Chap 3-1

Statistics For Managers5th Edition

Chapter 3Numerical Descriptive

Measures

Chap 3-2

Chapter Topics

Measures of central tendency 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

Chap 3-3

Summary Measures

Central Tendency

MeanMedian

Mode

Quartile

Geometric Mean

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of Variation

Range

Chap 3-4

Measures of Central Tendency

Central Tendency

Average Median Mode

Geometric Mean1

1

n

ii

N

ii

XX

n

X

N

1/

12

n

Gn XXXX

Chap 3-5

Mean (Arithmetic Mean)

Mean (arithmetic mean) of data values Sample mean

Population mean

1 1 2

n

ii n

XX X X

Xn n

1 1 2

N

ii N

XX X X

N N

Sample Size

Population Size

Chap 3-6

Mean (Arithmetic Mean)

The most common measure of central tendency

Most commonly used average

(continued)

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Mean = 5 Mean = 6

Chap 3-7

Mean of Grouped Data

=20660 = 33X

M • F)n=

M • F

45150175180110660

(F) (M)Class Frequency Mid-Point

10 but under 20 3 1520 but under 30 6 2530 but under 40 5 3540 but under 50 4 4550 but under 60 2 55

20

Chap 3-8

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

Chap 3-9

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

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Median = 5 Median = 5

Chap 3-10

Median of Group Data

Step 2: Assign a Value to the Median Term

MD = L+(MT - FP)

FMD

•(i)= 30+5

(10 - 9)•10 = 32

Median Class

Step 1: Locate Median Term MT = n2 = 20

2 = 10

3

6

5

4

(F)Class Frequency

10 but under 20

20 but under 30

30 but under 40

40 but under 50

50 but under 60 2

20

Chap 3-11

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

Chap 3-12

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

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 3-13

Weighted Mean

Used when observations differ in relative importance

Xw = X1W1 + X2W2 + …….. + XnWn

W1 + W2 + …….. + Wn

Chap 3-14

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 youraverage cost per share be?

Xw = 100(20) + 400(30)+500(40) =$34

100 + 400 +500

Chap 3-15

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

1/

1 2

n

G nX X X X

1/

1 21 1 1 1n

G nR R R R

Chap 3-16

ExampleAn investment of $100,000 declined to $50,000 at

the end of year one and rebounded to $100,000 at end of year two:

1 2 3$100,000 $50,000 $100,000X X X

1/ 2

1/ 2 1/ 2

Average rate of return:

( 50%) (100%)25%

2Geometric rate of return:

1 50% 1 100% 1

0.50 2 1 1 1 0%

G

X

R

Chap 3-17

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%

1Q 2Q 3Q

Data in Ordered Array: 11 12 13 16 16 17 18 21 22

1 1

1 9 1 12 13Position of 2.5 12.5

4 2Q Q

1Q 3Q

2Q

1

4i

i nQ

Chap 3-18

Measures of Variation

Variation

Variance Standard Deviation Coefficient of Variation

PopulationVariance

Sample

Variance

PopulationStandardDeviationSample

Standard

Deviation

Range

Interquartile Range

Chap 3-19

Range

Measure of variation Difference between the largest and the

smallest observations:

Ignores the way in which data are distributed

Largest SmallestRange X X

7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5

Chap 3-20

Measure of variation Also known as midspread

Spread in the middle 50% Difference between the first and third

quartiles

Not affected by extreme values

3 1Interquartile Range 17.5 12.5 5Q Q

Interquartile Range

Data in Ordered Array: 11 12 13 16 16 17 17 18 21

Chap 3-21

Average Deviation

X = Xn

=25

5= 5

X

1

3

6

9

6

25

- 4

- 2

+ 1

+ 4

+ 1

0

(X-X)

4

2

1

4

1

12

X-X

=AD 12

5= 2.4

n

X-X=

Chap 3-22

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:

2

1

1

n

ii

X XS

n

2

1

N

ii

X

N

Chap 3-23

Calculation Example:Sample Standard

Deviation

Sample Data (Xi) : 10 12 14 15 17 18 18 24

n = 8 Mean = X = 16

4.30957

130

18

16)(2416)(1416)(1216)(10

1n

)X(24)X(14)X(12)X(10S

2222

2222

A measure of the “average” scatter around the mean

Chap 3-24

Standard Deviation of Grouped Data

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 =

1n

[ M X2• F]

=292019

= 12.39

2

324

64

4

144

484

(M - X)972

384

20

576

968

2920

2•F(M - X)

Chap 3-25

Comparing Standard Deviations

Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21

11 12 13 14 15 16 17 18 19 20 21

Data B

Data A

Mean = 15.5 s = .9258

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5 s = 4.57

Data C

Chap 3-26

2

2 1

N

ii

X

N

Important measure of variation Shows variation about the mean

Sample variance:

Population variance:

2

2 1

1

n

ii

X XS

n

Variance

Chap 3-27

Coefficient of VariationUsed 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 100%

SCV

X

Chap 3-28

Comparing Coefficient of Variation

Stock A: Average price last year = $50 Standard deviation = $5

Stock B: Average price last year = $100 Standard deviation = $5

Coefficient of variation: Stock A:

Stock B:

$5100% 100% 10%

$50

SCV

X

$5100% 100% 5%

$100

SCV

X

Chap 3-29

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?

Using z scores to evaluate performance(Example)

Chap 3-30

Standard UnitsUsed to Compare Relative Positions of Individual

Observations in Two or More Data Sets

Sales person Bill

B= $2,500,000

= $500,000

XB= $4,000,000

Sales person Paula

P=$4,800,000

P= $600,000

XP= $6,000,000

ZB

XB - B

B= =

4,000,000 – 2,500,000500,000 =

+3

ZP =

XP - P

P

6,000,000 – 4,800,000600,000 =

+2=

Chap 3-31

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 = 2.29Generally, the higher the better.

Chap 3-32

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 = (10.5-2.5)/ 2.5 = 3.20 Doesn’t penalize for positive upside

returns which the Sharpe ratio does

Chap 3-33

Shape of a Distribution

Describes how data is distributed Measures of shape

Symmetric or skewed

Mean = Median =Mode Mean < Median < Mode Mode < Median < Mean

Right-SkewedLeft-Skewed Symmetric

Chap 3-34

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

Chap 3-35

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