1. Statistics For Managers 5th Edition
Chapter 3
Numerical Descriptive Measures

2. Chapter Topics Measures of central tendency Measure of variation Shape
Mean, median, mode, weighted mean, geometric mean, quartiles
Measure of variation
Range, interquartile range, average deviation, variance and standard deviation, coefficient of variation, standard units, Sharpe ratio, Sortino ratio
Shape

3. Coefficient of Variation
Summary Measures
Summary Measures
Central Tendency
Variation
Quartile
Mean
Mode
Coefficient of Variation
Median
Range
Variance
Standard Deviation
Geometric Mean

4. Measures of Central Tendency
Average
Median
Mode
Geometric Mean

5. Mean (Arithmetic Mean)
Mean (arithmetic mean) of data values
Sample mean
Population mean
Sample Size
Population Size

6. Mean (Arithmetic Mean)
(continued)
The most common measure of central tendency
Most commonly used average
Mean = 5
Mean = 6

7. Mean of Grouped Data = 20 660 33 X S (M • F) n (F) (M) Class Frequency
Mid-Point
10 but under 20 3 15
20 but under 30 6 25
30 but under 40 5 35
40 but under 50 4 45
50 but under 60 2 55 20
M • F 45
150
175
180
110
660 = 20
660 33 X
S (M • F) n

8. Advantages and Disadvantages of the Arithmetic Mean
Familiar and Easy to Understand
Easy to Calculate
Always Exists
Is Unique
Lends Itself to Further Calculation
Affected by Extreme Values

9. Median Robust measure of central tendency
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
Median = 5
Median = 5

10. Median of Group Data Median Class Step 1: Locate Median Term MT = n 2
Frequency
10 but under 20
20 but under 30
30 but under 40
40 but under 50
50 but under 60 3 6
Median Class 5 4 2 20
Step 1: Locate Median Term
MT = n 2 = 20
= 10
Step 2: Assign a Value to the Median Term
MD = L+
(MT - SFP)
FMD
•(i)
= 30+ 5
(10 - 9)
•10 = 32

11. Advantages and Disadvantages of the Median
Easy to Understand
Easy to Calculate
Always Exists
Is Unique
Not Affected by Extreme Values
Only Indicates Middle Value

12. 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
No Mode
Mode = 9

13. Weighted Mean Used when observations differ in relative importance
Xw = X1W1 + X2W2 + …….. + XnWn
W1 + W2 + …….. + Wn

14. Weighted Mean If you bought 100 shares of a stock at $20 per
share, 400 shares at $30 per share, and 500
shares at $40 per share, what would your
average cost per share be?
Xw = 100(20) + 400(30)+500(40) =$34

15. Geometric Mean
Useful in the measure of rate of change of a variable over time
Geometric mean rate of return
Measures the status of an investment over time

16. 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:

17. Quartiles Split Ordered Data into 4 Quarters Position of i-th Quartile
and Are Measures of Noncentral Location
= Median, A Measure of Central Tendency
25%
25%
25%
25%
Data in Ordered Array:

18. Measures of Variation Variation Variance Standard Deviation
Coefficient of Variation
Range
Population
Variance
Population
Standard
Deviation
Sample
Variance
Sample
Standard
Deviation
Interquartile Range

19. Range Measure of variation
Difference between the largest and the smallest observations:
Ignores the way in which data are distributed
Range = = 5
Range = = 5

20. Interquartile Range Measure of variation Also known as midspread
Spread in the middle 50%
Difference between the first and third quartiles
Not affected by extreme values
Data in Ordered Array:

21. Average Deviation X = S X n 25 5 = 5 = AD 12 5 = 2.4 n S X 1 3 6 9 25
- 4
- 2
+ 1
+ 4
(X-X) 4 2 1 12
X-X X =
S X n 25 5
= 5 = AD 12 5
= 2.4 n
X-X S

22. Standard Deviation Most important measure of variation
Shows variation about the mean
Has the same units as the original data
Sample standard deviation:
Population standard deviation:

23. Calculation Example: Sample Standard Deviation
Sample Data (Xi) :
n = Mean = X = 16
A measure of the “average” scatter around the mean

24. Standard Deviation of Grouped Data2
324 64 4
144
484
(M - X)
972
384 20
576
968
2920
2•F
(M - X)
10 but under 20 3 15
20 but under 30 6 25
30 but under 40 5 35
40 but under 50 4 45
50 but under 60 2 55 20
Classes F M
-18 -8 +2
+12
+22
(M - X)
s = 1 - n å [ ( M ) X 2
• F ] =
2920 19
12.39

25. Comparing Standard Deviations
Data A
Mean = 15.5
s = 3.338
Data B
Mean = 15.5
s = .9258
Data C
Mean = 15.5
s = 4.57

26. Variance Important measure of variation Shows variation about the mean
Sample variance:
Population variance:

27. Coefficient of Variation Used to Compare Relative Variation in Two or More Data Sets
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

28. Comparing Coefficient of Variation
Stock A:
Average price last year = $50
Standard deviation = $5
Stock B:
Average price last year = $100
Coefficient of variation:

29. 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, The industry in which sales rep Paula works has average annual sales of $4,800,000 with a standard deviation of $600, Last year Rep Bill’s sales were $4,000,000 and Rep Paula’s sales were $6,000, Which of the representatives would you hire if you had one sales position to fill?

30. Standard Units Used to Compare Relative Positions of Individual Observations in Two or More Data Sets
Sales person Bill
mB= $2,500,000
sB= $500,000
XB= $4,000,000
Sales person Paula
mP=$4,800,000
sP= $600,000
XP= $6,000,000 ZB
XB - mB sB =
4,000,000 – 2,500,000
500,000 +3 ZP =
XP - mP sP
6,000,000 – 4,800,000
600,000 +2

31. 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.

32. 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 = ( )/ 2.5 = 3.20
Doesn’t penalize for positive upside returns which the Sharpe ratio does

33. Shape of a Distribution
Describes how data is distributed
Measures of shape
Symmetric or skewed
Left-Skewed
Symmetric
Right-Skewed
Mean < Median < Mode
Mean = Median =Mode
Mode < Median < Mean

34. 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

35. 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