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**Unformatted text preview: **Business Statistics (BUSA 3101)
Dr. Lari H. Arjomand
lariarjomand@clayton.edu Slide 1 Chapters 2 & 3 (Part A) Descriptive Statistics:
Tabular and Graphical Presentations
s
s Summarizing Qualitative Data
Summarizing Quantitative Data
Typ es o f Dat a
Ty p
D at
Data
Data Numerical
Numerical Categorical
Categorical (Quantitative)
(Quantitative) Discrete
Discrete (Qualitative)
(Qualitative) Continuous
Continuous Slide 2 Summarizing Qualitative Data
s
s
s
s
s Frequency Distribution
Relative Frequency Distribution Percent Frequency Distribution
Bar Graph
Pie Chart Slide 3 Construction of a Frequency Distribution
Graph Raw data Question
to be
addressed Collect
Collect
data
data Organize
Organize
data
data Present
Present
data
data Draw
Draw
conclusion
conclusion Frequency
distribution Slide 4 Frequency Distribution A frequency distribution is a tabular summary of
A frequency distribution is a tabular summary of data showing the frequency (or number) of items
data showing the frequency (or number) of items in each of several nonoverlapping classes.
in each of several nonoverlapping classes. The objective is to provide insights about the data
The objective is to provide insights about the data that cannot be quickly obtained by looking only at
that cannot be quickly obtained by looking only at the original data.
the original data. Slide 5 Example: Marada Inn
Guests staying at Marada Inn were
asked to rate the quality of their accommodations as being excellent,
above average, average, below average, or
poor. The ratings provided by a sample of 20 guests are: Below Average Above Average Above Average Average Above Average Average Above Average Average Above Average Below Average Poor Excellent Above Average Average Above Average Above Average Below Average Poor Above Average Average Average Slide 6 Frequency Distribution Rating
Frequency 2
Poor 3
Below Average 5
Average 9
Above Average 1
Excellent
Total 20 Slide 7 Relative Frequency Distribution The relative frequency of a class is the fraction or
The relative frequency of a class is the fraction or proportion of the total number of data items
proportion of the total number of data items belonging to the class.
belonging to the class. A relative frequency distribution is a tabular
A relative frequency distribution is a tabular summary of a set of data showing the relative
summary of a set of data showing the relative frequency for each class.
frequency for each class. Slide 8 Percent Frequency Distribution The percent frequency of a class is the relative
The percent frequency of a class is the relative frequency multiplied by 100.
frequency multiplied by 100. A percent frequency distribution is a tabular
A percent frequency distribution is a tabular summary of a set of data showing the percent
summary of a set of data showing the percent frequency for each class.
frequency for each class. Slide 9 Relative Frequency and
Percent Frequency Distributions Relative
Frequency
Rating .10
Poor .15
Below Average .25
Average .45
Above Average .05
Excellent
Total 1.00 Percent
Frequency 10 15 25 .10(100) = 10 45 5 100
1/20 = .05 Slide 10 Bar Graph A bar graph is a graphical device for presenting qualitative 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. Slide 11 Bar Graph
Marada Inn Quality Ratings 10
9
Frequency 8
7
6
5
4
3
2
1
Poor Below Average Above Excellent
Average
Average Rating Slide 12 Pie Chart The pie chart is a commonly used graphical device for presenting relative frequency distributions for qualitative data.
s First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to the relative frequency for each class. s Since there are 360 degrees in a circle, a class with a relative frequency of .25 would consume .25(360) = 90 degrees of the circle. Slide 13 Pie Chart Marada Inn Quality Ratings
Excellent 5% Poor
10% Below Above
Average 45% Average 15%
Average 25% Slide 14 Example: Marada Inn
s Insights Gained from the Preceding Pie Chart • Onehalf of the customers surveyed gave Marada a quality rating of “above average” or “excellent” (looking at the left side of the pie). This might please the manager. • For each customer who gave an “excellent” rating, there were two customers who gave a “poor” rating (looking at the top of the pie). This should displease the manager. Slide 15 Summarizing Quantitative Data
s
s
s
s
s
s Frequency Distribution
Relative Frequency and Percent Frequency Distributions
Dot Plot
Histogram
Cumulative Distributions
Ogive N u m er ic al (Quantitative)
Nu
(Quantitative)
D at a Pr esen t at io n
Numerical
Numerical
Data
Data Ordered
Ordered
Array
Array Stem-&-Leaf
Stem-&-Leaf
Display
Display Frequency
Frequency
Distributions
Distributions HistoHistogram
gram Polygon
Polygon Ogive Ogive Slide 16 Frequency Distribution Table
Steps
s
s s
s
s
s 1 Determine range
2 Select number of classes
• Usually between 5 and 20 inclusive
3 Compute class intervals (width)
4 Determine class boundaries (limits)
5 Compute class midpoints
6 Count observations & assign to classes Slide 17 Example: Hudson Auto Repair
The manager of Hudson Auto
would like to have a better
understanding of the cost
of parts used in the engine
tuneups performed in the
shop. She examines 50
customer invoices for tuneups. The costs of parts,
rounded to the nearest dollar, are listed on the next
slide. Slide 18 Example: Hudson Auto Repair
s Sample of Parts Cost for 50 Tuneups 91
71
104
85
62 78
69
74
97
82 93
72
62
88
98 57
89
68
68
101 75
66
97
83
79 52
75
105
68
105 99
79
77
71
79 80
75
65
69
69 97
72
80
67
62 62
76
109
74
73 Slide 19 Frequency Distribution
s 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 Slide 20 Frequency Distribution (Continued)
s Guidelines for Selecting Width of Classes
•Use classes of equal width. •Approximate Class Width = Largest Data Value − Smallest Data Value
Number of Classes Slide 21 Example: Frequency Distribution
For Hudson Auto Repair, if we choose six classes: Approximate Class Width = (109 52)/6 = 9.5 ≅ 10
10 Frequency
Parts Cost ($) 5059 2 6069 13 7079 16 8089 7 9099 7 100109 5
Total 50 Slide 22 Solution Using SWStat+
s After putting your data into Excel, then create Data Area: Slide 23 Solution Using SWStat (cont.)
s
s SwStat
Statistics
Tabulations and Histograms
Statistics
Tabulations
Note that here we are using 10 classes—you can use 6 classes if you
Note
10
would like to do so.
would Slide 24 Solution Using SWStat (cont.)
s Results: Slide 25 Relative Frequency and
Percent Frequency Distributions
Parts Relative
Percent Cost ($)
Frequency Frequency 5059 .04 4 6069 .26
2/50 26 .04(100) 7079 .32 32 8089 .14 14 9099 .14 14 100109 .10 10
Total 1.00 100 Slide 26 Relative Frequency and
Percent Frequency Distributions
s Insights Gained from the Percent Frequency Distribution • Only 4% of the parts costs are in the $5059 class.
• 30% of the parts costs are under $70.
• The greatest percentage (32% or almost onethird) of the parts costs are in the $7079 class. • 10% of the parts costs are $100 or more. Slide 27 Dot Plot
s s
s One of the simplest graphical summaries of data is a dot plot.
A horizontal axis shows the range of data values.
Then each data value is represented by a dot placed above the axis. Slide 28 Dot Plot
Tuneup Parts Cost . . .. . . . . .. .. .. .. . . . . . ..... .......... .. . .. . . ... . .. .
50 60 70 80 90 100 110 Cost ($) Slide 29 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. Slide 30 Histogram
18 Tuneup Parts Cost 16 Frequency 14
12
10
8
6
4
2
50−59 60−69 70−79 80−89 90−99 100110 Parts
Cost ($) Slide 31 Histogram (Continued)
Symmetric
• Left tail is the mirror image of the right tail
• Example: heights and weights of people
.35 Relative Frequency s .30
.25
.20
.15
.10
.05
0 Slide 32 Histogram (Continued)
Moderately Skewed Left
• A longer tail to the left
• Example: exam scores
.35 Relative Frequency s .30
.25
.20
.15
.10
.05
0 Slide 33 Histogram (Continued)
Moderately Right Skewed
• A Longer tail to the right
• Example: housing values
.35 Relative Frequency s .30
.25
.20
.15
.10
.05
0 Slide 34 Histogram (Continued)
Highly Skewed Right
• A very long tail to the right
• Example: executive salaries
.35 Relative Frequency s .30
.25
.20
.15
.10
.05
0 Slide 35 Cumulative Distributions Cumulative frequency distribution − shows the
Cumulative frequency distribution − shows the number of items with values less than or equal to
number of items with values less than or equal to the upper limit of each class..
the upper limit of each class.. Cumulative relative frequency distribution – shows
Cumulative relative frequency distribution – shows the proportion of items with values less than or
the proportion of items with values less than or equal to the upper limit of each class.
equal to the upper limit of each class. Cumulative percent frequency distribution – shows
Cumulative percent frequency distribution – shows the percentage of items with values less than or
the percentage of items with values less than or equal to the upper limit of each class.
equal to the upper limit of each class. Slide 36 Cumulative Distributions
s Example: Hudson Auto Repair Cumulative Cumulative Cumulative
Relative
Percent Frequency
Frequency
Cost ($) Frequency 2 .04 < 59 4 15 .30 < 69 30 31 2 + 13 .62 < 79
15/50 62 .30(100) 38 .76 < 89 76 45 .90 < 99 90 1.00 < 109 50 100 Slide 37 Ogive
s An ogive is a graph of a cumulative distribution. s The data values are shown on the horizontal axis. s Shown on the vertical axis are the:
• cumulative frequencies, or
• cumulative relative frequencies, or
• cumulative percent frequencies s The frequency (one of the above) of each class is plotted as a point. s The plotted points are connected by straight lines. Slide 38 Ogive
s Example: Hudson Auto Repair
• Because the class limits for the partscost data are 50
59, 6069, and so on, there appear to be oneunit gaps from 59 to 60, 69 to 70, and so on. • These gaps are eliminated by plotting points halfway between the class limits. • Thus, 59.5 is used for the 5059 class, 69.5 is used for the 6069 class, and so on. Slide 39 Ogive with Cumulative Percent Frequencies Cumulative Percent Frequency Tuneup Parts Cost
Tuneup Parts Cost
100
80
60 (89.5, 76) 40
20 Parts
50 60 70 80 90 100 110 Cost ($) Slide 40 Frequency Distribution Table
Another Example
Raw Data: 24, 26, 24, 21, 27, 27, 30, 41, 32, 38 Class Frequency 15 but < 25 3 25 but < 35 5 35 but < 45 2 Slide 41 Frequency Distribution Table
Example (Continued)
Raw Data: 24, 26, 24, 21, 27, 27, 30, 41, 32, 38
24 26, 24 21
41, Class Midpoint Frequency 15 but < 25
Width 20 3 25 but < 35 30 5 35 but < 45 40 2 Boundaries (Upper + Lower Boundaries) / 2 Slide 42 Stated and True (or Real) Class Limits True Classes: Are those classes such that the upper true limit of a class is the same as the lower true limit of the next class. For comparison, the stated class limits and true class limits are given in the following table—next slide: Slide 43 Stated and True (or Real) Class Limits
Stated
$600$799
$800$999 True
$599.50 up to but not including $799.50
$799.50 up to but not including $999.50 In the first column of the above table the data were rounded to the nearest dollar. For example, $799.50 was rounded up to $800 and tailed in the second class. Any amount over $799 but under 799.50 was rounded down to $799 and included in the first class. Thus, the $600$799 class actually includes all data from $599.50 inclusive up to but not including $799.50. Slide 44 Relative Frequency & % Distribution Tables
Example (Continued) The relative frequency of a class is obtained by dividing the class frequency by the total frequency, which in the following
problem = 10. Relative Frequency
Relative
Distribution
Distribution Percentage
Percentage
Distribution
Distribution Class Prop. Class % 15 but < 25 .3 15 but < 25 30.0 25 but < 35 .5 25 but < 35 50.0 35 but < 45 .2 35 but < 45 20.0 Slide 45 Cumulative Percentage Distribution Table
Example (Continued)
Raw Data: 24, 26, 24, 21, 27, 27, 30, 41, 32, 38
Raw
Class Cumulative
Percentage Percentage
Percentage
less than lower
class boundary 15 but < 25
Lower
Lower
class
boundary 0.0 25 but < 35 30.0 35 but < 45 80.0 30% + 50%
30% 45 but < 55 100.0 80% + 20% Slide 46 End of Chapters 2 & 3, Part A Slide 47 ...

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