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Measurement Scales

The “richness” of the measure

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Status Report on Software Measurement

Shari Pfleeger, Ross Jeffrey, Bill Curtis, Barbara Kitchenham

IEEE Software March/April 97

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What is the status of Soft Measure?

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Scales

nominal ordinal interval ratio absolute

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Scales

defined in terms of allowed transformations– this is not convenient– research topic

» how to relate abstractions to scales

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Nominal

The weakest scale Classic example

– numbers on sports uniforms Transformation

– Any 1-1 mapping Stats

– mode, frequency, median, percentile

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Nominal Scales

Not valid - “Our team is better because the numbers on our uniforms total more than the numbers on your uniforms”

Not valid - “Ch 11, Ch 13, Ch27, and Ch49 equal 100% of your viewing needs”

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Ordinal

Gives an “ordering” Classic example

– class rank Transformation

– any monotonic transformation Statistics

– spearman correlation

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Class Rank

Not valid - “I am ranked 4th and you are ranked 8th, so I am twice as good as you are.”

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Interval

The size of the intervals are constant Classic example

– temparature Transformation

– aX + b Statistics

– mean, stand dev., pearson correlation

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Converting Temps

How do we convert from fahrenheit to celsius?– (F-32)*5/9 = C– 68 F = 36*(5/9) = 20 C– 50 F = 18* (5/9) = 10 C– 32 F = 0*(5/9) = 0 C

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Temperature

not valid - “it is twice as hot today as yesterday” - this is scale dependent - if it is true for fahrenheit, it is not true for celsius

valid - “the diurnal variation today is twice what it was yesterday” (the difference between max and min).

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Diurnal Variation

68 F - 32 F is twice 50 F - 32 F 20 C - 0 C is twice 10 C - 0 C

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Ratio

Classic example – length measurement

Transformation– aX

Statistics– geometric mean, coefficient of variation

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Ratio Scales

have a well-accepted zero convert from one to another by

multiplication

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Absolute

Counting Classic example

– marbles Transformation

– no Some practioners do not consider this a

scale separate from the ratio scale

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Classifying Scales

Grades Shoe Size Money LOC McCabe’s Cyclomatic Number

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Measurement Theory

circa 1900 - applied to physics 1940’s - applied to psychology, sociology 1990’s - applied to software measurement

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Measurement

“the process by which numbers or symbols are assigned to attributes of entities in the real world in such a way as to describe them according to clearly defined rules”

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Measure (Fenton)

a mapping from the document to the answer set that satisfies measurement theory

the value in the answer set that corresponds to a document

compare to “metric” which is just a mapping

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Terminology

entity is an object or event attribute is a feature or property of the

entity

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Representational TOM

empirical relation system– (C,R)

numerical relation system– (N,P)– M maps (C,R) to (N,P)

representation condition– x<y iff M(x)<M(y)

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Empirical

A set of entities, E A set of relationships, R

– often “less than” or “less than or equal”– note that not everything has to be related

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Relationships - R

The set of relationships R is mathematically defined as a subset of the crossproduct of the elements, ExE

Note that not every pair of elements has to be related and an element may or may not be related to itself

Since we are interested in comparing entities, “less than” or “less than or equal”, are good relationships

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Examples

less than - if a < b than b is not < a less than or equal - a is “less than or equal”

to a

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Numerical

A set of entities– also called the “answer set”– usually numbers - natural numbers, integers or

reals A set of relations

– usually already exists– often “less than” or “less than or equal”

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The Mapping

The representation condition– M(x) rel M(y) if x rel y– x rel y iff M(x) rel M(y)

Both have been used by classical measurement theory authors

Fenton prefers the second definition

Questions

Questions