Download - OPRE 6301-SYSM 6303 Chapter 04 - Students
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OPRE 6301/SYSM 6303Quantitative Introduction to Risk and
Uncertainty in Business
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Chapter FourNumerical Descriptive Techniques
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Measures of Central Location
Arithmetic Mean
Mean or Average
Population Mean:
Sample Mean:
xi = the data values labeled 1 through N or n
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N
xN
i
i 1
n
x
x
n
i
i 1
Measures of Central Location
Mean Time Spent on the Internet
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0 7 12 5 33 14 8 0 9 22
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Measures of Central Location
Median
The middle value
Median Time Spent on the Internet
Must first sort the date into ascending order
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0 0 5 7 8 9 12 14 22 33
Measures of Central Location
Median Long-Distance Telephone Bill
Recall our Long-Distance Telephone Bill
Data from Data File Xm03-01.xls
Interpretation: half the telephone bills are
below 26.905 and half are above 26.905
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Measures of Central Location
Mode
Observation that occurs most frequently
Mode Time Spent on the Internet
Best to sort the date into ascending order
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0 0 5 7 8 9 12 14 22 33
Measures of Central Location
Mode Long-Distance Telephone Bill
Recall our Long-Distance Telephone Bill
Data from Data File Xm03-01.xls
Two issues with the mode:
may not be “central”
may not be unique4-9
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Measures of Variability
Range
Largest observation – Smallest observation
Advantage: simple
Disadvantage: simple
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Measures of Variability
Range
Set 1
Set 2
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4 4 4 4 4 50
4 8 15 24 39 50
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Measures of Variability
Variance
Population Variance:
Sample Variance:
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N
xN
i
i
1
2
2
1
1
2
2
n
xx
s
n
i
i
Why dividing by n-1.pdf
Measures of Variability
Calculating Sample Variance
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8 4 9 11 3
ix xxi 2xxi
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Measures of Variability
Standard Deviation
Population St. Deviation:
Sample St. Deviation:
Recall our Long-Distance Telephone Bill
Data from Data File Xm03-01.xls4-17
2
2ss
Measures of Variability
Comparing Two Data Sets
Let’s explore the interpretation of variance
using Excel and Data File Xm04-08.xls
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Empirical Rule
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Empirical Rule - example
A histogram of returns on an investment is bell-shaped and has a mean of 10% and a s = 8%.
How is the distribution of the returns?
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Chebysheff’s Theorem
The proportion of observations
in any sample or population that lie
within k standard deviations of the meanis at least
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1for 1
12
kk
Chebysheff’s Theorem –example
Salaries of a computer store are positively
skewed. Mean =$28,000, std.dev=$3,000.
What can you say about these salaries?
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Measures of Variability
Coefficient of Variation
ratio of standard deviation to mean
Population Coeff. Of Variation:
Sample Coeff. Of Variation:
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CV
x
scv
Measures of Relative Standing
Percentile
the Pth percentile is the value for which
P% are less than that value and
(100-P)% are greater than that value
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Measures of Relative Standing
Quartiles
Values that divide our data set into fourths
Q1 = 25th percentile
Q2 = 50th percentile
Q3 = 75th percentile
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Measures of Relative Standing
The location of a percentile
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100
1P
nLP
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Measures of Relative Standing
Calculate the 25th, 50th and 75th Percentiles
for Time Spent on the Internet(Quartiles)
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0 0 5 7 8 9 12 14 22 33
Measures of Relative Standing
Interquartile Range
another measure of variability
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13 QQIQR
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Box Plot
Graph of five statistics
Minimum and maximum data values
1st, 2nd and 3rd Quartiles
Determine outliers using IQR
1.5 times IQR less than Q1
1.5 times IQR larger than Q3
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Measures of Linear Relationship
Covariance
Population Covariance:
Sample Covariance:
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N
yxN
i
yixi
xy
1
11
n
yyxx
s
n
i
ii
xy
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Measures of Linear Relationship
Calculating Covariance
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x y
4 24
9 27
2 18
mean 5 23
xxi yy i yyxx ii
Measures of Linear Relationship
Coefficient of Correlation
Population Correlation:
Sample Correlation:
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yx
xy
xy
yx
xy
xyss
sr
11 11 r
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Measures of Linear Relationship
Least Squares Method
Simple Linear Regression(simple => two variables only)
Two variables
Independent Variable
Dependent Variable
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Measures of Linear Relationship
Which variable is which?
Example:
Salary vs. Grocery Bill
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Measures of Linear Relationship
Let’s explore the Least Squares Method
using Data File Xm04-17.xls
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Measures of Linear Relationship
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Regression Line
+e
-e
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Measures of Linear Relationship
Least Squares Method
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xbby 10ˆ
bmxy
Measures of Linear Relationship
Least Squares Method
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21
x
xy
s
sb
xbyb 10