QMS210: Applied Statistics for
Business Winter 2025
Nursel S. Ruzgar
nruzgar@torontomu.ca
The copyright to this original work is held by Nursel Ruzgar and students registered in course
QMS210 can use this material for the purposes of this course, but no other use id permitted and
there can be no sale or transfer or use of the work for any other purpose without explicit
permission of Nursel Ruzgar. Copyright © Nursel Ruzgar. All rights reserved.
Lecture 01- Outline
What statistics is
◼ How statistics is fundamental to business
◼ The basic concepts and vocabulary of statistics
◼
◼
Data Types
◼
Measurement Scale
◼
Steam-and-leaf Plot
Rules and convention to construct a stem-and-Leaf Plot
◼ How do you construct a Stem-and-Leaf Plot with negative and
positive data values
◼ Interpretation of results using Stem-and-Leaf Plot
◼ Frequency Distribution
◼
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
2
What is Statistics? (Cont’d)
“Statistics is a way to get information from data”
Statistics
Data
Information
Data: Facts, especially
numerical facts, collected
together for reference or
information.
Information: Knowledge
communicated concerning
some particular fact.
Statistics is a tool for creating new understanding from a set of numbers.
Definitions: Oxford English Dictionary
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
3
Example
A student is somewhat apprehensive about the statistics
course because the student believes the myth that the
course is difficult. The professor provides last term’s marks to
the student. What information can the student obtain from this
list?
Statistics
Data
Information
List of last term’s marks.
95
89
70
65
78
57
:
New information about
the statistics class.
E.g. Median of all marks,
“Typical” mark, i.e. average,
Mark distribution, etc.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
4
Two Different Branches Of
Statistics Are Used In Business
Statistics
Transforms data into useful information for decision
makers.
Descriptive Statistics
Inferential Statistics
Collecting, summarizing,
visualizing, presenting and
analyzing data
Using data collected from a
small group to draw conclusions
about a larger group
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
5
Descriptive Statistics
◼
Collect data
◼
◼
Summarize, visualize, present data
◼
◼
e.g., Survey
e.g., Tables and graphs
Analyze data
◼
e.g., The sample mean
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
6
Inferential Statistics
◼
Estimation
◼
◼
e.g., Estimate the population
mean weight using the sample
mean weight
Hypothesis testing
◼ e.g., Test the claim that the
population mean weight is 120
pounds
Drawing conclusions about a large group of
individuals based on a smaller group.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
7
In Business, Statistics Plays A
Fundamental & Important Role
◼
To visualize & summarize business data
◼
◼
To draw conclusions from business data
◼
◼
Descriptive methods used to create charts & tables
Inferential methods used to reach conclusions about
a large group based on data from a smaller group
To make reliable forecasts about business
activities
◼
Inferential methods utilizing statistical models based
on business data
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
8
Types of Variables
▪
Categorical (qualitative) variables have values that
can only be placed into categories, such as “yes” and
“no.”
▪
Numerical (quantitative) variables have values that
represent quantities.
Discrete variables arise from a counting process
▪ Continuous variables arise from a measuring process
▪
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
9
Types of Variables
Variables
Categorical
Numerical
Examples:
◼
◼
◼
Marital Status
Political Party
Eye Color
(Defined categories)
Discrete
Continuous
Examples:
◼
◼
Number of Children
Defects per hour
(Counted items)
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
Examples:
Weight
◼
Voltage
(Measured characteristics)
◼
10
Types of data – analysis
Knowing the type of data is necessary to properly
select the technique to be used when analyzing data.
Type of analysis allowed for each type of data
Interval data – arithmetic calculations
◼ Ratio – arithmetic calculations
◼ Nominal data – counting the number of observation in each
category
◼ Ordinal data - computations based on an ordering process
◼
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
11
Nominal Data…
Examples:
Sport: tennis, soccer, swimming, etc.
Gender: Male, female,
Postal code: M4B5G7, N4E 4C9, etc.
Colors: Red, Blue, Green, etc.
◼
The only calculations that can be performed on nominal data
are based on the number of responses in each category.
To display nominal data we generally use bar, pie and Pareto
charts.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
12
Ordinal Data…
Examples:
◼Letter grades: A+, A, A-, B+, B, etc.
◼Rating of a service: very poor, poor, moderate, good, excellent,
◼Class standing: Freshman, Sophomore, Junior, Senior
◼
The only calculations that can be performed on ordinal data are based
on the number of values in each category.
◼
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
13
Interval Data…
◼
◼
◼
◼
The distinguishing feature of interval data is that
the data have units of measurement. Data must
be numeric and are quantitative. The value “0” is
only an arbitrary reference point.
Real numbers, i.e. weights, prices, distance, etc.
Also called as quantitative or numerical.
Arithmetic operations can be performed on
Interval Data, so it’s meaningful to talk about
2*Weight, or Price + $1.5, and so on.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
14
Ratio
Ratio data have the characteristics of interval data, but
the “0” value does mean the absence of the
characteristic being measured and the ratio of data
values is meaningful. Ratio data can be discrete or
continuous.
Examples:
Number of visits you have taken last 5 years (D)
Number of graduate students at Ryerson University (D)
Sales (in dollars) (C),
Size of your house (in square meters (C).
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
15
Ratio data
All types of calculations can be performed with ratio
data.
To display ratio data, we general use histograms,
polygons, ogives, stem and-leaf displays, and boxwhisker plots.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
16
A Step by Step Process For Examining &
Concluding From Data Is Helpful
In this book we will use DCOVA
Define the variables for which you want to reach
conclusions
◼ Collect the data from appropriate sources
◼ Organize the data collected by developing tables
◼ Visualize the data by developing charts
◼ Analyze the data by examining the appropriate
tables and charts (and in later chapters by using
other statistical methods) to reach conclusions
◼
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
17
Categorical Data Are Organized By
Utilizing Tables
Categorical
Data
Tallying Data
One
Categorical
Variable
Two
Categorical
Variables
Summary
Table
Contingency
Table
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
18
Organizing Categorical Data:
Summary Table
▪
A summary table indicates the frequency, amount, or percentage of items
in a set of categories so that you can see differences between categories.
Summary Table From A Survey of 1000 Banking Customers
Banking Preference?
Percent
ATM
16%
Automated or live telephone
2%
Drive-through service at branch
17%
In person at branch
41%
Internet
24%
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
19
A Contingency Table Helps Organize
Two or More Categorical Variables
◼
Used to study patterns that may exist between
the responses of two or more categorical
variables
◼
Cross tabulates or tallies jointly the responses
of the categorical variables
◼
For two variables the tallies for one variable are
located in the rows and the tallies for the
second variable are located in the columns
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
20
Contingency Table - Example
A random sample of 400
invoices is drawn.
◼ Each invoice is categorized
as a small, medium, or large
amount.
◼ Each invoice is also
examined to identify if there
are any errors.
◼ These data are then
organized in the contingency
table to the right.
◼
Contingency Table Showing
Frequency of Invoices Categorized
By Size and The Presence Of Errors
No
Errors
Errors
Small
Amount
170
20
190
Medium
Amount
100
40
140
Large
Amount
65
5
70
335
65
400
Total
Total
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
21
Contingency Table Based On
Percentage of Overall Total
No
Errors
Errors
Small
Amount
170
20
190
Medium
Amount
100
40
140
Large
Amount
65
Total
335
5
65
42.50% = 170 / 400
25.00% = 100 / 400
16.25% = 65 / 400
Total
No
Errors
Errors
Total
Small
Amount
42.50%
5.00%
47.50%
Medium
Amount
25.00%
10.00%
35.00%
Large
Amount
16.25%
1.25%
17.50%
Total
83.75%
16.25%
100.0%
70
400
83.75% of sampled invoices
have no errors and 47.50%
of sampled invoices are for
small amounts.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
22
Visualizing Categorical Data:
The Bar Chart
▪
In a bar chart, a bar shows each category, the length of which
represents the amount, frequency or percentage of values falling into
a category which come from the summary table of the variable.
Banking Preference
Banking Preference?
%
ATM
16%
Automated or live
telephone
2%
Drive-through service at
branch
17%
In person at branch
41%
Internet
24%
Internet
In person at branch
Drive-through service at branch
Automated or live telephone
ATM
0%
5% 10% 15% 20% 25% 30% 35% 40% 45%
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
23
Visualizing Categorical Data:
The Pie Chart
▪
The pie chart is a circle broken up into slices that represent categories.
The size of each slice of the pie varies according to the percentage in
each category.
Banking Preference
Banking Preference?
%
ATM
16%
ATM
16%
Automated or live
telephone
2%
Drive-through service at
branch
17%
In person at branch
41%
Internet
24%
24%
2%
17%
Automated or live
telephone
Drive-through service at
branch
In person at branch
Internet
41%
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
24
Visualizing Categorical Data:
The Pareto Chart
◼
Used to portray categorical data
◼
A vertical bar chart, where categories are
shown in descending order of frequency
◼
A cumulative polygon is shown in the same
graph
◼
Used to separate the “vital few” from the “trivial
many”
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
25
Visualizing Categorical Data:
The Pareto Chart (con’t)
100%
100%
80%
80%
60%
60%
40%
40%
20%
20%
0%
0%
In person Internet
at branch
Drivethrough
service at
branch
ATM
Cumulative %
(line graph)
% in each category
(bar graph)
Pareto Chart For Banking Preference
Automated
or live
telephone
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
26
Visualizing Categorical Data:
Side-By-Side Bar Charts
▪
The side-by side-bar chart represents the data from a contingency
table.
No
Errors
Errors
Total
Small
Amount
50.75%
30.77%
47.50%
Medium
Amount
29.85%
61.54%
35.00%
Errors
Large
Amount
19.40%
7.69%
17.50%
No Errors
Invoice Size Split Out By Errors
& No Errors
0.0%
Total
100.0%
100.0%
10.0%
20.0%
30.0%
40.0%
Large
Medium
50.0%
60.0%
70.0%
Small
100.0%
Invoices with errors are much more likely to be of
medium size (61.54% vs 30.77% and 7.69%)
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
27
Graphical Presentation of Quantitative Data
Stem-and-Leaf Displays
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
28
Stem-and-Leaf Display
◼
A simple way to see how the data are distributed
and where concentrations of data exist
METHOD: Separate the sorted data series
into leading digits (the stems) and
the trailing digits (the leaves)
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
29
Organizing Numerical Data:
Stem and Leaf Display
▪
A stem-and-leaf display organizes data into groups (called
stems) so that the values within each group (the leaves)
branch out to the right on each row.
Age of College Students
Age of
Surveyed
College
Students
Day Students
Day Students
16
17
17
18
18
18
19
19
20
20
21
22
22
25
27
32
38
42
Night Students
18
18
19
19
20
21
23
28
32
33
41
45
Stem
Leaf
Night Students
Stem Leaf
1
67788899
1
8899
2
0012257
2
0138
3
28
3
23
4
2
4
15
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
30
Stem-and-Leaf Displays
◼
A stem-and-leaf display conveys information about the
following aspects of the data:
identification of a typical or representative value
◼ extent of spread about the typical value
◼ presence of any gaps in the data
◼ extent of symmetry in the distribution of values
◼ number and location of peaks
◼ presence of any outlying values
◼
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
31
Interpretation of results using stem-and-leaf
plot
▪
A) Using the stem-and-leaf plot to detect the shape of the data
distribution.
◼
The symmetry of a data distribution can be classified in three ways:
◼
◼
◼
l) skewed to the left (the bulk of the data is located on the right side of the distribution; see
Graphic A),
(2) symmetrical (the bulk of the data is located in the middle of the distribution; see Graphic
B) and
(3)skewed to the right (the bulk of the data is located on the left side of the distribution; see
Graphic C).
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
32
Interpretation of results using stem-and-leaf
plot
▪
▪
▪
B) Using the stem-and-leaf plot to extract information
Example:
A random sample of employees was taken from a technology
company, and the hours of overtime claimed per month were and
recorded and summarized in the following stem-and-leaf plot.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
33
Interpretation of results using stem-and-leaf
plot
cont’d
◼
How many employees are being sampled?
◼
What percent of employees claimed at least 152 hours per month?
◼
What percent of employees claimed less than 130 hours per month?
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
36
Interpretation of results using stem-and-leaf
plot
cont’d
◼
What percent of employees claimed more than 163 hours per month?
◼
What percent of employees claimed at most 115 hours per month?
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
37
Frequency Distribution
The frequency distribution is a summary table in which the data
are arranged into numerically ordered classes.
◼ Number of classes: You should not have too few or too many
classes. For the questions you will solve in this course you should
use 5 to l0 classes in your final constructed frequency distribution.
◼ Notation for indicating classes: There are several possible notations
that are used to designate the classes. In this course we will use the
"and under" notation( e.g., 260 and under 270). The numerical
values used in designating the classes when using the "and under“
notation are called boundaries.
◼
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
38
Organizing Numerical Data:
Frequency Distribution Example
Example: A manufacturer of insulation randomly selects 20 winter
days and records the daily high temperature in degrees F.
24, 35, 17, 21, 24, 37, 26, 46, 58, 30, 32, 13, 12, 38, 41, 43, 44, 27, 53, 27
▪
Solution: Sort raw data in ascending order:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
▪
Find max value: 58, min value: 12, range: 58 - 12 = 46
Find relative
frequency or
percentage
distribution.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
39
Organizing Numerical Data: Relative &
Percent Frequency Distribution Example
𝑭𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 (𝒇)
𝑹𝒆𝒍𝒂𝒕𝒊𝒗𝒆 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚(𝒓𝒇) =
𝒏
𝑭𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 (𝒇)
𝑷𝒆𝒓𝒄𝒆𝒏𝒕𝒂𝒈𝒆 𝒐𝒇 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚(%) =
× 𝟏𝟎𝟎
𝒏
Class
10 and under 20
20 and under 30
30 and under 40
40 and under 50
50 and under 60
Total
Frequency
3
6
5
4
2
20
Relative
Frequency
Percentage
.15
.30
.25
.20
.10
1.00
15
30
25
20
10
100
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
40
Organizing Numerical Data: Cumulative
Frequency Distribution Example
Data in ordered array:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Class
Frequency Percentage
Cumulative Cumulative
Frequency Percentage
10 but less than 20
3
15%
3
15%
20 but less than 30
6
30%
9
45%
30 but less than 40
5
25%
14
70%
40 but less than 50
4
20%
18
90%
50 but less than 60
2
10%
20
100%
20
100
20
100%
Total
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
41
Visualizing Numerical Data: The Histogram
▪
A vertical bar chart of the data in a frequency distribution is
called a histogram.
▪
In a histogram there are no gaps between adjacent bars.
▪
The class boundaries (or class midpoints) are shown on the
horizontal axis.
▪
The vertical axis is either frequency, relative frequency, or
percentage.
▪
The height of the bars represent the frequency, relative
frequency, or percentage.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
42
Why Use a Frequency Distribution?
◼
It condenses the raw data into a more useful form
◼
It allows for a quick visual interpretation of the data
◼
It enables the determination of the major
characteristics of the data set including where the
data are concentrated / clustered
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
43
Organizing Numerical Data: Frequency
Distribution Example
Data in ordered array:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Class
10 and under 20
20 and under 30
30 and under 40
40 and under 50
50 and under 60
Total
Midpoints
15
25
35
45
55
Frequency
3
6
5
4
2
20
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
44
Visualizing Numerical Data:
The Histogram
10 and under 20
20 and under 30
30 and under 40
40 and under 50
50 and under 60
Total
Frequency
3
6
5
4
2
20
(In a percentage
histogram the vertical
axis would be defined to
show the percentage of
observations per class)
Relative
Frequency
Percentage
.15
.30
.25
.20
.10
1.00
15
30
25
20
10
100
8
Histogram: temprature
Frequency
Class
6
4
2
0
5
15 25 35 45 55 More
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
45
Visualizing Numerical Data:
The Polygon
▪
A percentage polygon is formed by having the midpoint of
each class represent the data in that class and then connecting
the sequence of midpoints at their respective class
percentages.
▪
The cumulative percentage polygon, or ogive, displays the
variable of interest along the X axis, and the cumulative
percentages along the Y axis.
▪
Useful when there are two or more groups to compare.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
46
Visualizing Numerical Data:
The Frequency Polygon
Class
Midpoint Frequency
Class
15
25
35
45
55
3
6
5
4
2
Frequency Polygon: Age Of Students
Frequency
10 but less than 20
20 but less than 30
30 but less than 40
40 but less than 50
50 but less than 60
(In a percentage
polygon the vertical axis
would be defined to
show the percentage of
observations per class)
7
6
5
4
3
2
1
0
5
15
25
35
45
55
65
Class Midpoints
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
47
Visualizing Numerical Data
By Using Graphical Displays
Numerical Data
Ordered Array
Stem-and-Leaf
Display
Frequency Distributions
and
Cumulative Distributions
Histogram
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
Polygon
Ogive
48
The height of a bar represents:
◼
The proportion of measurements falling in that class or
subinterval
◼
The probability that a single measurement, drawn at random
from the set, will belong to that class or subinterval
Relative Frequency Histograms
If the data are discrete, one class might be assigned for each
integer value taken on by the data
◼ A large number of integer values may need to be grouped into
classes
◼
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
50
Visualizing Numerical Data:
The Ogive (Cumulative % Polygon)
How to draw an Ogive
◼ Start with a frequency distribution table and add a
cumulative frequency column and a cumulative
relative frequency column. You may construct two
kinds of cumulative relative frequencies.
◼ cumulative relative frequency (crf) and
◼ cumulative percentage frequency (c%).
Both kinds of cumulative frequency will give you the
same answer. However, it is more common to use
c% to construct the ogive.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
51
Visualizing Numerical Data:
The Ogive (Cumulative % Polygon)
Cumulative relative frequency (crf)
◼ crf = relative frequency (rf) of the class + sum of
all previous class relative frequencies
Alternatively,
crf = Sum of all relative frequencies up to and
including the class.
1.
𝑐% = 100 ×
𝑐𝑓
𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑒𝑠
Or
𝑐% = 100 × 𝑐𝑟𝑓
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
52
Visualizing Numerical Data:
The Ogive (Cumulative % Polygon)
Steps to construct OGIVE
Step 1: Construct the cumulative frequency
distribution
Step 2: Draw the ogive
To plot the ogive, you need to construct the vertical
axis (y-axis) representing the c% and the horizontal
axis (x-axis) representing the upper boundary for
each class intervals. Start the graph at the first
boundary.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
53
Example
A soft drink manufacturer has to check periodically whether bottles marked as 2 litre
actually contain 2 litres. Have you ever noticed that some bottles are filled to the top and
some are not? To avoid customers‘ complaints, the manufacturer has to conduct some
investigations to ensure that the bottling machine is working properly and filling the
bottles with the right amount. A sample of 255 bottles was taken from the production
floor, and the volumes of these 255 bottles were recorded. The data that showed how
much soft drink was in each bottle were summarized in the following frequency
distribution table:
a. What percent of the bottles had less than
2000.00 ml?
b. What percent of the bottles had less than
2002.70 ml?
c. What percent of the bottles had at least
2001 .25 ml?
d. 30% of the bottles have at most
______ml.
e. 25% of the bottles have at least ____ml.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
54
Example
cont’d
Question( a) is easy to answer because 2000.00 ml is the upper
boundary of the second class and you just add the frequencies of the
first and second classes (i.e. 5 + l0). The answer is l5 bottles, which
is (15 / 255) x 100 = 5.9% of the bottles had lless than 2000.00 ml .
◼ Question( b) is not easy because 2002.7ml falls between 2002.50
and 2003.00. Similarly, for question (c), 2001.25ml falls between
2001.00 and 2001.50.
◼ To answer questions (b) and (c), you must plot an ogive.
◼
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
55
Example
◼
cont’d
Step 1: Construct the cumulative frequency distribution
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
56
Example
◼
◼
cont’d
Step 2: Draw the ogive
To plot the ogive, you need to construct the vertical axis (y-axis) representing the c% , and
the horizontal axis (x-axis) representing the upper boundary for each class intervals. Start the
graph at the first boundary.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
57
Example
cont’d
Join the points with a
straight line. In the
construction of an
ogive, an extra upper
boundary is added at the
beginning so the graph
starts with a crf of 0 or
0%.
◼ Use the ogive presented
earlier to answer the
following questions
part (b) to part (c).
◼
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
58
Example
◼
cont’d
b) What percent of bottles had less than 2002.70 ml?
To find the "less than
2002.7,“ go vertically
upward at 2002.10 of
the x-axis until you hit
the ogive line, and
then go across to
acquire the y value.
The y value is 89%;
that is, 89% of the
bottles had less than
2002.7ml.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
59
Example
◼
cont’d
c) What percent of bottles had at least 2001.25 ml?
To find the “at least 2001.25,"
go vertically upward at 2001.25
of the x-axis until you hit
the ogive line, and then go
across to acquire the y value.
The y value is 38%. However
38% indicates that 38% of the
bottles had less than 2001.25
ml. The question asks for the
percent of the bottles that had at
least 2001.25 ml. Therefore,
62% (=100%-38%) of bottles
had at least 2001 .25 ml.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
60
Example
◼
cont’d
d) 30% of the bottles have at most ____________ml?
Go across at 30%
of the y-axis until
you hit the ogive
line, and then go
downward to
acquire the x
value. The x value
is 2001.1 ml.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
61
Example
◼
cont’d
e) 25% of the bottles have at least ____________ml?
Go across at 75%
(=100%-25%) of
the y-axis until
you hit the ogive
line, and then go
downward to
acquire the x
value. The x value
is 2002.2 ml.
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
62
Example
◼
Given the following frequency distribution for a two-star hotel.
Draw histogram, percentage polygon, and ogive.
bin
Frequency
15 and under 25
25 and under 35
35 and under 45
45 and under 55
55 and under 65
65 and under 75
75 and under 85
85 and under 95
95 and under 105
1
5
2
5
9
12
5
1
1
105 and under 115
1
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
63
Example
Histogram
Histogram of Two-Star
14
12
10
Frequency
◼
cont’d
8
6
4
2
0
20
30
40
50
60
70
80
90
100
110
Midpoints
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
64
Example
bin
cont’d
Frequency
Percentage
15 and under 25
25 and under 35
35 and under 45
45 and under 55
55 and under 65
65 and under 75
75 and under 85
85 and under 95
95 and under 105
1
5
2
5
9
12
5
1
1
2.38%
11.90%
4.76%
11.90%
21.43%
28.57%
11.90%
2.38%
2.38%
2.38%
14.29%
19.05%
30.95%
52.38%
80.95%
92.86%
95.24%
97.62%
20
30
40
50
60
70
80
90
100
105 and under 115
1
2.38%
100.00%
110
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
Cumulative
Percentage.
Midpts.
65
Example
◼
cont’d
Percentage Polygon
Percentage Polygon
30%
25%
20%
15%
10%
5%
0%
20
30
40
50
60
70
80
90
100
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
110
66
Example
◼
cont’d
Cumulative Percentage Polygon (Ogive)
QMS210 W25 Lecture 01 Copyright © N. Ruzgar
67