Bounded scales: a new type of scale for educational practice and research

In Higher Education, achievement assessment of students (aka summative evaluation) is nearly always done by qualified humans, called assessors, by means of counting or scoring on bounded scales related to quality criteria that are specific to the assignment or test. We do not want to criticize or change this age-old tradition. On the contrary: we accept this practice as it is, not the least because a viable alternative is not yet in sight (for theoretical as well as practical reasons, we consider psychometric tests and learning analytics, though available, as non-viable). Moreover, assessment by well-qualified and well-trained assessors appears to be reasonably effective and robust, whereas it is often not possible to delegate it to some testing instrument or software (exceptions, of course, are the simplest forms of assessment like multiple-choice tests and short essays). Thus, human assessment as such, in its double connotation of “assessment of humans by humans”, is not a big issue in educational assessment practice and research. The real issue though is the deceptively simple-looking phrase “bounded scale.” Unfortunately, there is not yet a well-developed theory or methodology of measuring on bounded scales that is flexible and practical enough for recurrent or incidental use in educational assessment. Therefore, we set out to develop such a theory of measurement for bounded scales satisfying several technical and practical requirements:
1. It should enable assessors to add, multiply, and average bounded scores measured for any number of quality criteria.
2. It should be possible to convert scores from one bounded scale to another bounded scale, with the lower bound, upper bound, and neutral score as parameters.
3. There should be a systematic way of standardizing bounded scales, so that scores can be interpreted independently from the original measurement scale.
4. It should be easy to aggregate, or average, scores from a given bounded scale to get a single bounded score that represents the original bounded scores analogous to the way that the arithmetic mean represents a set of boundary-free decimal numbers.

The above list is not complete, but it highlights the key features of the concept/construct of bounded scale. It appears that the resulting scale type has all the characteristics of an algebraic module, with scores as module elements, and addition and scalar multiplication of scores as module operations. Moreover, using the two basic operations, it is rather straightforward to define the neutral score, the inverse of a given score, the subtraction operation, the absolute value of a score, the distance between two scores, and finally an n-place operation called quasi-arithmetic mean of scores. Thus, bounded scales conceived of as modules have a rich structure of their own. As modules are a generalisation of ordinary linear spaces, we may still talk about scores and their manifold manipulations as if they were decimal numbers with associated numeric operations, but when it comes to actual calculations, we must of course apply the new operations defined for bounded scales as genuine modules.
We will demonstrate the above construction for the percentage scale, i.e., the standard bounded scale of scores between 0 (0%) and 1 (100%) with a variable neutral score. This scale type is well-known and often used in educational assessment for its unique properties and relatively simple rules of calculation. For neutral scores of 0%, 50%, or 100%, there are simple (but different) formulae for score addition, multiplication, and averaging. However, for arbitrary neutral scores, we must introduce a so-called generating function with two auxiliary parameters (both dependent on the neutral score) to get the correct (generalized) definitions of the score operations. Furthermore, we will show how to get a complete characterisation of the signed percentage scale with signed scores between ‒1 (‒100%) and +1 (+100%), again with a variable neutral score, using the same approach as for the standard percentage scale, but slightly adapted formulae and auxiliary parameters.