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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
      ShoeSize
      Money
      LOC
      McCabe’s Cyclomatic Number




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   Measurement Theory
      circa1900 - 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




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posted:9/2/2012
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