Business Statistics **** Management Information Systems
Business Statistics
Third levelFirst mid-term: 1436-1437
Instructor: Dr. ZRELLI Houyem
Majmaah University ***** Faculty of Science and Humanities in Ghat
الرحيم الرحمان الله بسم
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Chapter 2;
Describing Data:Frequency Tables, Frequency
Distributions, and Graphic Presentation
Business Statistics **** Management Information Systems
Data Presentation
Data Presentation
QualitativeData
QuantitativeData
Continues Variables
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Discrete Variables
• Qualitative Data are nonnumerical– Major Discipline– Political Party– Gender– Eye color
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Describing Qualitative Data
• Summarized in two ways:– Class Frequency– Class Relative Frequency
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Describing Qualitative Data
• Class Frequency– A class is one of the categories into which
qualitative data can be classified– Class frequency is the number of observations in
the data set that fall into a particular class
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Describing Qualitative Data
Student Evaluation Student Evaluation
1 Very Good 12 Very Good
2 Good 13 Good
3 Good 14 Very Good
4 Excellent 15 Good
5 Very Good 16 Good
6 Excellent 17 Good
7 Excellent 18 Excellent
8 Good 19 Very Good
9 Excellent 20 Good
10 Good 21 Good
11 Excellent 22 Excellent
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Describing Qualitative Data Example: Student
evaluations
Evaluation Frequency
Good 10
Very Good 5
Excellent 7
Total 22
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Describing Qualitative Data Example: Student
evaluations
• Class Relative Frequency– Class frequency divided by the total number of
observations in the data set
class frequencyclass relative frequency =
n
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Describing Qualitative Data
• Class Percentage– Class relative frequency multiplied by 100
class percentage = (class relative frequency) 100
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Describing Qualitative Data
Evaluation Relative Frequency
Class Percentage
Good 10/22 = .455 45.5%
Very Good 5/22 = .227 22.7%
Excellent 7/22 = .318 31.8%
Total 22/22 = 1.00 100%
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Describing Qualitative Data Example: Student
evaluations
0
2
4
6
8
10
Good Very Good Excellent
Bar Graph: The categories (classes) of the qualitative variable are represented by bars, where the height of each bar is either the class frequency, class relative frequency or class percentage.
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Describing Qualitative Data Example: Student
evaluations
Good163.8
Very Good81.72
Excellent114.48
Pie Chart: The categories (classes) of the qualitative variable are represented by slices of a pie. The size of each slice is proportional to the class relative frequency.
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Describing Qualitative Data Example: Student
evaluations
Pareto Diagram: A bar graph with the categories (classes) of the qualitative variable (i.e., the bars) arranged in height in descending order from left to right.
Good Very Good Excellent
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Describing Qualitative Data Example: Student
evaluations
Quantitative data: Discrete Variable
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A discrete variable can have only countable number of
values
Examples:
Family size (x = 0, 1, 2, 3, … )
Number of patients (x = 0, 1, 2, 3, … )
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Example:
The following data represent the number of children of 16 Saudi women:
3, 5, 2, 4, 0, 1, 3, 5, 2, 3, 2, 3, 3, 2, 4, 1
- Variable = X = no. of children (discrete, quantitative)
- Sample size = n = 16
- The possible values of the variable are: 0, 1, 2, 3, 4, 5
Quantitative data: Discrete Variable
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Business Statistics **** Management Information Systems
no. of children(variable)
Frequency(no. of women)
Relative Freq. (R.F.) (=Freq /n)
Percentage Freq.(= R.F. * 100%)
0 1 0.0625 6.25%
1 2 0.125 12.5%
2 4 0.25 25%
3 5 0.3125 31.25%
4 2 0.125 12.5%
5 2 0.125 12.5%
Total n=16 1.00 100%
NoteTotal of the frequencies = n = e sample size·Relative frequency = frequency/nPercentage frequency = Relative frequency *100%
Simple frequency distribution of the no. of children
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Quantitative data: Discrete Variable
The most common form of graphs for discrete variables is the bar chart.
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Graphical representation of discrete variables
· Frequency bar chart
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Graphical representation of discrete variables
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Cumulative representation of discrete variables
No. Of children
Frequency Relative frequency
Cumulative relative frequency
0 1 0.0625 0
1 2 0.125 0.0625
2 4 0.25 0.1875
3 5 0.3125 0.4375
4 2 0.125 0.75
5 2 0.125 0.875
Total N=10 1.00 1
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Cumulative representation of discrete variables
modalities
Fi
1 2 3 4 5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Cumulative distribution
0.9
1
0
. ..
. .
.
.
A continuous variable can have any
value within a certain interval of values.
Quantitative data: Continuous variablesBusiness Statistics **** Management Information Systems
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Examples: - height (140 < x < 190)
- blood sugar level (10 < x < 15)
A continuous frequency distribution CANNOT be represented by a bar chart. It is most appropriately represented by a histogram
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Graphical representation of continuous variables
Example: Hudson Auto Repair
The manager of Hudson Autowould like to have a betterunderstanding of the costof parts used in the enginetune-ups performed in theshop. She examines 50customer invoices for tune-ups. The costs of
parts,rounded to the nearest dollar, are listed on the
nextslide.
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Example: Hudson Auto Repair Sample of Parts Cost for 50 Tune-ups
91 78 93 57 75 52 99 80 97 6271 69 72 89 66 75 79 75 72 76104 74 62 68 97 105 77 65 80 10985 97 88 68 83 68 71 69 67 7462 82 98 101 79 105 79 69 62 73
Including a line in the table for every possible cost is not a good idea.
Need to categorize.
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Frequency Distribution• Guidelines for Selecting Number of Classes
• Use 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
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Frequency Distribution• Guidelines for Selecting Width of Classes
Largest Data Value Smallest Data ValueNumber of Classes
•Use classes of equal width.
•Approximate Class Width =
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Frequency Distribution
For Hudson Auto Repair, if we choose six classes:
[50-60[ [60-70[ [70-80[ [80-90[ [90-100[ [100-110[
2 13 16 7 7 5Total 50
Parts Cost ($)Frequency
Approximate Class Width = (109 - 52)/6 = 9.5 10
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Relative Frequency and Percent Frequency Distributions
[50-60[ [60-70[ [70-80[ [80-90[ [90-100[ [100-110[
PartsCost ($)
.04 .26 .32 .14 .14 .10Total 1.00
RelativeFrequency
4 26 32 14 14 10 100
Percent Frequency
2/50 .04(100)
Prev
iew
cum
ulati
ve fr
eque
ncie
s he
re.
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• Only 4% of the parts costs are in the $50-60 class.
• The greatest percentage (32% or almost one-third) of the parts costs are in the $70-80 class.
• 30% of the parts costs are under $70.
• 10% of the parts costs are $100 or more.
Insights Gained from the Percent Frequency Distribution
Relative Frequency and Percent Frequency Distributions
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Histogram Another common graphical presentation of quantitative data is a histogram.
The variable of interest is placed on the horizontal axis. A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, relative frequency, or percent frequency.
Unlike a bar graph, a histogram has no natural separation between rectangles of adjacent classes.
In informal discussions bar graphs and histograms are often equated. In this class you should be careful to keep them straight.
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Histogram
22
44
66
88
1010
1212
1414
1616
1818
PartsCost ($) PartsCost ($)
Fre
qu
en
cy
Fre
qu
en
cy
50 60 70 80 90 100 11050 60 70 80 90 100 110
Tune-up Parts Cost
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• Symmetric– Left tail is the mirror image of the right tail– Examples: heights and weights of people
Histogram (Common categories)
Rela
tive F
req
uen
cyR
ela
tive F
req
uen
cy
.05.05
.10.10
.15.15
.20.20
.25.25
.30.30
.35.35
00
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Cumulative frequency distribution - shows the number of items with values less than or equal to the upper limit of each class..
Cumulative relative frequency distribution – shows the proportion of items with values less than or equal to the upper limit of each class.
Cumulative Distributions
Cumulative percent frequency distribution – shows the percentage of items with values less than or equal to the upper limit of each class.
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Cumulative Distributions• Hudson Auto Repair
[50-60[ [60-70[ [70-80[ [80-90[ [90-100[ [100-110[
Cost ($) CumulativeFrequency
CumulativeRelativeFrequency
CumulativePercent Frequency
2 15 31 38 45 50
0 .04
.30 .62 .76 .90 1.00
4 30 62 76 90 100
2 + 13
15/50 .30(100)
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Ogive
An ogive is a graph of a cumulative distribution. The data values are shown on the horizontal axis. Shown on the vertical axis are the:• cumulative frequencies, or• cumulative relative frequencies, or• cumulative percent frequencies
The frequency (one of the above) of each class is plotted as a point.
The plotted points are connected by straight lines.
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PartsCost ($) PartsCost ($)
2020
4040
6060
8080
100100
Cu
mu
lati
ve P
erc
en
t Fr
eq
uen
cyC
um
ula
tive P
erc
en
t Fr
eq
uen
cy
50 60 70 80 90 100 11050 60 70 80 90 100 110
(89.5, 76)
Ogive with Cumulative Percent Frequencies
Tune-up Parts Cost
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End of Chapter 2