Scales of Measurement -6
• Interval
The data have the properties of ordinal data, and
the interval between observations is expressed in
terms of a fixed unit of measure.
Example: SAT scores, temperature
Interval data are always numeric.
1
Scales of Measurement -7
• Interval
Example:
three students with SAT math scores of 620, 550,
and 470 can be ranked or ordered in terms of best
performance to poorest performance in math.
The differences between the scores are meaningful.
For instance, student 1 scored 620 - 550 = 70 points
more than student 2, while
student 2 scored 550 - 470 = 80 points more than
student 3.
2
Scales of Measurement -8
• Ratio
The data have all the properties of interval data
and the ratio of two values is meaningful.
Variables such as distance, height, weight, and time
use the ratio scale.
This scale must contain a zero value that indicates
that nothing exists for the variable at the zero point.
3
Scales of Measurement -9
• Ratio
Example:
Melissa’s college record shows 36 credit hours
earned, while Kevin’s record shows 72 credit hours
earned. Kevin has twice as many credit hours earned
as Melissa.
4
Categorical and Quantitative Data
Data can be further classified as being categorical
or quantitative.
The statistical analysis that is appropriate depends
on whether the data for the variable are categorical
or quantitative.
In general, there are more alternatives for statistical
analysis when the data are quantitative.
5
Quantitative Data
Quantitative data indicate how many or how much:
discrete, if measuring how many
continuous, if measuring how much
Quantitative data are always numeric.
Ordinary arithmetic operations are meaningful for
quantitative data.
6
Scales of Measurement
Data
Categorical
Numeric
Nominal
Ordinal
Quantitative
Non-numeric
Nominal
Ordinal
Numeric
Interval
Ratio
7
Cross-Sectional Data
Cross-sectional data are collected at the same or
approximately the same point in time.
Example: data detailing the number of building
permits issued in November 2012 in each of the
counties of Ohio
8
Time Series Data -1
Time series data are collected over several time
periods.
Example: data detailing the number of building
permits issued in Lucas County, Ohio in each of
the last 36 months
Graphs of time series help analysts understand
• what happened in the past,
• identify any trends over time, and
• project future levels for the time series
9
Time Series Data -2
• Graph of Time Series Data
10
1.3 Data Sources -1
• Existing Sources
Internal company records – almost any department
Business database services – Dow Jones & Co.
Government agencies - U.S. Department of Labor
Industry associations – Travel Industry Association
of America
Special-interest organizations – Graduate Management
Admission Council
Internet – more and more firms
11
Data Sources -2
• Statistical Studies - Observational
In observational (nonexperimental) studies no
attempt is made to control or influence the
variables of interest.
a survey is a good example
Studies of smokers and nonsmokers are
observational studies because researchers
do not determine or control
who will smoke and who will not smoke.
12
Data Sources -3
• Statistical Studies - Experimental
In experimental studies the variable of interest is
first identified. Then one or more other variables
are identified and controlled so that data can be
obtained about how they influence the variable of
interest.
13
Data Sources -4
• For example, a pharmaceutical firm might be
interested in conducting an experiment to
learn how a new drug affects blood pressure.
– Blood pressure is the variable of interest in the
study.
– The dosage level of the new drug is another
variable that is hoped to have a causal effect on
blood pressure.
14
Data Acquisition Considerations
Time Requirement
– Searching for information can be time consuming.
– Information may no longer be useful by the time
it is available.
Cost of Acquisition
– Organizations often charge for information even
when it is not their primary business activity.
Data Errors
– Using any data that happen to be available or
were acquired with little care can lead to
misleading information.
15
Data Acquisition Errors
• We should always be aware of the possibility of
data errors in statistical studies. Using erroneous
data can be worse than not using any data at all.
• An error in data acquisition occurs whenever the
data value obtained is not equal to actual value
that would be obtained with a correct procedure.
• For example,
– In writing the age of a 24-year-old person as 42.
– A respondent shown to be 22 years of age but
reporting 20 years of work experience.
16
1.4 Descriptive Statistics
• Most of the statistical information in
newspapers, magazines, company reports,
and other publications consists of data that
are summarized and presented in a form that
is easy to understand.
• Such summaries of data, which may be
tabular, graphical, or numerical, are referred
to as descriptive statistics.
17
Descriptive Statistics -1
• A tabular summary of the data set in Table
1.1
• Frequencies and Percent Frequencies for The
Fitch Credit Rating Outlook of 60 Nations
18
Three Lines Table
19
Three Lines Table
20
Descriptive Statistics -2
• Bar Chart for The Fitch Credit Rating
Outlook of 60 Nations
21
Descriptive Statistics -3
• A graphical summary of the data for
quantitative variable Per Capita GDP in
Table 1.1, called a histogram
22
1.5 Statistical Inference -1
Population - the set of all elements of interest in a
particular study
Sample - a subset of the population
Statistical inference - the process of using data obtained
from a sample to make estimates
and test hypotheses about the
characteristics of a population
Census - collecting data for the entire population
Sample survey - collecting data for a sample
23
Statistical Inference -2
• Example: Norris Electronics.
– Norris manufactures a high-intensity lightbulb.
– To increase the useful life of the lightbulb, the product
design group developed a new lightbulb filament. In
this case, the population is defined as all lightbulbs
that could be produced with the new filament.
– To evaluate the advantages of the new filament, 200
bulbs with the new filament were manufactured and
tested.
– Data collected from this sample showed the number of
hours each lightbulb operated before filament burnout.
24
Process of Statistical Inference
1. Population
consists of all bulbs
Manufactured with
the new filament.
Average lifetime is
unknown.
4. The sample average
is used to estimate the
population average.
2. A sample of 200
bulbs is manufactured
with the new filament.
3. The sample data
provide a sample
average parts cost
of $76 hours per bulb.
25
Statistics for Business
and Economics
Chapter 2
Descriptive Statistics: Tabular and
Graphical Displays
How to describe the unknown data?
27
2.1 Summarizing Categorical Data for a
categorical Variable
• Frequency Distribution
• Relative Frequency Distribution
• Percent Frequency Distribution
• Bar Chart
• Pie Chart
28
Frequency Distribution
A frequency distribution (次數分配) is a tabular
summary of data showing the number (frequency)
of observations in each of several non-overlapping
categories or classes.
The objective is to provide insights about the data
that cannot be quickly obtained by looking only at
the original data.
29
Frequency Distribution
• Example: Frequency Distribution of Soft
Drink Purchases
30
Relative Frequency Distribution
The relative frequency of a class is the fraction or
proportion of the total number of data items
belonging to the class.
A relative frequency distribution is a tabular
summary of a set of data showing the relative
frequency for each class.
31
Percent Frequency Distribution
The percent frequency of a class is the relative
frequency multiplied by 100.
A percent frequency distribution is a tabular
summary of a set of data showing the percent
frequency for each class.
32
Relative Frequency and
Percent Frequency Distribution
• Example: Relative and Percent Frequency
Distribution of Soft Drink Purchases
33
Bar Chart -1
• A bar chart is a graphical display for depicting
categorial data.
• On one axis (usually the horizontal axis), we specify
the labels that are used for each of the classes.
• A frequency, relative frequency, or percent
frequency scale can be used for the other axis
(usually the vertical axis).
• Using a bar of fixed width drawn above each class
label, we extend the height appropriately.
• The bars are separated to emphasize the fact that
each class is a separate category.
34
Bar Chart -2
• Example: Bar Graph of Soft Drink Purchases
35
Pie Chart -1
• The pie chart is a commonly used graphical
display for presenting relative frequency and
percent frequency distributions for categorical
data.
• First draw a circle; then use the relative
frequencies to subdivide the circle into sectors
that correspond to the relative frequency for
each class.
• Since there are 360 degrees in a circle, a class
with a relative frequency of 0.38 would consume
0.38(360) = 136.8 degrees of the circle.
36
Pie Chart -2
• Example: Pie Chart of Soft Drink Purchases
37
3-D Pie Chart and Comparison
• Example: Pie Chart of Soft Drink Purchases
•
38
2.2 Summarizing Quantitative Data
• Frequency Distribution
• Relative Frequency and Percent Frequency
Distributions
• Histogram
• Cumulative Distributions
39
Frequency Distribution -1
• The three steps necessary to define the
classes for a frequency distribution with
quantitative data are:
1. Determine the number of non-overlapping
classes.
2. Determine the width of each class.
3. Determine the class limits.
40
Frequency Distribution -2
• Guidelines for Determining the Number of
Classes
– Use between 5 and 20 classes.
• we recommend using between 5 and 20 classes.
– Data sets with a larger number of elements
usually require a larger number of classes.
– Smaller data sets usually require fewer classes.
The goal is to use enough classes to show the
variation in the data, but not so many classes
that some contain only a few data items.
41
Frequency Distribution -3
• Guidelines for Determining the Width of
Each Class
– Use classes of equal width.
– Approximate Class Width =
Largest Data Value - Smallest Data Value
Number of Classes
Making the classes the same
width reduces the chance of
inappropriate interpretations.
42
Frequency Distribution -4
• Note on Number of Classes and Class Width
– In practice, the number of classes and the appropriate
class width are determined by trial and error.
– Once a possible number of classes is chosen, the
appropriate class width is found.
– The process can be repeated for a different number of
classes.
– Ultimately, the analyst uses judgment to determine
the combination of the number of classes and class
width that provides the best frequency distribution for
summarizing the data.
43
Frequency Distribution -5
• Guidelines for Determining the Class Limits
– Class limits must be chosen so that each data item belongs
to one and only one class.
– The lower class limit identifies the smallest possible data
value assigned to the class.
– The upper class limit identifies the largest possible data
value assigned to the class.
– The appropriate values for the class limits depend on the
level of accuracy of the data.
An open-end class requires only a
lower class limit or an upper class limit.
44
Frequency Distribution -6
• Example: These data show the time in days
required to complete year-end audits for a
sample of 20 clients of Sanderson and
Clifford, a small public accounting firm with
the data rounded to the nearest day.
45
Frequency Distribution -7
• Example: Year-end audit times
1. Number of classes = 5
2. An approximate class width of (33 — 12)/5= 4.2.
3. We therefore decided to round up and use a class
width of 5 days in the frequency distribution.
4. Frequency Distribution
46
Relative Frequency and
Percent Frequency Distributions
• Example: Year-end audit times
47
Histogram -1
• Another common graphical display of
quantitative data is a histogram.
• The variable of interest is placed on the
horizontal axis.
• A rectangle is drawn above each class interval
with its height corresponding to the interval’s
frequency, relative frequency, or percent
frequency.
• Unlike a bar graph, a histogram has no natural
separation between rectangles of adjacent
classes.
48
Histogram -2
• Example: Histogram for The Audit Time
Data
49
Histograms Showing Skewness -1
• Histograms Showing Differing Levels of Skewness
50
Cumulative Distributions -1
Cumulative frequency distribution - shows the
number of items with values less than or equal to the
upper limit of each class..
Cumulative relative frequency distribution – shows
the proportion of items with values less than or
equal to the upper limit of each class.
Cumulative percent frequency distribution – shows
the percentage of items with values less than or
equal to the upper limit of each class.
51
Cumulative Distributions -2
• Example:
– Cumulative Frequency
– Cumulative Relative Frequency and Cumulative
Percent Frequency Distributions for the Audit
Data.
52
Stem-and-Leaf Display -1
• A stem-and-leaf display shows both the rank
order and shape of the distribution of the data.
• It is similar to a histogram on its side, but it has
the advantage of showing the actual data values.
• The first digits of each data item are arranged to
the left of a vertical line.
• To the right of the vertical line we record the last
digit for each item in rank order.
• Each line in the display is referred to as a stem.
• Each digit on a stem is a leaf.
53
Stem-and-Leaf Display -2
• Example: Number of Questions Answered
Correctly
54
Stem-and-Leaf Display -3
• Stem : The numbers to the left of the vertical
line
(6, 7, 8, 9, 10, 11, 12, 13, and 14).
• Leaf : each digit to the right of the vertical line.
55
Stem-and-Leaf Display -4
• Although the stem-and-leaf display may
appear to offer the same information as a
histogram, it has two primary advantages.
1. The stem-and-leaf display is easier to construct by
hand.
2. Within a class interval, the stem-and-leaf display
provides more information than the histogram
because the stem-and-leaf shows the actual (or
near actual) data.
56
Stem-and-Leaf Display
• Leaf Units
– A single digit is used to define each leaf.
– In the preceding example, the leaf unit was 1.
– Leaf units may be 100, 10, 1, 0.1, and so on.
– Where the leaf unit is not shown, it is assumed to
equal 1.
– The leaf unit indicates how to multiply the stemand-leaf numbers in order to approximate the
original data.
57
2.3 Summarizing Data for Two Variables
Using Tables -1
• Crosstabulation (交叉表)
58
Summarizing Data for Two Variables
Using Tables -2
• Thus far we have focused on methods that
are used to summarize the data for one
variable at a time.
• Often a manager is interested in tabular and
graphical methods that will help understand
the relationship between two variables.
• Crosstabulation is a method for summarizing
the data for two variables.
59
Crosstabulation -1
• A crosstabulation is a tabular summary of
data for two variables.
• Crosstabulation can be used when:
– one variable is qualitative and the other is
quantitative,
– both variables are qualitative, or
– both variables are quantitative.
• The left and top margin labels define the
classes for the two variables.
60
Crosstabulation -2
• Example: Data from Zagat’s Restaurant
Review
– Data on a restaurant’s quality rating and typical
meal price are reported.
– Quality rating is a qualitative variable with rating
categories of good, very good, and excellent.
– Meal price is a quantitative variable that ranges
from $10 to $49.
61
Crosstabulation -3
• Example: : Data from Zagat’s Restaurant
The data for the first 10 restaurants
62
Crosstabulation -4
• Example: Data from Zagat’s Restaurant
Crosstabulation of Quality Rating and Meal
Price for 300 Los Angeles Restaurants
63
2.4 Summarizing Data for Two Variables
Using Graphical Displays -1
• Scatter Diagram and Trendline
64
Summarizing Data for Two Variables
Using Graphical Displays -2
• In most cases, a graphical display is more
useful than a table for recognizing patterns
and trends.
• Displaying data in creative ways can lead to
powerful insights.
• Scatter diagrams and trendlines are useful in
exploring the relationship between two
variables.
65
Scatter Diagram and Trendline
• A scatter diagram is a graphical presentation of
the relationship between two quantitative
variables.
• One variable is shown on the horizontal axis and
the other variable is shown on the vertical axis.
• The general pattern of the plotted points
suggests the overall relationship between the
variables.
• A trendline provides an approximation of the
relationship.
66
Scatter Diagram -4
• Example: The Stereo and Sound Equipment
Store
– Consider the advertising/sales relationship for a
stereo and sound equipment store in San
Francisco. On 10 occasions during the past three
months.
67
Scatter Diagram -5
• Example: Scatter Diagram and Trendline for
The Stereo and Sound Equipment Store.
68
Scatter Diagram -6
• The scatter diagram indicates a positive
relationship between the number of
commercials and sales.
• Higher sales are associated with a higher
number of commercials.
• The relationship is NOT perfect in that all
points are not on a straight line. However,
the general pattern of the points and the
trendline suggest that the overall relationship
is positive.
69
Scatter Diagram -1
• A Positive Relationship
No Apparent Relationship
y
y
x
x
• A Negative Relationship
y
• A Negative Relationship
x
70
Choosing the Type of Graphical Display
• Displays used to show the distribution of
data:
Bar Chart
Pie Chart
Dot Plot
Histogram
Stem-and-Leaf Display
• Displays used to make comparisons:
Side-by-Side Bar Chart
Stacked Bar Chart
• Displays used to show relationships:
Scatter Diagram
Trendline
71
Tabular and Graphical Displays
Data
Categorical Data
Quantitative Data
Tabular
Displays
Graphical
Displays
Tabular
Displays
Graphical
Displays
•Frequency
Distribution
•Rel. Freq. Dist.
•Percent Freq.
Distribution
•Crosstabulation
•Bar Chart
•Pie Chart
•Side-by-Side
Bar Chart
• Stacked
Bar Chart
•Frequency
Distribution
•Rel. Freq. Dist.
•% Freq. Dist.
•Cum. Freq. Dist.
•Cum. Rel. Freq.
Distribution
•Cum. % Freq.
Distribution
•Crosstabulation
•Dot Plot
•Histogram
•Stem-andLeaf Display
•Scatter
Diagram
72
End of Chapter 2
• Chapter 1 HW:
• Chapter 2 HW:
73