Measurement theoryMeasurement theory- for the interested student- for the interested student

Erland JonssonErland Jonsson

Department of Department of Computer Science and EngineeringComputer Science and EngineeringChalmers University of TechnologyChalmers University of Technology

ContentsContents11

What is measurement?What is measurement? Relational systemsRelational systems Scale of measurement Scale of measurement Measurement scalesMeasurement scales SummarySummary

1. This presentation is partly based on Norman E Fenton: Software Metrics, 1. This presentation is partly based on Norman E Fenton: Software Metrics, 1st. ed. 1st. ed.

What is measurement What is measurement ??

Definition of Definition of measurementmeasurement 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.

General requirements General requirements on measurement on measurement operationsoperations 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

Specific requirements Specific requirements on measurement on measurement operationsoperations 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

Relational systemsRelational systems- from measurement theory - from measurement theory

Relational systems Relational systems

There are two types of relational systemsThere are two types of relational systems: : – the the empiricalempirical relational system relational system– the the formalformal relational system relational system

These two relational systems gives the These two relational systems gives the theoretical basis for theoretical basis for defining defining measurement scalesmeasurement scales

Empirical relational Empirical relational system system

Let Let A A = {= {a,b,ca,b,c,…..,,…..,zz} } be the be the target set target set and and the chosen the chosen keykey propertyproperty

Ex. Ex. AA is the set of schoolchildren in a class is the set of schoolchildren in a classand and is the is the property of their height. property of their height.

Now let Now let A = {a,b,c,…..,z} A = {a,b,c,…..,z} be the be the model ofmodel of A A which describes each child in terms of which describes each child in terms of the property height the property height

The empirical relational system comprises The empirical relational system comprises this this model setmodel set together with all the together with all the operationsoperations and and relationsrelations defined over the defined over the setset

Empirical relational Empirical relational system system (con’t)(con’t)

We can now attempt to describe the We can now attempt to describe the empirical relational systemempirical relational system as an as an ordered ordered set set E = (A,R,O) = (A, {r1,r2,r3},{o})E = (A,R,O) = (A, {r1,r2,r3},{o}), where , where

RR is a set of relations: is a set of relations:r1r1 = taller than = taller thanr2r2 = the same height as = the same height asr3r3 = heads and shoulders above, and = heads and shoulders above, and

OO is a set of binary operations: is a set of binary operations:oo is the operator ”standing on the head of” is the operator ”standing on the head of”

Formal relational Formal relational system system

We can now attempt to describe the We can now attempt to describe the formal formal relational systemrelational system as a as a nested set nested set F = (A’,R’,O’)F = (A’,R’,O’)

The formal relational system The formal relational system F F must be capable must be capable of of expressing all of the relations and operationsexpressing all of the relations and operations of the empirical relational system of the empirical relational system EE

The mapping from The mapping from E E to to FF must actually must actually represent all of the observations, preserving all represent all of the observations, preserving all the relations and operations of the relations and operations of E E

If this is true we say that the mapping A -> A’ is If this is true we say that the mapping A -> A’ is a a homomorphism homomorphism andand F is homomorphic to EF is homomorphic to E

Scaling and scale Scaling and scale types types - from measurement theory - from measurement theory

Scale of measurementScale of measurement

Assume that:Assume that:

1.1. The set The set AA models the target set models the target set AA, wrt to , wrt to

We have the empirical system We have the empirical system E = E = (A,R,O)(A,R,O) and and the formal system the formal system F = (A’,R’,O’)F = (A’,R’,O’)

3.3. m m is a homomorphic mapping from is a homomorphic mapping from EE to to F F

Then Then S = (E, F, m) S = (E, F, m) is called the is called the scale of scale of measurement measurement for the key property for the key property

Scale of measurementScale of measurement

Now if Now if FF is defined over some subset of real is defined over some subset of real numbers, then: numbers, then: – measurement maps the key property of each measurement maps the key property of each

object of the target set into a number object of the target set into a number – further, if the mapping is further, if the mapping is homomorphichomomorphic then: then:

1. the 1. the measured data will be representative measured data will be representative of of the key property of the corresponding object, the key property of the corresponding object, andand2. empirical 2. empirical relations and operations relations and operations on the on the properties properties will have correct representations will have correct representations on on the corresponding numbers the corresponding numbers

Scale of measurementScale of measurement

The The homomorphismhomomorphism assures that these assures that these formal formal conclusions, drawn in the number domain conclusions, drawn in the number domain will will have have corresponding conclusions in the empirical corresponding conclusions in the empirical domain domain and thus that the purpose of the and thus that the purpose of the measurement is fulfilledmeasurement is fulfilled

Or in more general terms: Or in more general terms: Our theoretical conclusions will be valid to the Our theoretical conclusions will be valid to the real world and let us draw corresponding real world and let us draw corresponding conclusions for itconclusions for it

A homomorphism is seldom unique, e.g cost can A homomorphism is seldom unique, e.g cost can be expressed in EUROs or in SEKbe expressed in EUROs or in SEK

Measurement scalesMeasurement scales

Mesurement theory distinguishes five types of Mesurement theory distinguishes five types of scalescale::

- - nominalnominal scale scale

- - ordinalordinal scale scale - - intervalinterval scale scale - - ratioratio scale scale - - absoluteabsolute scale scale

Here they are given in an ascending order of Here they are given in an ascending order of ”strength””strength”, in the sense that each is permitting , in the sense that each is permitting less freedom of choice and imposing stricter less freedom of choice and imposing stricter conditions than the previous oneconditions than the previous one

Nominal scaleNominal scale

The The nominal scale nominal scale can be used to denote can be used to denote membership of a class for purposes such as membership of a class for purposes such as labellinglabelling or colour matching or colour matching

There are There are no operations no operations between between EE and and FF The The only relation is equivalence only relation is equivalence One-to-one mapping One-to-one mapping

Ordinal scaleOrdinal scale

The The ordinal scale ordinal scale is used when measurement is used when measurement expresses expresses comparitive judgement comparitive judgement

The scale is preserved under any montonic, The scale is preserved under any montonic, transformation: transformation:

x>=y x>=y xxyy,,where where is an admissible transformation is an admissible transformation

Used for grading goods or rating candidates Used for grading goods or rating candidates

Interval scaleInterval scale

The The interval scale interval scale is used when is used when measuring measuring ”distance” ”distance” between pairs of items of a class between pairs of items of a class according to the chosen attribute according to the chosen attribute

The scale is preserved under positive linear The scale is preserved under positive linear transformation: transformation:

xxmmwherewhere Used for measuring e.g. temperature in Used for measuring e.g. temperature in

centigrade or Fahrenheit (but not Kelvin) or centigrade or Fahrenheit (but not Kelvin) or calendar time calendar time

Ratio scaleRatio scale

The The ratio scale ratio scale denotes the denotes the degreedegree in relation in relation to a standard. It must preserve the origin. to a standard. It must preserve the origin.

It is the most frequently used scaleIt is the most frequently used scale The scale is preserved under the The scale is preserved under the

transformation: transformation: xxmmwherewhere

Used for measuring e.g. mass, length, elapsed Used for measuring e.g. mass, length, elapsed time and temperature in Kelvintime and temperature in Kelvin

Absolute scaleAbsolute scale

The The absolute scale absolute scale is a ratio scale which is a ratio scale which includes a ”standard” unit. includes a ”standard” unit.

The scale is only preserved under the identity The scale is only preserved under the identity transformation: transformation:

xxx,x,which means that it is not transformable which means that it is not transformable

Used for Used for counting itemscounting items of a class of a class

Meaningfulness Meaningfulness

MeaningfulnessMeaningfulness means that the scale means that the scale measurement should be appropriate to the measurement should be appropriate to the type of property measured, such that type of property measured, such that once measurement has been performed – and once measurement has been performed – and data expressed on some scale - data expressed on some scale - sensible sensible conclusions can be drawnconclusions can be drawn from it from it

Example 1: Point A is twice as far as point B Example 1: Point A is twice as far as point B (meaningless, since distance is a ratio scale, (meaningless, since distance is a ratio scale, but position is not) but position is not)

Example 2: Point A is twice as far from point X Example 2: Point A is twice as far from point X as point B (is meaningful) as point B (is meaningful)

Summary 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