MeasureMeasure

MeasureMeasure

Definition:Definition:

It is the demonstration of the existence of an It is the demonstration of the existence of an homomorphism between an empirical relational homomorphism between an empirical relational structure and a numerical relational structurestructure and a numerical relational structure. .

Numerical relational structure

<N, O>

MeasureMeasure

Empirical relational

structure

<M, R>

M = M = objectsobjects

R = relationsR = relations

N = N = numbersnumbers

O = operationsO = operations

HomomorphismHomomorphism

Empirical relational structureEmpirical relational structureExample: LengthExample: Length

aa bb cc

M = M = (chalk a, chalk b, chalk c)(chalk a, chalk b, chalk c)

O = ( « concatenation », « longer than »)O = ( « concatenation », « longer than »)

b c ab c a

« « If we place chalks b and c over each If we place chalks b and c over each other, then the result will be greater other, then the result will be greater than chalk athan chalk a » »

cc

Numerical relational structureNumerical relational structure Example: LengthExample: Length

N = N = (x, y, z) (x, y, z) O = (+ «  addition », > « greater than »)O = (+ «  addition », > « greater than »)

y+z>x = 5+3>7y+z>x = 5+3>7

N = N = (x, y, z) (x, y, z)

O = (+ «  addition », > « greater than »)O = (+ «  addition », > « greater than »)

y+z>x = 4+2>9y+z>x = 4+2>9

= (7, 5, 3)= (7, 5, 3)

= (9, 4, 2)= (9, 4, 2)

HomomorphismHomomorphism

To link objects with numbersTo link objects with numbers

To link relations with operationsTo link relations with operations

ExampleExample

( ) 5

( ) 2

a x

b y

5 2 (Ordre)

7 5 2 (additif)

( ) ( ) ( )a b a b x y AdditiveAdditiveAssumption

s

OrderOrder ( ) ( )a b x y

(Order)

(Additive)

HomomorphismHomomorphism

To link objects with numbersTo link objects with numbers

To link relations with operationsTo link relations with operations

,

, ; ,

, , , = Structure relationnelle empirique

M R

a b M R

a b

,

, ; ,

, , , = Structure relationnelle numérique

N O

x y N O

x y

, , ,

, , ,

a b

x y

Numerical relational structure

Empirical relational structure

ScalesScales

The freedom available to construct my scale will determine its type.

The less the freedom in choice of scale, the more powerful it will be

RatioRatio

OrdinalOrdinal

IntervalInterval NominalNominal

Power

Power

ParametricParametric Non parametricNon parametric

Ordinal scaleOrdinal scale

aa bb cc

Definition: uses number to order objectsDefinition: uses number to order objects

( ) 20

( ) 15

( ) 2

20 15 2

a

b

c

( ) 3

( ) 2

( ) 1

3 2 1

a

b

c

( ) ( ) ( )a b c

Nonlinearity assumptionNonlinearity assumption

Ordinal scaleOrdinal scaleExampleExample

Time

Perf

orm

ance

Interval scaleInterval scaleDefinition: uses number to order objects Definition: uses number to order objects and the distance and the distance between each attribute is constant.between each attribute is constant.

Example: conversion of Example: conversion of Celsius (Celsius (xx) into) into Fahrenheit ( Fahrenheit (yy))

yy=9/5*=9/5*xx+32+32

Interval of 5Interval of 5ºCºC

xx11=5 and x=5 and x22=10=10

OrOr

xx11=20 and x=20 and x22=25=25

Linearity assumption: fLinearity assumption: f((xx)=m)=mxx+b+b

Interval scaleInterval scaleExample: conversion of Celsius (Example: conversion of Celsius (xx) into Fahrenheit () into Fahrenheit (yy))

yy=9/5*=9/5*xx+32+32

Interval of 5ºInterval of 5ºCC

xx11=5 and x=5 and x22=10=10(x(x22-x-x11=10-5=5)=10-5=5)

=> y=> y11=41 and y=41 and y22=50=50(y(y22-y-y11=50-41=9)=50-41=9)

OrOrxx11=20 and x=20 and x22=25=25(x(x22-x-x11=25-20=5)=25-20=5)

=> y=> y11=68 and =68 and yy22=77=77(y(y22-y-y11=77-68=9)=77-68=9)

WarningWarning

If we double the ºC we do not double the ºF

RatioRatioDefinition: uses number to order objects, the distance between Definition: uses number to order objects, the distance between each attribute is constant each attribute is constant and the zero is “meaningful”and the zero is “meaningful”..

Example: the distance Example: the distance traveled (y) in function of traveled (y) in function of time (x)time (x)

yy=100*=100*xx

Linearity assumption: fLinearity assumption: f((xx)=m)=mxx

Time (hours)

Dis

tance

(K

m)