OPRE 6301-SYSM 6303 Chapter 04 Slides PDF

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OPRE 6301/SYSM 6303 Statistics and Data Analysis

4-1

Chapter Four Numerical Descriptive Techniques

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1

Measures of Central Location Arithmetic Mean Mean or Average N

x

i

Population Mean:



i 1

N n

x

i

Sample Mean:

x

i 1

n

xi = the data values labeled 1 through N or n 4-3

Measures of Central Location Mean Time Spent on the Internet 0

7

12

5

33

14

8

0

9

22

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2

Measures of Central Location Mean Long-Distance Telephone Bill Recall our Long-Distance Telephone Bill Data from Data File Xm03-01.xls

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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 0

0

5

7

8

9

12

14

22

33

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3

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 4-7

Measures of Central Location Mode Observation that occurs most frequently Mode Time Spent on the Internet Best to sort the date into ascending order 0

0

5

7

8

9

12

14

22

33

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4

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 unique 4-9

Histogram Shapes Symmetrical

Skewed

Positively or Right Skewed

Negatively or Left Skewed

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5

Measures o Variability Range Largest observation – Smallest observation Advantage: simple Disadvantage: simple

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Measures o Variability Range Set 1 4

4

4

4

4

50

39

50

Set 2 4

8

15

24

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6

Measures of Variability Variance N

Population Variance:

 x   

2

i

 2

i1

N n

Sample Variance:

 x  x 

2

i

s2 

i 1

n 1

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Measures of Variability Calculating Sample Variance 8 xi

4

9

xi  x 

11

3

xi  x 2

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7

Measures of Variability Understanding the Sample Variance 8

4

9

11

3

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Measures o Variability Variance of Long-Distance Telephone Bill Recall our Long-Distance Telephone Bill Data from Data File Xm03-01.xls

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8

Measures of Variability Standard Deviation Population St. Deviation:

  2

Sample St. Deviation:

s  s2

Recall our Long-Distance Telephone Bill Data from Data File Xm03-01.xls 4-17

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 bellshaped and has a mean of 10% and a s = 8%. How is the distribution of the returns?

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Chebysheff’ Theorem The proportion of observations in any sample or population that lie within k standard deviations of the mean is at least 1

1 for k  1 k2

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Cheby

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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11

Measures of Variability Coefficient of Variation ratio of standard deviation to mean Population Coeff. Of Variation:

CV 

 

Sample Coeff. Of Variation:

cv 

s x

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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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12

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

LP  n  1

P 100

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Measures of Relative Standing Calculate the 25th, 50th and 75th Percentiles for Time Spent on the Internet (The Quartiles) 0

0

5

7

8

9

12

14

22

33

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Measures of Relative Standing Calculate the 25th, 50th and 75th Percentiles for Time Spent on the Internet(Quartiles) 0

0

5

7

8

9

12

14

22

33

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Measures of Relative Standing Calculate the 25th, 50th and 75th Percentiles for Time Spent on the Internet(Quartiles) 0

0

5

7

8

9

12

14

22

33

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Measures of Relative Standing Calculate the 25th, 50th and 75th Percentiles for Time Spent on the Internet(Quartiles) 0

0

5

7

8

9

12

14

22

33

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Measures of Relative Standing Interquartile Range another measure of variability

IQR  Q3  Q1

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Measures o Relative Standing IQR of Long-Distance Telephone Bills Recall our Long-Distance Telephone Bill Data from Data File Xm03-01.xls

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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 4-33

Box Plot Recall our Long-Distance Telephone Bill Data from Data File Xm03-01.xls

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Box Plot Using Box Plots for the Comparison of Multiple Data Sets Let’s explore this technique using Excel and Data File Xm04-15.xls

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Measures of Linear Relationship Covariance N

Population Covariance:

  x  yi   y 

x

i

 xy 

i 1

N n

Sample Covariance:

 x  xy  y  i

s xy 

i

i 1

n 1

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18

Measures of Linear Relationship Calculating Covariance

mean

x

y

4

24

9

27

2

18

5

23

xi  x  yi  y  xi  x yi  y 

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Measures of Linear Relationship Coefficient of Correlation Population Correlation:

 xy 

Sample Correlation:

rxy 

 1    1

 xy  x y sxy s x sy

 1  r  1

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Measures of Linear Relationship Covariance Coefficient of Correlation Let’s explore these calculations using Excel and Data File Xm04-08

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Measures of Linear Relationship Least Squares Method Simple Linear Regression (simple => two variables only) Two variables Independent Variable Dependent Variable 4-40

20

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

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Measures of Linear Relationship Regression Line +e

-e

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Measures of Linear Relationship Least Squares Method

yˆ  b0  b1x y  mx  b

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Measures of Linear Relationship Least Squares Method

b1 

sxy s2x

b0  y  b1x 4-45

Measures of Linear Relationship Least Squares Method Let’s explore the Least Squares Method using Excel and Data File Xm04-17

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