lecture 4: measure of dispersion - university of new...

Lecture 4 Measure of Dispersion

Donglei Du(dduunbedu)

Faculty of Business Administration University of New Brunswick NB Canada FrederictonE3B 9Y2

Donglei Du (UNB) ADM 2623 Business Statistics 1 59

Table of contents1 Measure of Dispersion scale parameter

IntroductionRangeMean Absolute DeviationVariance and Standard DeviationRange for grouped dataVarianceStandard Deviation for Grouped DataRange for grouped data

2 Coefficient of Variation (CV)3 Coefficient of Skewness (optional)

Skewness Risk4 Coefficient of Kurtosis (optional)

Kurtosis Risk5 Chebyshevrsquos Theorem and The Empirical rule

Chebyshevrsquos TheoremThe Empirical rule

6 Correlation Analysis7 Case study


Layout1 Measure of Dispersion scale parameter








Introduction

Dispersion (aka variability scatter or spread)) characterizes howstretched or squeezed of the data

A measure of statistical dispersion is a nonnegative real number thatis zero if all the data are the same and increases as the data becomemore diverse

Dispersion is contrasted with location or central tendency andtogether they are the most used properties of distributions

There are many types of dispersion measures

RangeMean Absolute DeviationVarianceStandard Deviation


Range

Range = maxminusmin


Example

Example Find the range of the following data 10 7 6 2 5 4 8 1

Range = 10minus 1 = 9

Example the weight example (weightcsv)

The R code

gtweight lt- readcsv(weightcsv)

gtsec_01Alt-weight$Weight01A2013Fall

gtrange(sec_01A)

[1] 130 189



Properties of Range

Only two values are used in its calculation

It is influenced by an extreme value (non-robust)

It is easy to compute and understand


Mean Absolute Deviation

The Mean Absolute Deviation of a set of n numbers

MAD =|x1 minus micro|+ + |xn minus micro|

n

|x ‐ ||x1 ‐ |

|x ‐ |

|x3 ‐ |

|x4 ‐ |

x1 x4x2 x3

|x2 |


Example

Example A sample of four executives received the following bonuseslast year ($000) 140 150 170 160

Problem Determine the MAD

Solution

x =14 + 15 + 17 + 16

4=

62

4= 155

MAD =|14minus 155|+ |15minus 155|+ |17minus 155|+ |16minus 155|

4

=4

4= 1


Example


The R code



gtmad(sec_01A center=mean(sec_01A)constant = 1)

[1] 65


Properties of MAD

All values are used in the calculation

It is not unduly influenced by large or small values (robust)

The absolute values are difficult to manipulate


Variance

The variance of a set of n numbers as population

Var = σ2 =(x1 minus micro)2 + + (xn minus micro)2

n(conceptual formula)

=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n

n(computational formula)

The variance of a set of n numbers as sample

S2 =(x1 minus micro)2 + + (xn minus micro)2

nminus 1(conceptual formula)

=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n

nminus 1(computational formula)


Standard Deviation

The Standard Deviation is the square root of variance

Other notations σ for population and S for sample


Example Conceptual formula

Example The hourly wages earned by three students are $10 $11$13

Problem Find the mean variance and Standard Deviation

Solution Mean and variance

micro =10 + 11 + 13

3=

34

3asymp 1133333333333

σ2 asymp (10minus 1133)2 + (11minus 1133)2 + (13minus 1133)2

3

=1769 + 01089 + 27889

3=

46668

3= 15556

Standard Deviationσ asymp 1247237


Example Computational formula


Problem Find the variance and Standard Deviation

Solution

Variance

σ2 =(102 + 112 + 132)minus (10+11+13)2

3

3

=390minus 1156

3

3=

390minus 38533

3=

467

3= 1555667


If the above is sample then σ2 asymp 233335 and σ asymp 1527531


Example


The R code



Sample Variance

gtvar(sec_01A)

[1] 1463228

if you want the population variance

gtvar(sec_01A)(length(sec_01A)-1)(length(sec_01A))

[1] 1439628

Sample standard deviation

gtsd(sec_01A)

[1] 120964

if you want the population SD

gtsd(sec_01A)sqrt((length(sec_01A)-1)(length(sec_01A)))

[1] 1199845


Note

Conceptual formula may have accumulated rounding error

Computational formula only has rounding error towards the end


Properties of Variancestandard deviation


It is not extremely influenced by outliers (non-robust)

The units of variance are awkward the square of the original unitsTherefore standard deviation is more natural since it recovers heoriginal units


Range for grouped data

The range of a sample of data organized in a frequency distribution iscomputed by the following formula

Range = upper limit of the last class - lower limit of the first class


VarianceStandard Deviation for Grouped Data

The variance of a sample of data organized in a frequency distributionis computed by the following formula

S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1

where fi is the class frequency and xi is the class midpoint for Classi = 1 k


Example

Example Recall the weight example from Chapter 2

class freq (fi) mid point (xi) fixi fix2i

[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350

Solution The VarianceStandard Deviation are

S2 =1 561 350minus 98102

62

62minus 1asymp 1500793

S asymp 1225069

The real sample varianceSD for the raw data is 1463228120964














Coefficient of Variation (CV)

The Coefficient of Variation (CV) for a data set defined as the ratioof the standard deviation to the mean

CV =

σmicro for populationSx for sample

It is also known as unitized risk or the variation coefficient

The absolute value of the CV is sometimes known as relative standarddeviation (RSD) which is expressed as a percentage

It shows the extent of variability in relation to mean of the population

It is a normalized measure of dispersion of a probability distribution orfrequency distributionCV is closely related to some important concepts in other disciplines

The reciprocal of CV is related to the Sharpe ratio in FinanceWilliam Forsyth Sharpe is the winner of the 1990 Nobel Memorial Prizein Economic Sciences

The reciprocal of CV is one definition of the signal-to-noise ratio inimage processing


Example

Example Rates of return over the past 6 years for two mutual fundsare shown below

Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14

Problem Which fund has higher riskSolution The CVrsquos are given as

CVA =1319

985= 134

CVB =1029

842= 122

There is more variability in Fund A as compared to Fund BTherefore Fund A is riskierAlternatively if we look at the reciprocals of CVs which are theSharpe ratios Then we always prefer the fund with higher Sharperatio which says that for every unit of risk taken how much returnyou can obtainDonglei Du (UNB) ADM 2623 Business Statistics 25 59

Properties of Coefficient of Variation


It is only defined for ratio level of data

The actual value of the CV is independent of the unit in which themeasurement has been taken so it is a dimensionless number

For comparison between data sets with different units orwidely different means one should use the coefficient of variationinstead of the standard deviation










Coefficient of Skewness

Skewness is a measure of the extent to which a probability distributionof a real-valued random variable rdquoleansrdquo to one side of the mean Theskewness value can be positive or negative or even undefined

The Coefficient of Skewness for a data set

Skew = E[(Xminusmicro

σ

)3 ]=micro3

σ3

where micro3 is the third moment about the mean micro σ is the standarddeviation and E is the expectation operator


Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk

Skewness risk is the risk that results when observations are not spreadsymmetrically around an average value but instead have a skeweddistribution

As a result the mean and the median can be different

Skewness risk can arise in any quantitative model that assumes asymmetric distribution (such as the normal distribution) but is appliedto skewed data

Ignoring skewness risk by assuming that variables are symmetricallydistributed when they are not will cause any model to understate therisk of variables with high skewness

Reference Mandelbrot Benoit B and Hudson Richard L The(mis)behaviour of markets a fractal view of risk ruin and reward2004










Coefficient of Excess Kurtosis Kurt

Kurtosis (meaning curved arching) is a measure of the rdquopeakednessrdquoof the probability distribution

The Excess Kurtosis for a data set

Kurt =E[(X minus micro)4]

(E[(X minus micro)2])2minus 3 =

micro4

σ4minus 3

where micro4 is the fourth moment about the mean and σ is the standarddeviationThe Coefficient of excess kurtosis provides a comparison of the shapeof a given distribution to that of the normal distribution

Negative excess kurtosis platykurticsub-Gaussian A low kurtosisimplies a more rounded peak and shorterthinner tails such asuniform Bernoulli distributionsPositive excess kurtosis leptokurticsuper-Gaussian A high kurtosisimplies a sharper peak and longerfatter tails such as the Studentrsquost-distribution exponential Poisson and the logistic distributionZero excess kurtosis are called mesokurtic or mesokurtotic such asNormal distribution


Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk

Kurtosis risk is the risk that results when a statistical model assumesthe normal distribution but is applied to observations that do notcluster as much near the average but rather have more of a tendencyto populate the extremes either far above or far below the average

Kurtosis risk is commonly referred to as rdquofat tailrdquo risk The rdquofat tailrdquometaphor explicitly describes the situation of having moreobservations at either extreme than the tails of the normaldistribution would suggest therefore the tails are rdquofatterrdquo

Ignoring kurtosis risk will cause any model to understate the risk ofvariables with high kurtosis










Chebyshevrsquos Theorem

Given a data set with mean micro and standard deviation σ for anyconstant k the minimum proportion of the values that lie within kstandard deviations of the mean is at least

1minus 1

k2

2

11k

+k‐k

kk

+k k


Example

Given a data set of five values 140 150 170 160 150

Calculate mean and standard deviation micro = 154 and σ asymp 10198

For k = 11 there should be at least

1minus 1

112asymp 17

within the interval

[microminus kσ micro+ kσ] = [154minus 11(10198) 154 + 11(10198)]

= [1428 1652]

Let us count the real percentage Among the five three of which(namely 150 150 160) are in the above interval Therefore thereare three out of five ie 60 percent which is greater than 17percent So the theorem is correct though not very accurate


Example

Example The average of the marks of 65 students in a test was704 The standard deviation was 24

Problem 1 About what percent of studentsrsquo marks were between656 and 752

Solution Based on Chebychevrsquos Theorem 75 since656 = 704minus (2)(24) and 752 = 704 + (2)(24)




Example







The Empirical rule

We can obtain better estimation if we know more

For any symmetrical bell-shaped distribution (ie Normaldistribution)

About 68 of the observations will lie within 1 standard deviation ofthe meanAbout 95 of the observations will lie within 2 standard deviation ofthe meanVirtually all the observations (997) will be within 3 standarddeviation of the mean


The Empirical rule


Comparing Chebyshevrsquos Theorem with The Empirical rule

k Chebyshevrsquos Theorem The Empirical rule

1 0 682 75 953 889 997


An Approximation for Range

Based on the last claim in the empirical rule we can approximate therange using the following formula

Range asymp 6σ

Or more often we can use the above to estimate standard deviationfrom range (such as in Project Management or QualityManagement)

σ asymp Range

6

Example If the range for a normally distributed data is 60 given aapproximation of the standard deviation

Solution σ asymp 606 = 10


Why you should check the normal assumption first

Model risk is everywhere

Black Swans are more than you think










Coefficient of Correlation Scatterplot for the weight vsheight collected form the guessing game in the first class

Coefficient of Correlation is a measure of the linear correlation(dependence) between two sets of data x = (x1 xn) andy = (y1 yn)

A scatter plot pairs up values of two quantitative variables in a dataset and display them as geometric points inside a Cartesian diagram


Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code

gtweight_height lt- readcsv(weight_height_01Acsv)

gtplot(weight_height$weight_01A weight_height$height_01A

xlab=Weight ylab=Height)


Coefficient of Correlation r

Pearsonrsquos correlation coefficient between two variables is defined asthe covariance of the two variables divided by the product of theirstandard deviations

rxy =

sumni=1(Xi minus X)(Yi minus Y )radicsumn

i=1(Xi minus X)2radicsumn

i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2


Coefficient of Correlation Interpretation


Coefficient of Correlation Example

For the the weight vs height collected form the guessing game in thefirst class

The Coefficient of Correlation is

r asymp 02856416

Indicated weak positive linear correlation

The R code

gt weight_height lt- readcsv(weight_height_01Acsv)

gt cor(weight_height$weight_01A weight_height$height_01A

use = everything method = pearson)

[1] 02856416










Case 1 Wisdom of the Crowd

The aggregation of information in groups resultsin decisions that are often better than could havebeen made by any single member of the group

We collect some data from the guessing game in the first class Theimpression is that the crowd can be very wise

The question is when the crowd is wise We given a mathematicalexplanation based on the concepts of mean and standard deviation

For more explanations please refer to Surowiecki James (2005)The Wisdom of Crowds Why the Many Are Smarter Than the Fewand How Collective Wisdom Shapes Business Economies Societiesand Nations


Case 1 Wisdom of the Crowd Mathematical explanation

Let xi (i = 1 n) be the individual predictions and let θ be thetrue value

The crowdrsquos overall prediction

micro =

nsumi=1

xi

n

The crowdrsquos (squared) error

(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n

= Biasminus σ2 = Bias - Variance

Diversity of opinion Each person should have private informationindependent of each other


Case 2 Friendship paradox Why are your friends morepopular than you

On average most people have fewer friends thantheir friends have Scott L Feld (1991)

References Feld Scott L (1991) rdquoWhy your friends have morefriends than you dordquo American Journal of Sociology 96 (6)1464-147


Case 2 Friendship paradox An example

References [hyphen]httpwwweconomistcomblogseconomist-explains201304economist-explains-why-friends-more-popular-paradox



Person degree

Alice 1Bob 3

Chole 2Dave 2

Individualrsquos average number of friends is

micro =8

4= 2

Individualrsquos friendsrsquo average number of friends is

12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225


Case 2 Friendship paradox Mathematical explanation

Let xi (i = 1 n) be the degrees of a network


micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro


Case 2 A video by Nicholas Christakis How socialnetworks predict epidemics

Here is the youtube link


Measure of Dispersion scale parameter

Introduction

Range


Variance and Standard Deviation





Coefficient of Skewness (optional)

Skewness Risk

Coefficient of Kurtosis (optional)

Kurtosis Risk

Chebyshevs Theorem and The Empirical rule

Chebyshevs Theorem

The Empirical rule

Correlation Analysis

Case study

Table of contents1 Measure of Dispersion scale parameter
















Introduction







Range

Range = maxminusmin


Example




The R code



gtrange(sec_01A)

[1] 130 189



Properties of Range








n

|x ‐ ||x1 ‐ |

|x ‐ |

|x3 ‐ |

|x4 ‐ |

x1 x4x2 x3

|x2 |


Example



Solution

x =14 + 15 + 17 + 16

4=

62

4= 155


4

=4

4= 1


Example


The R code




[1] 65


Properties of MAD





Variance




=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n





=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n



Standard Deviation








micro =10 + 11 + 13

3=

34

3asymp 1133333333333


3

=1769 + 01089 + 27889

3=

46668

3= 15556






Solution

Variance

σ2 =(102 + 112 + 132)minus (10+11+13)2

3

3

=390minus 1156

3

3=

390minus 38533

3=

467

3= 1555667




Example


The R code



Sample Variance

gtvar(sec_01A)

[1] 1463228



[1] 1439628


gtsd(sec_01A)

[1] 120964



[1] 1199845


Note















S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1



Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study









Introduction







Range

Range = maxminusmin


Example




The R code



gtrange(sec_01A)

[1] 130 189



Properties of Range








n

|x ‐ ||x1 ‐ |

|x ‐ |

|x3 ‐ |

|x4 ‐ |

x1 x4x2 x3

|x2 |


Example



Solution

x =14 + 15 + 17 + 16

4=

62

4= 155


4

=4

4= 1


Example


The R code




[1] 65


Properties of MAD





Variance




=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n





=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n



Standard Deviation








micro =10 + 11 + 13

3=

34

3asymp 1133333333333


3

=1769 + 01089 + 27889

3=

46668

3= 15556






Solution

Variance

σ2 =(102 + 112 + 132)minus (10+11+13)2

3

3

=390minus 1156

3

3=

390minus 38533

3=

467

3= 1555667




Example


The R code



Sample Variance

gtvar(sec_01A)

[1] 1463228



[1] 1439628


gtsd(sec_01A)

[1] 120964



[1] 1199845


Note















S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1



Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Range

Range = maxminusmin


Example




The R code



gtrange(sec_01A)

[1] 130 189



Properties of Range








n

|x ‐ ||x1 ‐ |

|x ‐ |

|x3 ‐ |

|x4 ‐ |

x1 x4x2 x3

|x2 |


Example



Solution

x =14 + 15 + 17 + 16

4=

62

4= 155


4

=4

4= 1


Example


The R code




[1] 65


Properties of MAD





Variance




=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n





=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n



Standard Deviation








micro =10 + 11 + 13

3=

34

3asymp 1133333333333


3

=1769 + 01089 + 27889

3=

46668

3= 15556






Solution

Variance

σ2 =(102 + 112 + 132)minus (10+11+13)2

3

3

=390minus 1156

3

3=

390minus 38533

3=

467

3= 1555667




Example


The R code



Sample Variance

gtvar(sec_01A)

[1] 1463228



[1] 1439628


gtsd(sec_01A)

[1] 120964



[1] 1199845


Note















S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1



Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Example




The R code



gtrange(sec_01A)

[1] 130 189



Properties of Range








n

|x ‐ ||x1 ‐ |

|x ‐ |

|x3 ‐ |

|x4 ‐ |

x1 x4x2 x3

|x2 |


Example



Solution

x =14 + 15 + 17 + 16

4=

62

4= 155


4

=4

4= 1


Example


The R code




[1] 65


Properties of MAD





Variance




=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n





=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n



Standard Deviation








micro =10 + 11 + 13

3=

34

3asymp 1133333333333


3

=1769 + 01089 + 27889

3=

46668

3= 15556






Solution

Variance

σ2 =(102 + 112 + 132)minus (10+11+13)2

3

3

=390minus 1156

3

3=

390minus 38533

3=

467

3= 1555667




Example


The R code



Sample Variance

gtvar(sec_01A)

[1] 1463228



[1] 1439628


gtsd(sec_01A)

[1] 120964



[1] 1199845


Note















S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1



Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Properties of Range








n

|x ‐ ||x1 ‐ |

|x ‐ |

|x3 ‐ |

|x4 ‐ |

x1 x4x2 x3

|x2 |


Example



Solution

x =14 + 15 + 17 + 16

4=

62

4= 155


4

=4

4= 1


Example


The R code




[1] 65


Properties of MAD





Variance




=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n





=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n



Standard Deviation








micro =10 + 11 + 13

3=

34

3asymp 1133333333333


3

=1769 + 01089 + 27889

3=

46668

3= 15556






Solution

Variance

σ2 =(102 + 112 + 132)minus (10+11+13)2

3

3

=390minus 1156

3

3=

390minus 38533

3=

467

3= 1555667




Example


The R code



Sample Variance

gtvar(sec_01A)

[1] 1463228



[1] 1439628


gtsd(sec_01A)

[1] 120964



[1] 1199845


Note















S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1



Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study




n

|x ‐ ||x1 ‐ |

|x ‐ |

|x3 ‐ |

|x4 ‐ |

x1 x4x2 x3

|x2 |


Example



Solution

x =14 + 15 + 17 + 16

4=

62

4= 155


4

=4

4= 1


Example


The R code




[1] 65


Properties of MAD





Variance




=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n





=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n



Standard Deviation








micro =10 + 11 + 13

3=

34

3asymp 1133333333333


3

=1769 + 01089 + 27889

3=

46668

3= 15556






Solution

Variance

σ2 =(102 + 112 + 132)minus (10+11+13)2

3

3

=390minus 1156

3

3=

390minus 38533

3=

467

3= 1555667




Example


The R code



Sample Variance

gtvar(sec_01A)

[1] 1463228



[1] 1439628


gtsd(sec_01A)

[1] 120964



[1] 1199845


Note















S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1



Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Example



Solution

x =14 + 15 + 17 + 16

4=

62

4= 155


4

=4

4= 1


Example


The R code




[1] 65


Properties of MAD





Variance




=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n





=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n



Standard Deviation








micro =10 + 11 + 13

3=

34

3asymp 1133333333333


3

=1769 + 01089 + 27889

3=

46668

3= 15556






Solution

Variance

σ2 =(102 + 112 + 132)minus (10+11+13)2

3

3

=390minus 1156

3

3=

390minus 38533

3=

467

3= 1555667




Example


The R code



Sample Variance

gtvar(sec_01A)

[1] 1463228



[1] 1439628


gtsd(sec_01A)

[1] 120964



[1] 1199845


Note















S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1



Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Example


The R code




[1] 65


Properties of MAD





Variance




=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n





=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n



Standard Deviation








micro =10 + 11 + 13

3=

34

3asymp 1133333333333


3

=1769 + 01089 + 27889

3=

46668

3= 15556






Solution

Variance

σ2 =(102 + 112 + 132)minus (10+11+13)2

3

3

=390minus 1156

3

3=

390minus 38533

3=

467

3= 1555667




Example


The R code



Sample Variance

gtvar(sec_01A)

[1] 1463228



[1] 1439628


gtsd(sec_01A)

[1] 120964



[1] 1199845


Note















S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1



Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Properties of MAD





Variance




=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n





=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n



Standard Deviation








micro =10 + 11 + 13

3=

34

3asymp 1133333333333


3

=1769 + 01089 + 27889

3=

46668

3= 15556






Solution

Variance

σ2 =(102 + 112 + 132)minus (10+11+13)2

3

3

=390minus 1156

3

3=

390minus 38533

3=

467

3= 1555667




Example


The R code



Sample Variance

gtvar(sec_01A)

[1] 1463228



[1] 1439628


gtsd(sec_01A)

[1] 120964



[1] 1199845


Note















S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1



Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Variance




=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n





=

nsumi=1

x2i minus

(nsum

i=1xi

)2

n



Standard Deviation








micro =10 + 11 + 13

3=

34

3asymp 1133333333333


3

=1769 + 01089 + 27889

3=

46668

3= 15556






Solution

Variance

σ2 =(102 + 112 + 132)minus (10+11+13)2

3

3

=390minus 1156

3

3=

390minus 38533

3=

467

3= 1555667




Example


The R code



Sample Variance

gtvar(sec_01A)

[1] 1463228



[1] 1439628


gtsd(sec_01A)

[1] 120964



[1] 1199845


Note















S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1



Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Standard Deviation








micro =10 + 11 + 13

3=

34

3asymp 1133333333333


3

=1769 + 01089 + 27889

3=

46668

3= 15556






Solution

Variance

σ2 =(102 + 112 + 132)minus (10+11+13)2

3

3

=390minus 1156

3

3=

390minus 38533

3=

467

3= 1555667




Example


The R code



Sample Variance

gtvar(sec_01A)

[1] 1463228



[1] 1439628


gtsd(sec_01A)

[1] 120964



[1] 1199845


Note















S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1



Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study





micro =10 + 11 + 13

3=

34

3asymp 1133333333333


3

=1769 + 01089 + 27889

3=

46668

3= 15556






Solution

Variance

σ2 =(102 + 112 + 132)minus (10+11+13)2

3

3

=390minus 1156

3

3=

390minus 38533

3=

467

3= 1555667




Example


The R code



Sample Variance

gtvar(sec_01A)

[1] 1463228



[1] 1439628


gtsd(sec_01A)

[1] 120964



[1] 1199845


Note















S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1



Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study




Solution

Variance

σ2 =(102 + 112 + 132)minus (10+11+13)2

3

3

=390minus 1156

3

3=

390minus 38533

3=

467

3= 1555667




Example


The R code



Sample Variance

gtvar(sec_01A)

[1] 1463228



[1] 1439628


gtsd(sec_01A)

[1] 120964



[1] 1199845


Note















S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1



Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Example


The R code



Sample Variance

gtvar(sec_01A)

[1] 1463228



[1] 1439628


gtsd(sec_01A)

[1] 120964



[1] 1199845


Note















S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1



Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Note















S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1



Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study












S2 =

ksumi=1

fix2i minus

(ksumi1fixi

)2

n

nminus 1



Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Example



[130 140) 3 135 405 54675

[140 150) 12 145 1740 252300

[150 160) 23 155 3565 552575

[160 170) 14 165 2310 381150

[170 180) 6 175 1050 183750

[180 190] 4 185 740 136900

62 9810 1561350


S2 =1 561 350minus 98102

62


S asymp 1225069

















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study















CV =









Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Example


Fund A 83 -60 189 -57 236 20

Fund B 12 -48 64 102 253 14


CVA =1319

985= 134

CVB =1029

842= 122




















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study



















σ

)3 ]=micro3

σ3



Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Example


The R code



Skewness

gtskewness(sec_01A)

[1] 04267608


Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Skewness Risk




















micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study














micro4

σ4minus 3




Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Example


The R code



kurtosis

gtkurtosis(sec_01A)

[1] -006660501


Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Kurtosis Risk















1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study











1minus 1

k2

2

11k

+k‐k

kk

+k k


Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Example




1minus 1

112asymp 17

within the interval


= [1428 1652]



Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Example







Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

Example







The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

The Empirical rule





The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

The Empirical rule




1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study



1 0 682 75 953 889 997




Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study



Range asymp 6σ


σ asymp Range

6




















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

















Scatterplot Example

160 170 180 190

130

140

150

160

170

180

190

Weight

Hei

ght

The R code







rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study



rxy =



i=1(Yi minus Y )2

=

nnsumi=1

xiyi minusnsumi=1

xinsumi=1

yiradicn

nsumi=1

xi minus(n

nsumi=1

xi

)2radicn

nsumi=1

yi minus(n

nsumi=1

yi

)2







r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study






r asymp 02856416


The R code




[1] 02856416



















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study


















micro =

nsumi=1

xi

n


(microminus θ)2 =

nsumi=1

xi

nminus θ

2

=

nsumi=1

(xi minus θ)2

nminus

nsumi=1

(xi minus micro)2

n












Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study









Person degree

Alice 1Bob 3

Chole 2Dave 2


micro =8

4= 2


12 + 22 + 22 + 32

1 + 2 + 2 + 3=

18

8= 225





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study




micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro





micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study




micro =

nsumi=1

xi

n


nsumi=1

x2i

nsumi=1

xi

= micro+σ2

micro






Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study





Introduction

Range








Skewness Risk


Kurtosis Risk


Chebyshevs Theorem

The Empirical rule


Case study

lecture 4: measure of dispersion - university of new...

Documents