1. scale2 1 Measurement Scales The “richness” of the measure

2. scale2 2 Status Report on Software Measurement Shari Pfleeger, Ross Jeffrey, Bill Curtis, Barbara Kitchenham IEEE Software March/April 97

3. scale2 3 What is the status of Soft Measure?

4. scale2 4 Scales u nominal u ordinal u interval u ratio u absolute

5. scale2 5 Scales u defined in terms of allowed transformations –this is not convenient –research topic »how to relate abstractions to scales

6. scale2 6 Nominal u The weakest scale u Classic example –numbers on sports uniforms u Transformation –Any 1-1 mapping u Stats –mode, frequency, median, percentile

7. scale2 7 Nominal Scales u Not valid - “Our team is better because the numbers on our uniforms total more than the numbers on your uniforms” u Not valid - “Ch 11, Ch 13, Ch27, and Ch49 equal 100% of your viewing needs”

8. scale2 8 Ordinal u Gives an “ordering” u Classic example –class rank u Transformation –any monotonic transformation u Statistics –spearman correlation

9. scale2 9 Class Rank u Not valid - “I am ranked 4th and you are ranked 8th, so I am twice as good as you are.”

10. scale2 10 Interval u The size of the intervals are constant u Classic example –temparature u Transformation –aX + b u Statistics –mean, stand dev., pearson correlation

11. scale2 11 Converting Temps u 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

12. scale2 12 Temperature u 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 u valid - “the diurnal variation today is twice what it was yesterday” (the difference between max and min).

13. scale2 13 Diurnal Variation u 68 F - 32 F is twice 50 F - 32 F u 20 C - 0 C is twice 10 C - 0 C

14. scale2 14 Ratio u Classic example –length measurement u Transformation –aX u Statistics –geometric mean, coefficient of variation

15. scale2 15 Ratio Scales u have a well-accepted zero u convert from one to another by multiplication

16. scale2 16 Absolute u Counting u Classic example –marbles u Transformation –no u Some practioners do not consider this a scale separate from the ratio scale

17. scale2 17 Classifying Scales u Grades u Shoe Size u Money u LOC u McCabe’s Cyclomatic Number

18. scale2 18 Measurement Theory u circa 1900 - applied to physics u 1940’s - applied to psychology, sociology u 1990’s - applied to software measurement

19. scale2 19 Measurement u “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”

20. scale2 20 Measure (Fenton) u a mapping from the document to the answer set that satisfies measurement theory u the value in the answer set that corresponds to a document u compare to “metric” which is just a mapping

21. scale2 21 Terminology u entity is an object or event u attribute is a feature or property of the entity

22. scale2 22 Representational TOM u empirical relation system –(C,R) u numerical relation system –(N,P) –M maps (C,R) to (N,P) u representation condition –x<y iff M(x)<M(y)

23. scale2 23 Empirical u A set of entities, E u A set of relationships, R –often “less than” or “less than or equal” –note that not everything has to be related

24. scale2 24 Relationships - R u The set of relationships R is mathematically defined as a subset of the crossproduct of the elements, ExE u Note that not every pair of elements has to be related and an element may or may not be related to itself u Since we are interested in comparing entities, “less than” or “less than or equal”, are good relationships

25. scale2 25 Examples u less than - if a < b than b is not < a u less than or equal - a is “less than or equal” to a

26. scale2 26 Numerical u A set of entities –also called the “answer set” –usually numbers - natural numbers, integers or reals u A set of relations –usually already exists –often “less than” or “less than or equal”

27. scale2 27 The Mapping u The representation condition –M(x) rel M(y) if x rel y –x rel y iff M(x) rel M(y) u Both have been used by classical measurement theory authors u Fenton prefers the second definition