1. Measure

2. Measure Definition: Definition: It is the demonstration of the existence of an homomorphism between an empirical relational structure and a numerical relational structure.

3. Numerical relational structure Measure  Empirical relational structure M = objects R = relations N = numbers O = operations Homomorphism

4. Empirical relational structure Example: Length abc M = (chalk a, chalk b, chalk c) O = ( « concatenation », « longer than ») b c a « If we place chalks b and c over each other, then the result will be greater than chalk a » c

5. Numerical relational structure Example: Length N = (x, y, z) O = (+ « addition », > « greater than ») y+z>x = 5+3>7 y+z>x = 5+3>7 N = (x, y, z) O = (+ « addition », > « greater than ») y+z>x = 4+2>9 y+z>x = 4+2>9 = (7, 5, 3) = (9, 4, 2)

6. Homomorphism To link objects with numbers To link relations with operations Example Additive Assumptions Order (Order) (Additive)

7. Homomorphism To link objects with numbers To link relations with operations Numerical relational structure Empirical relational structure

8. Scales 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 Ratio Ordinal Interval Nominal Power Parametric Non parametric

9. Ordinal scale abc Definition: uses number to order objects Nonlinearity assumption

10. Ordinal scale Example Time Performance

11. Interval scale Definition: uses number to order objects and the distance between each attribute is constant. Example: conversion of Celsius (x) into Fahrenheit (y) y=9/5*x+32 Interval of 5ºC x 1 =5 and x 2 =10 Or x 1 =20 and x 2 =25 Linearity assumption: f(x)=mx+b

12. Interval scale Example: conversion of Celsius (x) into Fahrenheit (y) y=9/5*x+32 Interval of 5ºC x 1 =5 and x 2 =10 (x 2 -x 1 =10-5=5) => y 1 =41 and y 2 =50 (y 2 -y 1 =50-41=9) Or x 1 =20 and x 2 =25 (x 2 -x 1 =25-20=5) => y 1 =68 and y 2 =77 (y 2 -y 1 =77-68=9) Warning If we double the ºC we do not double the ºF

13. Ratio Definition: uses number to order objects, the distance between each attribute is constant and the zero is “meaningful”. Example: the distance traveled (y) in function of time (x) y=100*x Linearity assumption: f(x)=mx Time (hours) Distance (Km)