Slide 2.Slide 2.11Descriptive StatisticsDescriptive Statistics

MathematicalMathematicalMarketingMarketing

Chapter 2: Descriptive Statistics

We will be comparing the univariate and matrix formulae for common statistical quantities.

Slide 2.Slide 2.22Descriptive StatisticsDescriptive Statistics

MathematicalMathematicalMarketingMarketing

The Sample Mean Vector

n

iix

n

1x

.

xxx

xxx

xxx

111n

1

n

1

xxx

nm2n1n

m22221

m11211

n1

m21

X1

x

Scalar Matrix

Slide 2.Slide 2.33Descriptive StatisticsDescriptive Statistics

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Deviation Scores

m21

m21

m21

m21

nm2n1n

m22221

m11211

1n

xxx

xxx

xxx

1

1

1

ddd

ddd

ddd

X

xxxX

X1XD

xxd ii

Scalar Matrix

Slide 2.Slide 2.44Descriptive StatisticsDescriptive Statistics

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Sum of Squares – Conceptual Formula

A = DD

n

i

2

i

n

i

2

i

d

or)xx(a

Scalar Matrix

Slide 2.Slide 2.55Descriptive StatisticsDescriptive Statistics

MathematicalMathematicalMarketingMarketing

Sum of Squares – Hand Calculator Version

imim2iim1iim

im2i2i2i1i2i

im1i2i1i1i1i

xxxxxx

xxxxxx

xxxxxx

XXB

2

im2iim1iim

im2i

2

2i1i2i

im1i2i1i

2

1i

xxxxx

xxxxx

xxxxx

n

1

BA

.n

xxa

2n

iin

i

2

i

Scalar Matrix

Slide 2.Slide 2.66Descriptive StatisticsDescriptive Statistics

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The Variance-Covariance Matrix

a1n

1s2

AS1n

1

,dd1n

1

n

yxyx

1n

1s

ii y

n

ix

n

ii

n

iin

iiixy

Scalar Matrix

Slide 2.Slide 2.77Descriptive StatisticsDescriptive Statistics

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The Variance-Covariance Matrix

It’s a key matrix It summarizes the relationship between each pair of variables.

Its order is m · m (where m is the number of vars)

It has lots of names variance matrix, covariance matrix, variance-covariance matrix

Slide 2.Slide 2.88Descriptive StatisticsDescriptive Statistics

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The Variance-Covariance Matrix

mm2m1m

m22221

m11211

sss

sss

sss

S

Diagonal entries could be called

2

is

The S matrix is symmetric

There are m (m-1) / 2 unique off diagonal elements.

There are m (m+1) / 2 unique elements.

Slide 2.Slide 2.99Descriptive StatisticsDescriptive Statistics

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The Diag(·) Function

2

m

2

2

2

1

s00

0s0

00s

)(Diag

SΔ

Slide 2.Slide 2.1010Descriptive StatisticsDescriptive Statistics

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The Square Root of a Diagonal Matrix

2

m

2

2

2

1

2/1

s/100

0s/10

00s/1

Δ

A unique square root of a diagonal matrix may exist. For other square or rectangularmatrices, the square root is not unique.

Slide 2.Slide 2.1111Descriptive StatisticsDescriptive Statistics

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Z Scores

s

d

s

xxz ii

i

2

m

nm

2

2

2n

2

1

1n

2

m

m2

2

2

22

2

1

21

2

m

m1

2

2

12

2

1

11

2

m

2

2

2

1

nm2n1n

m22221

m11211

2/1

s

d

s

d

s

d

s

d

s

d

s

ds

d

s

d

s

d

s/100

0s/10

00s/1

ddd

ddd

ddd

DΔZ

Scalar Matrix

Slide 2.Slide 2.1212Descriptive StatisticsDescriptive Statistics

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The Correlation Matrix

1rr

r1r

rr1

1n

1

2m1m

m221

m112

2/12/1

SΔΔ

ZZR