Measurement theory - for the interested student

1. Measurementtheory- for the interestedstudent Erland Jonsson DepartmentofComputer Science and EngineeringChalmers University ofTechnology

2. Contents1 • What is measurement? • Relational systems • Scale of measurement • Measurementscales • Summary 1. This presentation is partly based on Norman E Fenton: Software Metrics, 1st. ed.

3. What is measurement ?

4. Definition of measurement Definition: Measurement is the process of empirical, objective encoding of some property of a selected class of entities in a formal system of symbols (A. Kaposi based on Finkelstein) Cp Metrology is the field of knowledge con-cerned with measurement. Metrology can be split up into theoretical, methodology, technology and legal aspects.

5. General requirements on measurement operations Operations of measurement involve collecting and recording data from observation It means identifying the class of entities to which the measurement relates Measurements must be independent of the views and preferences of the measurerer Measurements must not be corrupted by an incidental, unrecorded circumstance, which might influence the outcome

6. Specific requirements on measurement operations Measurement must be able to characterize abstract entities as well as to describe properties of real-world objects The result of measurement may be captured in terms of any well-defined formal system, i.e. not necessarily involving numbers

7. Relational systems- from measurementtheory

8. Relational systems • There are twotypes of relational systems: • the empiricalrelational system • the formalrelational system • Thesetworelational systems gives the theoretical basis for definingmeasurementscales

9. Empiricalrelational system • LetA = {a,b,c,…..,z} be the target set and kthe chosen keyproperty • Ex. A is the set of schoolchildren in a classand kis the property of theirheight. • NowletA = {a,b,c,…..,z} be the model of A whichdescribeseachchild in terms of the propertyheight • The empiricalrelational system comprises this model settogether with all the operations and relationsdefined over the set

10. Empiricalrelational system (con’t) • Wecannowattempt to describe the empiricalrelational system as an ordered set E = (A,R,O) = (A, {r1,r2,r3},{o}), where • R is a set of relations:r1 = tallerthanr2 = the same height asr3 = heads and shouldersabove, and • O is a set of binary operations:o is the operator ”standing on the head of”

11. Formal relationalsystem • Wecannowattempt to describe the formal relational system as a nested set F = (A’,R’,O’) • The formal relational system F must be capable of expressing all of the relations and operations of the empiricalrelational system E • The mapping from E to F must actuallyrepresent all of the observations, preserving all the relations and operations of E • If this is truewesay that the mapping A -> A’ is a homomorphismandF is homomorphic to E

12. Scaling and scaletypes- from measurementtheory

13. Scale of measurement • Assume that: 1. The set Amodels the target set A, wrt to k 2. Wehave the empirical system E = (A,R,O) and the formal system F = (A’,R’,O’) 3. m is a homomorphicmapping from E to F • ThenS = (E, F, m) is called the scale of measurementfor the key propertyk

14. Scale of measurement • NowifF is defined over some subset of real numbers, then: • measurementmaps the key property of eachobject of the target set into a number • further, if the mapping is homomorphicthen: 1. the measured data will be representative of the key property of the correspondingobject, and2. empiricalrelations and operations on the propertieswillhavecorrect representations on the correspondingnumbers

15. Scale of measurement • The homomorphismassures that theseformal conclusions, drawn in the numberdomainwillhavecorrespondingconclusions in the empiricaldomainand thus that the purpose of the measurement is fulfilled • Or in more general terms: Our theoreticalconclusionswill be valid to the real world and letusdrawcorrespondingconclusions for it • A homomorphism is seldom unique, e.gcostcan be expressed in EUROs or in SEK

16. Measurementscales • Mesurementtheorydistinguishesfivetypes of scale: - nominalscale - ordinalscale- intervalscale- ratioscale- absolutescale • Herethey are given in an ascending order of ”strength”, in the sense that each is permitting less freedom of choice and imposingstricterconditionsthan the previousone

17. Nominal scale • The nominal scalecan be used to denotemembership of a class for purposessuch as labelling or colourmatching • There are no operations betweenE and F • The only relation is equivalence • One-to-onemapping

18. Ordinalscale • The ordinal scale is used when measurement expresses comparitive judgement • The scale is preserved under any montonic, transformation: x>=y  f(x)>=f(y), where f is an admissible transformation • Used for grading goods or rating candidates

19. Interval scale • The interval scaleis usedwhenmeasuring ”distance” between pairs of items of a classaccording to the chosen attribute • The scale is preserved under positive linear transformation: f(x) = am + b, where a>0 • Used for measuringe.g. temperature in centigrade or Fahrenheit (but not Kelvin) or calendar time

20. Ratioscale • The ratioscaledenotes the degree in relation to a standard. It must preserve the origin. • It is the mostfrequentlyusedscale • The scale is preserved under the transformation: f(x) = am , where a>0 • Used for measuringe.g. mass, length, elapsed time and temperature in Kelvin

21. Absolute scale • The absolute scaleis a ratioscalewhichincludes a ”standard” unit. • The scale is onlypreserved under the identity transformation: f(x) = x,whichmeans that it is not transformable • Used for countingitems of a class

22. Meaningfulness • Meaningfulnessmeans that the scalemeasurementshould be appropriate to the type of propertymeasured, such that oncemeasurement has beenperformed – and data expressed on somescale - sensible conclusionscan be drawn from it • Example 1: Point A is twice as far as point B (meaningless, sincedistance is a ratioscale, but position is not) • Example 2: Point A is twice as far from point X as point B (is meaningful)

23. Summary • We have given a brief and heuristic overview of a few concepts from measurement theory • We have described a number of scales • We have defined ”scale of measurement” as a homomorphic mapping from an empirical relational system to a formal relational system