Relative Class Frequency
Class frequencies can be converted to relative class
frequencies to show the fraction of the total number of
observations in each class.
A relative frequency captures the relationship between a
class total and the total number of observations.
Type
Number of
Listings
Fraction
Relative
Frequency
Percent
Apartment
58
58/98
0.5918
59.18%
House
26
26/98
0.2653
26.53%
Townhouse
14
14/98
0.1429
14.29%
1.000
100.00%
98
TABLE 2–2 Relative Frequency Table of Halifax Real Estate Listings by Type
LO
1
© 2012 McGraw-Hill Ryerson Limited
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LO
2
GRAPHIC PRESENTATION OF
QUALITATIVE DATA
© 2012 McGraw-Hill Ryerson Limited
7
Bar Chart
A graph in which the classes are reported on the horizontal
axis and the class frequencies on the vertical axis. The
class frequencies are proportional to the heights of the
bars.
Number of Halifax Real Estate Listings by Type
70
Number of Listing
60
50
40
30
20
10
0
Apartment
House
Dwelling Type
Townhouse
CHART 2–1 Listings by Type from the Halifax Area.
LO
2
© 2012 McGraw-Hill Ryerson Limited
8
Bar Chart
The horizontal axis shows the variable of interest and the
vertical axis shows the amount, number or fraction of each
of the possible outcomes.
A distinguishing characteristic of a bar chart is that there is
a distance or gap between the bars.
Can depict any of the levels of measurement
• nominal
• ordinal
• interval
• ratio
LO
2
© 2012 McGraw-Hill Ryerson Limited
9
Bar Chart in Excel
Bar Chart : Income Based on Educational Level
LO
2
© 2012 McGraw-Hill Ryerson Limited
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Pie Chart
A chart that shows the proportion or percent that
each class represents of the total number of frequencies.
TABLE 2–3 OLG Lottery Proceeds
Use of Profits
Percent Share
(%)
Prizes
51.8
Province of
Ontario
30.3
Retailers
7.1
Operating
Expenses
8.4
Government of
Canada
2.4
100.0
LO
2
© 2012 McGraw-Hill Ryerson Limited
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Pie Chart
Each slice of the pie represents the relative share of the
each component.
OLC use of Profit
8.4
2.4
Prizes
7.1
Province of Ontario
Retailers
Operating Expenses
30.3
51.8
Government of
Canada
CHART 2–2 Pie Chart of OLG Use of Profits
LO
2
© 2012 McGraw-Hill Ryerson Limited
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Pie Chart in Excel
LO
2
© 2012 McGraw-Hill Ryerson Limited
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Example – Ease of Navigation on Ski Lodge
Website
A company is test marketing its new web site and is
interested in how easy its web page design is to navigate. It
randomly selected 200 regular Internet users and asked
them to perform a search task on the web page. Each
person was asked to rate the ease of navigation as poor,
good, excellent, or awesome. The results are shown in the
following table:
Awesome
100
Good
30
Excellent
60
Poor
10
1. What level of measurement is used for ease of navigation?
2. Draw a bar chart for the survey results.
3. Draw a pie chart for the survey results.
LO
2
© 2012 McGraw-Hill Ryerson Limited
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Solution - Ease of Navigation on Ski Lodge
Website
1. The data are measured on an ordinal scale. That is, the
scale ranks the ease of navigation from a low of “poor”
to a high of “awesome.” Also, the interval between each
rating is unknown so it is impossible, for example, to
conclude that a rating of good is twice the value of a
poor rating.
LO
2
© 2012 McGraw-Hill Ryerson Limited
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Solution - Ease of Navigation on Ski Lodge
Website
Frequency
Awesome
100
50%
Excellent
60
30%
Good
30
15%
Poor
10
5%
200
100%
Total
2.
120
Percent (%)
3.
Ease of Navigation on Ski Lodge
Website
Ease of Navigation on Ski Lodge
Website
Frequency
100
Poor
5%
80
Awesome
Good
15%
60
Excellent
Good
40
Excellent
30%
20
Awesome
50%
Poor
0
Awesome
LO
2
Excellent
Good
Poor
© 2012 McGraw-Hill Ryerson Limited
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You Try It Out!
A social welfare NGO decided to plant fruit trees near a city
park. Below are the number of each fruit tree to be planted.
(a)
(b)
(c)
(d)
LO
2
Is the data quantitative or qualitative? Why?
What is the table called?
Develop a bar chart to depict the information.
Develop a pie chart using the relative frequencies.
© 2012 McGraw-Hill Ryerson Limited
Fruit Plant
Number
Mango
45
Pineapple
36
Strawberry
57
Apple
62
Total
200
17
LO
3
CONSTRUCTING FREQUENCY
DISTRIBUTIONS: QUANTITATIVE
DATA
© 2012 McGraw-Hill Ryerson Limited
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Frequency distribution
A grouping of data into mutually exclusive classes showing
the number of observations in each class.
1.
2.
3.
4.
5.
LO
3
Decide on the number of classes
Determine the class interval or width
Set the individual class limits
Tally the list prices into the classes
Count the number of items in each class
© 2012 McGraw-Hill Ryerson Limited
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Example – Constructing Frequency
Distributions:
S&P/TSX composite index historical prices reports for 42
days from Sep Oct 11, 2011 to Dec 6, 2011 is given in the
below table.
What is the highest list volume? What is the lowest list
volume? Around what value do the list volumes tend to
cluster?
lowest
LO
3
highest
234,979,276
186,585,706
222,262,031
199,379,802
239,914,383
202,470,683
294,246,336
114,691,255
264,218,706
205,585,826
330,546,204
204,172,672
188,982,337
48,895,537
182,342,563
190,776,204
236,790,466
158,833,777
195,798,266
189,951,023
136,413,027
258,625,081
286,073,224
222,924,810
201,004,892
199,922,254
165,644,472
210,858,029
231,505,190
206,240,504
282,704,137
195,617,196
192,236,938
295,017,949
185,069,468
263,435,512
222,462,949
208,877,067
222,082,015
290,368,405
196,843,135
316,413,039
© 2012 McGraw-Hill Ryerson Limited
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Solution – Constructing Frequency Distributions:
Step 1: Decide on the number of classes. A useful recipe to
determine the number of classes (k) is the “2 to the
k rule.”
There were 42 days. So n = 42. If we try k = 5,
which means we would use 5 classes, then 25 = 32,
somewhat less than 42. Hence, 5 is not enough
classes. If we let k = 6, then 26= 64, which is
greater than 42. So the recommended number of
classes is 6.
LO
3
© 2012 McGraw-Hill Ryerson Limited
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Solution – Constructing Frequency Distributions:
Step 2: Determine the class interval or width. The classes
all taken together must cover at least the distance
from the lowest value in the raw data up to the
highest value.
Or i  H  L
k
Where:
i is the class interval
H is the highest observed value
L is the lowest observed value
K is the number of classes
LO
3
© 2012 McGraw-Hill Ryerson Limited
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Solution – Constructing Frequency Distributions:
($330,546,204- $48,895,537)/6 = $46941778
Round up to some convenient number, such as a
multiple of 10 or 100. Use a class width of
$50 000 000 as it will be easily understood.
Step 3: Set the individual class limits.
$0 to under $50 000 000
200 000 000 to under 250 000 000
50 000 000 to under 100 000 000
250 000 000 to under 300 000 000
100 000 000 to under 150 000 000 300 000 000 to under 350 000 000
150 000 000 to under 200 000 000
LO
3
© 2012 McGraw-Hill Ryerson Limited
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Solution – Constructing Frequency Distributions:
Step 4: Tally the list prices into the classes.
Class ($)
Tallies
0 to 50 000 000
50 000 000 to under 100 000 000
100 000 000 to under 150 000 000
150 000 000 to under 200 000 000
200 000 000 to under 250 000 000
250 000 000 to under 300 000 000
300 000 000 to under 350 000 000
LO
3
© 2012 McGraw-Hill Ryerson Limited
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Solution – Constructing Frequency Distributions:
Step 5: Count the number of items in each class.
List Volume
Frequency
0 to 50 000 000
1
50 000 000 to under 100 000 000
0
100 000 000 to under 150 000 000
2
150 000 000 to under 200 000 000
14
200 000 000 to under 250 000 000
15
250 000 000 to under 300 000 000
8
300 000 000 to under 350 000 000
2
Total
42
Frequency Distribution of List Volume
LO
3
© 2012 McGraw-Hill Ryerson Limited
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You Try It Out!
The profit earned, in dollars, for the first quarter of last year
by the 10 distributors of a refrigerator company in the city is
given below :
$2130
3250
2657
4000
3843
5000
3500
6500
5900
4567
(a) What are the values such as $2130 and $3250 called?
(b) Using $2000 up to $2500 as the first class, $2500up to $3000 as the
second class, and so forth, organize the quarterly commissions into a
frequency distribution.
(c) What are the numbers in the right column of your frequency
distribution called?
(d) Describe the distribution of quarterly commissions, based on the
frequency distribution.
LO
3
© 2012 McGraw-Hill Ryerson Limited
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Class Intervals and Class Midpoints
Class midpoint
Class interval
The midpoint is halfway
between the lower limits of
two consecutive classes.
To determine the class
interval, subtract the lower
limit of the class from the
lower limit of the next class.
It is computed by adding the
lower limits of consecutive
classes and dividing the
result by two.
LO
3
You can also determine the
class interval by finding the
difference between
consecutive midpoints.
© 2012 McGraw-Hill Ryerson Limited
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A Software Example
The following is a frequency distribution, produced by
MegaStat, showing the List Volume of S&P/TSX
composite index historical prices. (in Millions)
List Volume
Lower
LO
3
Cumulative
Upper
Midpoint
Width
Frequency
Percent
Frequency Percent
0
<
50
250
50
1
2.38%
1
2.38%
50
<
100
750
50
0
0.00%
1
2.38%
100
<
150
125
50
2
4.76%
3
7.14%
150
<
200
175
50
14
33.33%
17
40.48%
200
<
250
225
50
15
35.71%
32
76.19%
250
<
300
275
50
8
19.05%
40
95.24%
300
<
350
325
50
2
4.76%
42
100.00%
42
100.00%
© 2012 McGraw-Hill Ryerson Limited
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You Try It Out!
Jack had 83 customers in his book store last Sunday. The
customers spent between $43.50 and $450. Jack wants to
construct a frequency distribution of the amount spent by his
customers for that day.
(a) How many classes would you use?
(b) What class interval would you suggest?
(c) What actual classes would you suggest?
LO
3
© 2012 McGraw-Hill Ryerson Limited
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Relative Frequency Distribution
A relative frequency distribution converts the frequency to a
percent.
To convert a frequency distribution to a relative frequency
distribution, each of the class frequencies is divided by the
total number of observations.
LO
3
© 2012 McGraw-Hill Ryerson Limited
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Example - Relative Frequency Distribution
Find the relative frequency Distribution for the given data of
S&P/TSX composite index historical prices.
LO
3
© 2012 McGraw-Hill Ryerson Limited
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Solution - Relative Frequency Distribution
The relative frequency distribution is:
LO
3
List Volume
Frequency
Relative
Frequency
Found by
$0 to $50 000 000
1
0.024
1/42
50 000 000 to under 100 000 000
0
0
0/42
100 000 000 to under 150 000 000
2
0.048
2/42
150 000 000 to under 200 000 000
14
0.333
14/42
200 000 000 to under 250 000 000
15
0.357
15/42
250 000 000 to under 300 000 000
8
0.190
8/42
300 000 000 to under 350 000 000
2
0.048
2/42
Total
42
1.000
42/42
© 2012 McGraw-Hill Ryerson Limited
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You Try It Out!
Refer to the above table, which shows the relative frequency
for the List Volume of S&P/TSX composite index historical
prices.
(a) How many list volume were listed for $50 000 000 to under $10 000
000?
(b) What percent of list volume were listed for $200 000 000 to under
$250 000 000?
(c) What percent of the list volume were listed at $250 000 000 or more?
LO
3
© 2012 McGraw-Hill Ryerson Limited
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LO
4
GRAPHIC PRESENTATION OF A
FREQUENCY DISTRIBUTION
© 2012 McGraw-Hill Ryerson Limited
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Graphic Presentation of a Frequency Distribution
Three charts that will help portray a frequency distribution
graphically are :
1.Histogram
2.Frequency Polygon
3.Cumulative Frequency Polygon
LO
4
© 2012 McGraw-Hill Ryerson Limited
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Histogram
A bar graph in which the classes are marked on the
horizontal axis and the class frequencies on the vertical
axis. The class frequencies are represented by the heights
of the bars. The bars are drawn adjacent to each other.
LO
4
© 2012 McGraw-Hill Ryerson Limited
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Histogram Characteristics
Very similar to the bar chart showing the distribution of
qualitative data.
Classes are marked on the horizontal axis and the class
frequencies on the vertical axis.
Quantitative data is usually measured using scales that are
continuous, not discrete.
The horizontal axis represents all possible values, and the
bars are drawn adjacent to each other.
LO
4
© 2012 McGraw-Hill Ryerson Limited
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Example - Histogram
Construct a histogram for the frequency distribution given
below. What conclusions can you reach based on the
information presented in the histogram?
List Volume (in Million)
Midpoint
Frequency
0 to 50 000 000
250
1
50 000 000 to under 100 000 000
750
0
100 000 000 to under 150 000 000
1250
2
150 000 000 to under 200 000 000
1750
14
200 000 000 to under 250 000 000
2250
15
250 000 000 to under 300 000 000
2750
8
300 000 000 to under 350 000 000
3250
2
Total
LO
4
© 2012 McGraw-Hill Ryerson Limited
42
38
Solution - Histogram
Histogram of Frequency Distribution of S&P/TSX
composite index historical prices
16
15
14
14
List Volume
12
10
8
8
6
4
2
2
2
1
0
0
250
LO
4
750
1250
1750
Mid Point
2250
© 2012 McGraw-Hill Ryerson Limited
2750
3250
39
Solution - Histogram
Based on the above histogram, we conclude:
1. The lowest list volume is between $0 and under $50 000
000. The highest list volume is between $300 000 000
to under $350 000 000.
2. The largest class frequency is the $200 000 000 to
under $250 000 000 class. A total of 15 of the 42
volumes are within this volume range.
3. Eighty-two of the list volume, or 83.7 percent, had a list
volume between $0 and to under $800 000.
LO
4
© 2012 McGraw-Hill Ryerson Limited
40
Histogram In Excel
Follow the commands to create the frequency distribution.
A check automatically appears in the box to the left of the
word Histogram. This will produce a histogram with
percents on the Y-axis.
To change this default to frequencies:
1. Right click inside the plot area of the histogram. Choose
Source Data.
2. Click the Series tab. You need to change the range in
the Values: box.
3. In the Values: box, change the range from the Percent
column to the Frequency column. Click OK. Edit the
chart titles.
LO
4
© 2012 McGraw-Hill Ryerson Limited
41
Frequency Polygon Characteristics
Shows the shape of a distribution and is similar to a
histogram
Consists of line segments connecting the points formed by
the intersections of the class midpoints and the class
frequencies
The midpoint of each class is scaled on the X-axis and the
class frequencies on the Y-axis
To complete the frequency polygon, midpoints are added to
both ends of the X-axis to “anchor” the polygon at zero
frequencies
LO
4
© 2012 McGraw-Hill Ryerson Limited
42
Example – Frequency Polygon
Construct a frequency polygon for the frequency
distribution given below.
LO
4
List Volume ($ million)
Midpoint
Frequency
0 to 50
50 to under 100
100 to under 150
150 to under 200
200 to under 250
250 to under 300
300 to under 350
Total
250
750
1250
1750
2250
2750
3250
1
0
2
14
15
8
2
42
© 2012 McGraw-Hill Ryerson Limited
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Solution - Frequency Polygon
16
15
14
14
12
10
8
8
6
4
2
2
2
1
0
0
250
750
1250
1750
2250
2750
3250
Frequency Polygon of S&P/TSX composite index historical prices
LO
4
© 2012 McGraw-Hill Ryerson Limited
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Advantages
LO
Histogram
Frequency Polygon
Depicts each class as a
rectangle, with the height of
the rectangular bar
representing the number in
each class
Allows us to compare
directly two or more
frequency distributions by
constructing one on top of
the other
4
© 2012 McGraw-Hill Ryerson Limited
45
Frequency Polygon In Excel
To produce a frequency polygon using MegaStat, follow the
commands for the histogram, but select Polygon
LO
4
© 2012 McGraw-Hill Ryerson Limited
46
You Try It Out!
The result of an exam is shown in the following frequency
distribution.
Marks (% Percentage)
Number of Students
50 to under 55
15
55 to under 60
10
60 to under 65
16
65 to under 70
8
70 to under 75
9
(a) Portray the Marks as a histogram.
(b) Portray the Marks as a relative frequency polygon.
(c) Summarize the important facets of the distribution (such as classes
with the highest and lowest frequencies).
LO
4
© 2012 McGraw-Hill Ryerson Limited
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Cumulative Frequency Distribution
It is also called an ogive.
There are two types :
1. Less-than cumulative frequency distribution
2. More-than cumulative frequency distribution
LO
4
© 2012 McGraw-Hill Ryerson Limited
48
Example – Less-Than Cumulative Frequency
Distribution
The frequency distribution of the listings from S&P/TSX
composite index historical prices is given below. Construct
a less-than cumulative frequency polygon. Fifty percent of
the volumes were listed for less than what amount?
Twenty-five of the list volumes were less than what
amount?
List Volume
Frequency
$0 to $50 000 000
1
50 000 000 to under 10 000 000
0
10 000 000 to under 150 000 000
2
150 000 000 to under 200 000 000
14
200 000 000 to under 250 000 000
15
250 000 000 to under 300 000 000
8
300 000 000 to under 350 000 000
2
Total
LO
4
42
© 2012 McGraw-Hill Ryerson Limited
49
Solution – Less-Than Cumulative Frequency
Distribution
List Volume
($ million)
0 to 50
Frequency
Found by
1
Cumulative
frequency
1
50 to under 100
0
1
1+0
100 to under 150
2
3
1+0+2
150 to under 200
14
17
1+0+2+14
200 to under 250
15
32
1+0+2+14+15
250 to under 300
8
40
1+0+2+14+15+8
300 to under 350
2
42
1+0+2+14+15+8+2
Total
42
1
Less-than Cumulative Frequency Distribution of the listings from S&P/TSX composite
index historical prices.
LO
4
© 2012 McGraw-Hill Ryerson Limited
50
Solution – Less-Than Cumulative Frequency
Distribution
To begin the plotting, 1 listings were less than $5 000 000, so the first
point is X = 500 and Y = 1. The coordinates for the next point are X =
1000 and Y =3. The rest of the points are plotted as follows:
LO
4
List Price ($
million)
Cumulative
Frequency
Less than 50
1
50 to under 100
1
100 to under 150
3
150 to under 200
17
200 to under 250
32
250 to under 300
40
300 to under 350
42
© 2012 McGraw-Hill Ryerson Limited
51
Solution – Less-Than Cumulative Frequency
Distribution
Cumulative Frequency
Less-Than Cumulative Frequency Distribution
LO
4
100
90
80
70
60
50
40
30
20
10
0
40
42
32
17
1
Less
than
$500
1
500 to
under
1000
3
100 to
under
1500
1500 to 2000 to 2500 to 3000 to
under
under
under
under
2000
2500
3000
3500
List Volume
© 2012 McGraw-Hill Ryerson Limited
52
Example – More-Than Cumulative Frequency
Distribution
The frequency distribution of the listings from the S&P/TSX
composite index historical prices is repeated from below
table.
List Volume
Frequency
0 to 50 000 000
50 000 000 to under 100 000 000
100 000 000 to under 150 000 000
150 000 000 to under 200 000 000
200 000 000 to under 250 000 000
250 000 000 to under 300 000 000
300 000 000 to under 350 000 000
Total
1
0
2
14
15
8
2
42
Construct a more-than cumulative frequency polygon. Fifty
percent of the volumes were listed for less than what
amount? Twenty-five of the list volumes were less than
what amount?
LO
4
© 2012 McGraw-Hill Ryerson Limited
53
Solution – More-Than Cumulative Frequency
Distribution
To begin the plotting, 42 listings were 0 or more, so the first
point is X = 0 and Y = 42. The coordinates for the next
point are X = 500 and Y = 0. The rest of the points are
plotted as follows.
List Price
Cumulative
0 or more
50 000 000 or more
100 000 000 or more
150 000 000 or more
200 000 000 or more
250 000 000 or more
300 000 000 or more
350 000 000 or more
LO
4
© 2012 McGraw-Hill Ryerson Limited
Frequency
42
0
41
39
25
10
2
0
54
Solution – More-Than Cumulative Frequency
Distribution
More-Than Cumulative Frequency Distribution
100
Cumulative Frequency
90
80
70
60
50
40
42
25
20
10
0
0 or
more
4
39
30
10
LO
41
2
0
0
500 or 1000 or 1500 or 2000 or 2500 or 3000 or 3500 or
more
more
more
more
more
more
more
List Volume
© 2012 McGraw-Hill Ryerson Limited
55
Cumulative Frequency Polygon In Excel
To produce a less-than cumulative frequency polygon
(Ogive) using MegaStat, follow the commands for the
histogram, but select Ogive
LO
4
© 2012 McGraw-Hill Ryerson Limited
56
You Try It Out!
A sample of number of transactions performed per hour by
20 employees in a bank is given in following table
No. of Transactions per Hour
Number of Employees
4 to under 9
4
9 to under 13
7
13 to under 17
6
17 to under 21
3
(a) What is the table called?
(b) Develop a less-than and more-than cumulative frequency distribution
and portray the distribution in cumulative frequency polygons.
(c) Based on the cumulative frequency polygon, how many employees
performed10 transactions per hour or less? Half of the employees
performed how many transactions per hour? Four employees
performed how many transactions or less?
LO
4
© 2012 McGraw-Hill Ryerson Limited
57
LO
5
STEM-AND-LEAF DISPLAYS
© 2012 McGraw-Hill Ryerson Limited
58
Stem & Leaf Displays
A statistical technique to present a set of data
Each numerical value is divided into two parts
The leading digit(s) becomes the stem and the trailing digit
the leaf
• The stems are located along the vertical axis
• The leaf values are stacked against each other along
the horizontal axis
Advantage over a frequency distribution is:
• The identity of each observation is not lost
• The digits themselves give a picture of the distribution
• The cumulative frequencies are also reported
LO
5
© 2012 McGraw-Hill Ryerson Limited
59
Constructing a Stem & Leaf Display
Suppose the seven observations in a 90 up to 100 class
are: 96, 94, 93, 94, 95, 96, and 97.
The stem value is the leading digit or digits, in this case 9.
The leaves are the trailing digits.
The stem is placed to the left of a vertical line and the leaf
values to the right.
Finally, we sort the values within each stem from smallest
to largest.
9|6434567
9|3445667
LO
5
© 2012 McGraw-Hill Ryerson Limited
60
Example - Stem & Leaf Displays
Listed in the table below is the number of 30-second radio
advertising spots purchased by each of the 45 members of
the Greater Hilltown Automobile Dealers Association last
year. Organize the data into a stem-and-leaf display.
Around what values do the number of advertising spots
tend to cluster? What is the fewest number of spots
purchased by a dealer? The largest number purchased?
96
93
88
118
128
95
113
96
108
94
148
156
139
142
94
105
125
155
155
103
112
127
117
120
112
135
132
111
125
102
107
139
136
119
97
87
119
133
125
143
120
103
113
124
138
Number of Advertising Spots Purchased by Members of the Greater Hilltown Automobile Dealers Association
LO
5
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Solution - Stem & Leaf Displays
From the data in the table, we note that the smallest
number of spots purchased is 88. So we will make the first
stem value 8. The largest number is 156, so we will have
the stem values begin at 8 and continue to 15. The first
number in the table is 96, which will have a stem value of 9
and a leaf value of 6. Moving across the top row, the
second value is 93 and the third is 88.
LO
5
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Solution - Stem & Leaf Displays
After organizing all the data, the final table of stem-and-leaf
chart looks as follows:
Stem
LO
5
Leaf
8
78
9
3445667
10
234578
11
122337899
12
00455578
13
3566899
14
238
15
556
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Stem & Leaf Displays in Excel
Stem & Leaf In Excel
LO
5
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You Try It Out!
The male-female ratios for 22 different areas in the city are
as shown below :
5.4
6.2
8.9
10.1 11.3
7.9
4.8
10.3
8.7
9.4
12.0
10.5
5.9
5.4
8.6
6.9
9.8
7.0
12.4
8.9
7.7
10.3
Organize this information into a stem-and-leaf display.
(a) How many values are less than 7.0?
(b) List the values in the 8.0 up to 9.0 category.
(c) What is the middle (median) value?
(d) What are the largest and the smallest price-earnings ratios?
LO
5
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Chapter Summary
I. A frequency table is a grouping of qualitative data into
mutually exclusive classes showing the number of
observations in each class.
II. A relative frequency table shows the fraction or percent
of the number of observations in each class.
III. A bar chart is a graphic representation of a frequency
table.
IV. A pie chart shows the proportion each distinct class
represents of the total number of frequencies.
V. A frequency distribution is a grouping of data into
mutually exclusive classes showing the number of
observations in each class.
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Chapter Summary
A. The steps in constructing a frequency distribution
are:
1. Decide on the number of classes.
2. Determine the class interval .
3. Set the individual class limits.
4. Tally the raw data into the classes.
5. Count the number of tallies in each class.
B. The class frequency is the number of observations
in each class.
C. The class interval is the difference between the
limits of two consecutive classes.
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Chapter Summary
D. The class midpoint is halfway between the limits of
consecutive classes.
VI. A relative frequency distribution shows the percent of
observations in each class.
VII. There are three methods for graphically portraying a
frequency distribution.
A. A histogram portrays the number of frequencies in
each class in the form of a rectangle.
B. A frequency polygon consists of line segments
connecting the points formed by the intersections of
the class midpoints and the class frequency.
C. A cumulative frequency polygon shows the number
of observations below or above given values.
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Chapter Summary
VIII. A stem-and-leaf display is an alternative to a
frequency distribution.
A. The leading digit is the stem and the trailing digit, the
leaf.
B. The advantages of the stem-and-leaf chart over a
frequency distribution include
1. The identity of each observation is not lost.
2. The digits themselves give a picture of the
distribution.
3. The cumulative frequencies are also shown.
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Chapter Summary
V. There are many charts used in business.
A. A line chart is ideal for showing the trend or sales of
income over time.
B. Showing changes in nominal scale data.
C. Pie charts are useful for showing the percent that
various components compose of the total.
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