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```Statistics for Managers
4th Edition
Chapter 3
Numerical Descriptive Measures
Chap 3-1
Chapter Topics

Measures of central tendency


Measure of variation


Mean, median, mode, geometric mean, Quartile
Range, interquartile range, average deviation,
variance and standard deviation, coefficient of
variation, standard units, Sharpe ratio, Sortino
ratio
Shape
Chap 3-2
Chapter Topics

(continued)
Ethical considerations
Chap 3-3
Summary Measures
Summary Measures
Central Tendency
Mean
Quartile
Mode
Median
Range
Variation
Coefficient of
Variation
Variance
Geometric Mean
Standard Deviation
Chap 3-4
Measures of Central Tendency
Central Tendency
Average
Median
Mode
n
X 
X
i 1
N

i 1
Geometric Mean
X G   X1  X 2 
n
X
i
 Xn 
1/ n
i
N
Chap 3-5
Mean (Arithmetic Mean)

Mean (arithmetic mean) of data values

Sample mean
Sample Size
n
X

X
i 1
i
n
X1  X 2 

n
 Xn
Population mean
Population Size
N

X
i 1
N
i
X1  X 2 

N
 XN
Chap 3-6
Mean (Arithmetic Mean)
(continued)


The most common measure of central
tendency
Affected by extreme values (outliers)
0 1 2 3 4 5 6 7 8 9 10
Mean = 5
0 1 2 3 4 5 6 7 8 9 10 12 14
Mean = 6
Chap 3-7
Mean of Grouped Data
Class
10 but under 20
20 but under 30
30 but under 40
40 but under 50
50 but under 60
(F)
(M)
Frequency Mid-Point M • F
45
3
15
150
6
25
175
5
35
180
4
45
110
2
55
660
20
S M • F)
660 = 33
X =
=
n
20
Chap 3-8
Median


Robust measure of central tendency
Not affected by extreme values
0 1 2 3 4 5 6 7 8 9 10
Median = 5

0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5
In an ordered array, the median is the
“middle” number


If n or N is odd, the median is the middle number
If n or N is even, the median is the average of the
two middle numbers
Chap 3-9
Median of Group Data
Class
10 but under 20
20 but under 30
30 but under 40
40 but under 50
50 but under 60
(F)
Frequency
3
6
5
4
2
Median Class
20
Step 1: Locate Median Term
MT =
n =
2
20
2
= 10
Step 2: Assign a Value to the Median Term
(10 - 9)
(MT - SFP)
•(i)= 30+
MD = L+
FMD
5
•10 = 32
Chap 3-10
Mode






A measure of central tendency
Value that occurs most often
Not affected by extreme values
Used for either numerical or categorical data
There may may be no mode
There may be several modes
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
0 1 2 3 4 5 6
No Mode
Chap 3-11
Geometric Mean

Useful in the measure of rate of change of a
variable over time
X G   X1  X 2 

 Xn 
1/ n
Geometric mean rate of return

Measures the status of an investment over time
RG  1  R1   1  R2  
 1  Rn  
1/ n
1
Chap 3-12
Example
An investment of \$100,000 declined to \$50,000 at the
end of year one and rebounded to \$100,000 at end of
year two:
X1  \$100,000
X 2  \$50,000
X 3  \$100,000
Average rate of return:
(50%)  (100%)
X
 25%
2
Geometric rate of return:
RG  1   50%    1  100%   
1/ 2
  0.50    2  
1/ 2
1
 1  1  1  0%
1/ 2
Chap 3-13
Quartiles

Split Ordered Data into 4 Quarters
25%
25%
 Q1 

25%
 Q2 
Position of i-th Quartile
25%
Q3 
i  n  1
 Qi  
4
Data in Ordered Array: 11 12 13 16 16 17 18 21 22
1 9  1
Position of Q1 
 2.5
4
Q1
12  13


 12.5
2
Q1 and Q3 Are Measures of Noncentral Location
 Q = Median, A Measure of Central Tendency
2

Chap 3-14
Measures of Variation
Variation
Variance
Range
Population
Variance
Sample
Variance
Interquartile Range
Standard Deviation
Coefficient
of Variation
Population
Standard
Deviation
Sample
Standard
Deviation
Chap 3-15
Range


Measure of variation
Difference between the largest and the
smallest observations:
Range  X Largest  X Smallest

Ignores the way in which data are distributed
Range = 12 - 7 = 5
Range = 12 - 7 = 5
7
8
9
10
11
12
7
8
9
10
11
12
Chap 3-16
Interquartile Range


Measure of variation


Difference between the first and third
quartiles
Data in Ordered Array: 11 12 13 16 16 17
17 18 21
Interquartile Range  Q3  Q1  17.5  12.5  5

Not affected by extreme values
Chap 3-17
Average Deviation
X
(X-X)
X-X
1
-4
4
3
-2
2
6
+1
1
9
+4
4
6
+1
1
25
0
12
X = S X = 25 = 5
n
5
S X-X
n
12
=
= 2.4
5
Chap 3-18
Variance


Important measure of variation

Sample variance:
n
S 
2

 X
i 1
X
i
2
n 1
Population variance:
N
 
2
 X
i 1
i

N
2
Chap 3-19
Standard Deviation



Most important measure of variation
Has the same units as the original data

Sample standard deviation:
n
S

Population standard deviation:

 X
i 1
X
i
2
n 1
N
 X
i 1
i

2
N
Chap 3-20
Comparing Standard Deviations
Data A
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = 3.338
Data B
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = .9258
Data C
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = 4.57
Chap 3-21
Coefficient of Variation

Measures relative variation

Always in percentage (%)

Shows variation relative to mean


Is used to compare two or more sets of data
measured in different units
S
CV  
X

100%

Chap 3-22
Comparing Coefficient
of Variation

Stock A:



Stock B:



Average price last year = \$50
Standard deviation = \$5
Average price last year = \$100
Standard deviation = \$5
Coefficient of variation:

Stock A:

Stock B:
S
CV  
X

 \$5 
100%  
100%  10%

 \$50 
S
CV  
X

 \$5 
100%  
100%  5%

 \$100 
Chap 3-23
Using z
scores to evaluate performance
(Example)
The industry in which sales rep Bill works has
average annual sales of \$2,500,000 with a
standard deviation of \$500,000.
The industry in which sales rep Paula works has
average annual sales of \$4,800,000 with a
standard deviation of \$600,000.
Last year Rep Bill’s sales were \$4,000,000 and
Rep Paula’s sales were \$6,000,000.
Which of the representatives would you hire if
you had one sales position to fill?
Chap 3-24
Standard Units
Sales person Bill
Sales person Paula
B= \$2,500,000
P=\$4,800,000
B= \$500,000
P= \$600,000
XB= \$4,000,000
XP= \$6,000,000
ZB =
ZP
=
XB - B
4,000,000
–
2,500,000
=
= +3
B
500,000
XP - P
6,000,000
–
4,800,000
=
= +2
P
600,000
Chap 3-25
SHARPE RATIO

Sharpe ratio = (Prr – RFrr)/Srr

Where:



Prr = Annualized average return on the portfolio
RFrr = Annualized average return on risk free
proxy
Srr = Annualized standard deviation of average
returns
Sharpe R = (10.5 – 2.5)/ 3.5 = 2.29
Generally, the higher the better.
Chap 3-26
SORTINO RATIO

Sortino Ratio = (Prr – RFrr)/Srr(downside)

Where:





Prr = Annualized rate of return on portfolio
RFrr= Annualized risk free annualized rate of
return on portfolio
Srr(downside) = downside semi-standard
deviation
Sortino = (10.5-2.5)/ 2.5 = 3.20
Doesn’t penalize for positive upside returns
which the Sharpe ratio does
Chap 3-27
Shape of a Distribution

Describes how data is distributed

Measures of shape

Symmetric or skewed
Left-Skewed
Mean < Median < Mode
Symmetric
Mean = Median =Mode
Right-Skewed
Mode < Median < Mean
Chap 3-28
Ethical Considerations
Numerical descriptive measures:



Should document both good and bad results
Should be presented in a fair, objective and
neutral manner
Should not use inappropriate summary
measures to distort facts
Chap 3-29
Chapter Summary

Described measures of central tendency

Mean, median, mode, geometric mean

Discussed quartile

Described measure of variation


Range, interquartile range, average deviation, variance,
and standard deviation, coefficient of variation, standard
units, Sharp ratio, Sortino ratio
Illustrated shape of distribution

Symmetric, skewed
Chap 3-30
```
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