SFWR ENG 3S03: Software Testing

Transcription

1 (Slide 1 of 64) Dr. Ridha Khedri Department of Computing and Software, McMaster University Canada L8S 4L7, Hamilton, Ontario and Acknowledgments: Material based on [FP97, Chapter 2] and [Zus97, Chapter 4]

2 (Slide 2 of 64) 1 2 Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering 3 and Defining Attributes Direct and indirect measurement for Prediction 4 Nominal Scale Ordinal Scale Interval Scale Ratio Scale and

3 Basics of (Slide 3 of 64) We use measurement every day, to understand, control and improve what we do and how we do it We examine measurement in more depth In our daily life, to measure we use both tools and principles that we now take for granted se sophisticated measuring devices and techniques have been developed over time, based on the growth of understanding of the attributes we are measuring and

4 Basics of (Slide 4 of 64) As we understood more about an attributes and the relationships between them, we develop a framework for describing them tools for measuring them Unfortunately, we have no deep understanding of software attributes Many questions that are relatively easy to answer for non-software entities are difficult for software and

5 Basics of (Slide 5 of 64) Consider the following questions: Does a count of the number of bugs found in a system during integration testing measure the quality of the system? For instance, is it meaningful to talk about doubling a design s quality? If not, how do we compare two different designs? Is it sensible to compute average productivity for a group of developers, or the average quality of a set of modules? How can we measure how quality such as security or privacy? and

6 Basics of (Slide 6 of 64) To answer these questions, we must establish the basics of a theory of measurement We begin by examining formal measurement theory We see how the concepts of measurement theory apply to software This theory tells us how to measure how to analyze and depict data and how to tie the results back to our original questions

7 Basics of (Slide 7 of 64) In any measurement activity, there are rules to be followed help us to be consistent in our measurement provide us with a basis for interpreting data laying the groundwork for developing and reasoning about all kinds of measurement This rule-based approach is common in many sciences (theory) Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

8 Basics of (Slide 8 of 64) Relationship between theory and measurement We can use rules about measurement to codify our initial understanding, and then expand our horizons as we analyze our software However, there are several theories for field Euclidean and non-euclidean Psychology ories: provide a set of guiding principles and concepts that describe and explain human development Classical mechanics and quantum mechanics are the two major sub-fields of mechanics Depending on the theory chosen, there are also several theories of measurement Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

9 Basics of (Slide 9 of 64) Empirical relations representational theory of measurement formalizes our intuition about the studied entities data we obtain as measures should represent attributes of the entities Manipulation of the data should preserve relationships that we observe among the entities Thus, our intuition is the starting point for all measurement Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

10 Basics of (Slide 10 of 64) Empirical relations We tend to understand things by comparing them, not by assigning numbers to them We observe that certain people are taller than others without actually measuring them Our observation reflects a set of rules that we are imposing on the set of people We form pairs of people and define a binary relation taller than on them Given any two people, x and y, we can compare x and y using taller than taller than is an empirical relation for height Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

11 Basics of (Slide 11 of 64) Empirical relations A (binary) empirical relation is one for which there is a reasonable consensus about which pairs are in the relation We can define more than one empirical relation on the same set (e.g., much taller than, almost the same hight ) Empirical relations NEED NOT be binary We can think of these relations as mappings from the empirical, real world to a formal mathematical world Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

12 Basics of (Slide 12 of 64) Empirical relations For example: Any measure of height should assign a higher number to Jack than to Jill Example HOWEVER, the representation condition of measurement needs to preserve intuition and observation Suppose we are evaluating the four best-selling wordprocessing programs: A, B, C, and D. We ask 100 independent computer users to rank these programs according to their functionality, and the results are shown in the following table. Each cell of the table represents the percentage of respondents who preferred the row s program to the column s program. Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

13 Basics of (Slide 13 of 64) Empirical relations Table: More functionality A B C D A B C D greater functionality than def cell (x, y) exceeds 60% Table: More User-friendly A B C D A B C D user-friendliness relation: no real consensus Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

14 Basics of Empirical relations Table: Historical advances in temperature measurement ă 2000 BC Rankings hotter than 1600 AD First thermometer measuring hotter than 1720 AD Fahrenheit scale 1742 AD Celsius scale 1854 AD Absolute zero (Kelvin scale) We can use relatively unsophisticated relationships that require no measuring tools With more accumulated knowledge, we may build more sophisticated ways and tools (Slide 14 of 64) Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

15 Basics of (Slide 15 of 64) Empirical relations Definition () A measurement is a mapping from the empirical world to the formal, relational world. Consequently, a measure is a value or a symbol assigned to an entity by this mapping in order to characterize an attribute. Sometimes, the empirical relations for an attribute are not yet agreed upon (e.g., personal preference, no common understanding) y enable us to establish the basis for empirical relations, so that formal measurement may be possible in the future Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

16 Basics of (Slide 16 of 64) Rules of the Mapping Mathematical World mapping Real World Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

17 Basics of (Slide 17 of 64) Rules of the Mapping A measure must specify the domain and range as well as the rule for performing the mapping. Homeland Security Advisory System was a color-coded terrorism threat advisory scale (red, orange, yellow, blue, green) = (severe, high, significant, general, low) risk (Severe, High, Elevated, Guarded, Low) Why not p4, 3, 2, 1, 0q, p7777, 777, 77, 7, 0q or (,,,, )? Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

18 Basics of Rules of the Mapping A bijective mapping does not exist in the area of (software) measurement Why? s values can be identical for different objects re are four types of mappings: Injective, but not surjective (Not a measurement mapping) Not injective, but surjective (Could be a measurement mapping) Bijective (Not a measurement mapping) Neither Injective nor surjective ( most of the measurement mappings) (Slide 18 of 64) Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

19 Basics of (Slide 19 of 64) Rules of the Mapping Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

20 Basics of (Slide 20 of 64) Rules of the Mapping Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

21 Basics of (Slide 21 of 64) Representation Condition of Each relation in the empirical relational system corresponds via the measurement to an element in a carrier set in the mathematical world (could a set of numbers) We want to have preservation of operation and relations This rule is called the representation condition Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

22 Basics of (Slide 22 of 64) Representation Condition of Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

23 Basics of (Slide 23 of 64) Representation Condition of mapping that we call a measure is sometimes called a representation or homomorphism We use the notion of representation to define validity: any measure that satisfies the representation condition is a valid measure How can we build more sophisticated measures from simple ones? Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

24 Basics of Direct Product Structure Definition (Direct Product) Let tm i u I `M i, tf i u f PF, tr i u RPR i P I (, be an I -indexed family of L-structures. direct product Π I M i of the family is defined as follows: support set if Π I M i (i.e., the Cartesian Product of M i ) the product are defined componentwise Given R P R, the relation R Π on Π I M i is defined as follows: px 1,, x m q P R Π i P I : px 1 piq,, x m piqq P R i q, where m is the arity mprq of R and px 1,, x m q P pπ I A i q m. (Slide 24 of 64) Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

25 Basics of Direct Product Structure Clearly, Π I M i `Π I M i, tf Π u f PF, tr Π u RPR as it is defined has the same language as L as each of the structures in the family tm i u I. set I can be empty: the empty product Π H has a support with one element e. R H tpe,, equ j i, j P I : M i N M j q, then Π I M i N I denoted N I. N I def Π I M i is called I -direct power of the L-structure N. (Slide 25 of 64) Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

26 Basics of (Slide 26 of 64) Examples of Specific used in Software Engineering Empirical relations Rules of the Mapping Representation Condition of Direct Product Structure Examples of Specific used in Software Engineering and

27 Basics of and (Slide 27 of 64) Software Engineers use several types of models (e.g., cost-estimation models, quality models, capability-maturity models) A model is an abstraction of reality, allowing us to strip away detail and view an entity or concept from a particular perspective Cost models permit us to examine only those project aspects that contribute to the project s final cost come in many different forms (e.g., equations, mappings, or diagrams) and Defining Attributes Direct and indirect measurement for Prediction

28 Basics of and (Slide 28 of 64) As far as a model is concerned, the world can be divided into three parts: 1 Things whose effects are neglected 2 Things that affect the model but whose behavior the model is not designed to study 3 Things the model is designed to study the behavior of When we use a model to draw conclusion, we make a deductive process: If the assumptions are true, the conclusion must be true and Defining Attributes Direct and indirect measurement for Prediction

29 Basics of and (Slide 29 of 64) Example To measure length of programs using lines of code, we need a model of a program. model would specify how a program differs from a subroutine, whether or not to treat separate statements on the same line as distinct lines of code, whether or not to count comment lines, whether or not to count data declarations, and so on. model would also tell us what to do when we have programs written in a combination of different languages. It might distinguish delivered operational programs from those under development, and it would tell us how to handle situations where different versions run on different platforms. and Defining Attributes Direct and indirect measurement for Prediction

30 Basics of and (Slide 30 of 64) Defining Attributes When measuring, there is always a danger that we focus too much on the formal, mathematical system, and not enough on the empirical one We should give careful thought to the relationships among entities and their attributes in the real world and Defining Attributes Direct and indirect measurement for Prediction

31 Basics of and Example Defining Attributes Our intuition tells us that the complexity of a program can affect the time it takes to code it, test it, and fix it We suspect that complexity can help us to understand when a module is prone to contain faults However, there are few researchers who have built models of exactly what it means for a module to be complex Many software developers define program complexity as the cyclomatic number proposed by McCabe (Slide 31 of 64) and Defining Attributes Direct and indirect measurement for Prediction

32 Basics of and (Slide 32 of 64) Defining Attributes and Defining Attributes Direct and indirect measurement for Prediction

33 Basics of and (Slide 33 of 64) Defining Attributes On the basis of empirical research, McCabe claimed that modules with high values of η were those most likely to be fault-prone and unmaintainable He proposed a threshold value of 10 for each module Any module with η greater than 10 should be redesigned to reduce η Limitations: cyclomatic number presents only a partial view of complexity re are many programs that have a large number of decisions but are easy to understand, code and maintain (ηpgq d ` 1) A more complete model of complexity is needed and Defining Attributes Direct and indirect measurement for Prediction

34 Basics of and (Slide 34 of 64) Direct and indirect measurement Definition (Direct measurement) Direct measurement of an attribute of an entity involves no other attribute or entity. Model length of a physical object can be measured without reference to any other object or attribute Density of a physical object can be measured only indirectly in terms of mass and volume density mass volume and Defining Attributes Direct and indirect measurement for Prediction

35 Basics of and (Slide 35 of 64) Direct and indirect measurement following direct measures are commonly used in software engineering: Length of source code (measured by lines of code) Duration of testing process (measured by elapsed time in hours) Number of defects discovered during the testing process (measured by counting defects) Time a programmer spends on a project (measured by months worked) and Defining Attributes Direct and indirect measurement for Prediction

36 Basics of and (Slide 36 of 64) Direct and indirect measurement and Defining Attributes Direct and indirect measurement for Prediction

37 Basics of and (Slide 37 of 64) Direct and indirect measurement Direct measurement to assess a product [PFP94] and Defining Attributes Direct and indirect measurement for Prediction

38 Basics of and (Slide 38 of 64) Direct and indirect measurement Indirect measurement to assess a product [PFP94] and Defining Attributes Direct and indirect measurement for Prediction

39 Basics of and (Slide 39 of 64) for Prediction When wemeasure, we usually mean that we wish to assess some entity that already exists In many circumstances, we would like to predict an attribute of some entity distinction between measurement for assessment and prediction is not always clear-cut In general, measurement for prediction always requires some kind of mathematical model and Defining Attributes Direct and indirect measurement for Prediction

40 Basics of and (Slide 40 of 64) for Prediction Example (Reliability model) A well known reliability model is based on an exponential distribution for the time to the ith failure of the product. This distribution is described by the formula F ptq 1 e pn i`1qat, where t is time N represents the number of faults initially residing in the program a represents the overall rate of occurrence of failures re are many ways that the model parameters N and a can be estimated (e.g., Maximum Likelihood Estimation) and Defining Attributes Direct and indirect measurement for Prediction

41 Basics of (Slide 41 of 64) re are differences among the different kind of mappings differences among the mappings can restrict the kind of analysis we can do We discuss the notion of a measurement scale and then we use the scale to understand which analyses are appropriate scale = measurement mapping (M) + the empirical and numerical relation systems (i.e., dom, ran ) and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

42 Basics of (Slide 42 of 64) re are three important questions concerning representations and scales: How do we determine when one numerical (or, symbolic) relation system is preferable to another? How do we know if a particular empirical relation system has a representation in a given numerical relation system? (This question is about the representation problem) What do we do when we have several different possible representations (and hence many scales) in the same numerical relation system? (This question is about the uniqueness problem) and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

43 Basics of (Slide 43 of 64) In general, there are many different representations for a given empirical relation system ps, tr i i P I uq, where I is a set of indices Higher is the size of I, the fewer are the representations ps, tr i i P I uq is richer than ps, tq i i P I uq i P I : Q i Ď R i q richer the empirical relation system, the more restrictive the set of representations, and so the more sophisticated the scale of measurement and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

44 Basics of (Slide 44 of 64) We classify measurement scales as one of five major types: Nominal richer than Ordinal richer than Interval richer than ratio richer than Absolute re are other scales that can be defined (e.g., logarithmic scale) What is a scale? Example: We may measure the length of physical objects by using a mapping from length to inches re are equally acceptable measures in feet, meters, furlongs, miles, etc. All of the acceptable measures are very closely related (we can convert one to another) A Scale is defined by a homomorphism and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

45 Basics of (Slide 45 of 64) What is a scale type?it defined by the notion of admissible transformation A mapping from one acceptable measure to another is called an admissible transformation (also called rescaling) Scales and scale types lead directly to the notion of meaningfulness Meaningfulness A statement with measurement values is meaningful iff its truth or falsity value is invariant to admissible transformations. and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

46 Basics of (Slide 46 of 64) Another widely discussed notion in the area of software measurement is wholeness Wholeness whole is equally or more big than the sum of the parts. pp, q be a structure of flowgraphs with the concatenation operation Let P, Q P P the set of flowgraphs. Let µ be a software measure (e.g., ν linearly independent paths in the graph) requirement of wholeness property translates into requiring µpp Qq ě µppq ` µpqq and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

47 Basics of (Slide 47 of 64) Nominal Scale Definition (Nominal Scale) Let pp, «q be an empirical relational system, where P is a non-empty countable set and where «is an equivalence relation on P. Let pr, q be numerical mathematical structure with R its carrier set and is its identity relation. Let µ : P ÝÑ R be a real value function. system `pp, «q, pr, q, µ is a nominal scale q p, q P P : p «q ðñ µppq µpqq q Meaningful Operations admissible transformation is only a one-to-one transformation and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

48 Basics of (Slide 48 of 64) Ordinal Scale Definition (Ordinal Scale) Let pp, Áq be an empirical relational system, where P is a non-empty countable set and where Á is an empirical relation describing ranking properties on P. Let pr, ďq be numerical mathematical structure with R its carrier set and ď is its partial order. Let µ : P ÝÑ R be a real value function. system `pp, Áq, pr, ěq, µ is an ordinal scale iff q p, q P P : p Á q ðñ µppq ě µpqq q q, r p, q, r P P : p Á q ^ q Á r ùñ p Á r q (Transitivity) q p, q P P : p Á q _ q Á p q (Completeness) and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

49 Basics of (Slide 49 of 64) Interval Scale Interval Scale empirical conditions behind the interval scale are not intuitive in the area of software measurement because they consider empirical distances If we consider the empirical relation equally or more difficult to maintain, then we have to consider distances of maintainability Among others, ratio scale measures can be transformed to interval measures by admissible transformations, if users require that and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

50 Basics of (Slide 50 of 64) Interval Scale For this reason, the interval scale in the area of software measure is obsolete In physics, examples are the transformations from Kelvin to Celsius or Fahrenheit It is the movement of the zero point (In physics distances are well known) But, the definition of distances objects of software engineering is more difficult and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

51 Basics of (Slide 51 of 64) Interval Scale An interval scale can be defined by an algebraic difference structure Example concept of an algebraic difference structure is a set of objects and a 2-arity relation on A ˆ A pa, bq Á pc, dq def difference between a and b is ě to the difference between c and d pbeer, wineq Á pcoffee, teaq def my preference to beer over wine is equal or greater than my preference to coffee over tea and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

52 Basics of Interval Scale (Slide 52 of 64) ps, ěq is a weak y x, y P S : x ě y _ y ě x q y, z x, y, z P S : x ě y ^ y ě z ùñ px ě zq q (completeness or (transitivity) A sequence pa 1, a 2,, a i, q is bounded if there exists a real number M such i is an index : a i ď M q and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

53 Basics of (Slide 53 of 64) Interval Scale Definition (Algebraic Difference Structure) Let pa ˆ A, Áq be an algebraic difference structure iff, for all a, b, c, a 1, b 1, and c 1 P A and all sequences a 1, a 2,, a i, P A, the following five axioms satisfied: 1 pa ˆ A, Áq is a weak order 2 pa, bq Á pc, dq ùñ pd, cq Á pb, aq 3 pa, bq Á pa 1, b 1 q ^ pb, cq Á pb 1, c 1 q ùñ pa, cq Á pa 1, c 1 q 4 pa, bq Á pc, dq ^ pc, dq Á pa, aq ùñ Dpd 1, d 2 d 1, d 2 P A : pa, d 1 q Á pc, dq ^ pc, dq Á pd 2, bq q 5 If a sequence a 1, a 2,, a i, P A is strictly bounded sequence, then it is finite. and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

54 Basics of (Slide 54 of 64) Interval Scale Definition (Interval Scale) An Interval Structure is the system `pa ˆ A, Áq, pr ˆ R, ěq, µ, where 1 pa ˆ A, Áq is an algebraic difference structure 2 µ be a real valued function on A such b, c, d a, b, c, d P A : pa, bq Á pc, dq ðñ µpaq µpbq ě µpcq µpdq q 3 If another real value function g satisfies property (2), then there exist real value numbers α, β ą 0, such that gpxq αµpxq ` β holds Le last condition can be written as follows b, c, d a, b, c, d P A : pa, bq Á pc, dq ðñ gpaq gpbq ě gpcq gpdq q : Dpα, β α, β ą 0 : gpxq αµpxq ` β for all x P A q q and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

55 Basics of (Slide 55 of 64) Interval Scale axioms of the interval scale show us that no single elements of a set A are used, but pairs of elements of A se pairs are treated as differences (intervals) of the elements of A relation Á is here no ranking order of elements of A, but an order of differences (intervals) of A Difference structures can be applied, if we can formulate differences or intervals empirically Axiom 2 of the difference structure shows that the set A ˆ A contains positive pa, bq and negative pb, aq differences and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

56 Basics of (Slide 56 of 64) Ratio Scale Definition (Ratio Scale) Let pp, Á, q be an empirical relational system, where P is a non-empty countable set, Á is an empirical relation describing ranking properties on P, and is a binary operation on P. Let pr, ě, `q be numerical mathematical structure with R its carrier set, ě is its partial order, and ` is the addition on R. Let µ : P ÝÑ R be a real value function. system `pp, Á, q, pr, ě, `q, µ is a ratio scale iff q p, q P P : p Á q ðñ µppq ě µpqq q q p, q P P : µpp qq µppq ` µpqq q and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

57 Basics of (Slide 57 of 64) Ratio Scale orem Let `pp, Á, q, pr, ě, `q, µ be a structure such that its relational part is an ordinal scale. A real value function ν : P ÝÑ R satisfies (1) and (2) of the above definition iff Dpα α P R ^ α 0 p P P : νppq αµppq q q νppq αµppq is the admissible transformation for ratio scale An additive measure assume an extensive additive structure and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

58 Basics of Admissible Transformation (revisited) Definition (Admissible Transformation) Let pa, B, µq be a scale, where A (resp. B) is the carrier set (or underlying set) of A (resp. B). A mapping g : A ÝÑ R is an admissible transformation iff pa, B, gq is also a scale. Example pa, B, µq def `pp, Á, q, pr, ě, `q, LOC, where LOC is function that take a program and returns the number of lines of code question is whether a measure LOC exists that pa, B, µq def `pp, Á, q, pr, ě, `q, LOC, where LOC ppq LOCppq 1000, for p P P???? Yes. gpxq µpxq. LOC is denoted KLOC (Slide 58 of 64) and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

59 Basics of Admissible Transformation (revisited) (Slide 59 of 64) Table: Scale classification using admissible transformation Scale Nominal Ordinal Admissible Transformation g Any one-to-one g is a strictly increasing monotonic function Interval gpxq aµpxq`b a 1 x `b, a, a 1 ą 0 Ratio gpxq aµpxq a 1 x, a, a 1 ą 0 Scales are defined by a homomorphism Scale types are defined by admissible transformations and Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

60 Basics of (Slide 60 of 64) Admissible Transformation (revisited) Nominal Metrical Ordinal Interval n affine and n Euclidean Ratio n linear Figure: General Classification of Scales Nominal Scale Ordinal Scale Interval Scale Ratio Scale Admissible Transformation (revisited)

61 Basics of (Slide 61 of 64) Table: List of what can be computed on a scale You can compute... Nominal Ordinal Interval Ratio Frequency distribution Yes Yes Yes Yes Median and percentiles NO Yes Yes Yes Add or Substruct NO NO Yes Yes Mean, standard deviation, NO NO Yes Yes standard error of the mean Ratio, or coefficient of variation NO NO NO Yes and

62 Basics of (Slide 62 of 64) and Figure: Summary of measurement scales and statistics relevant to each

63 References I (Slide 63 of 64) Norman E. Fenton and Shari Lawrence Pfleeger, Software metrics: A rigorous and practical approach, second ed., PWS Publishing Company, S.L. Pfleeger, N.E. Fenton, and S. Page, Evaluating software engineering standards, IEEE Computer 27 (1994), no. 9, Horst Zuse, A framework of software measurement, Walter de Gruyter, and

64 (Slide 64 of 64) and