Abstract:

In Higher Education, achievement assessment of students (aka summative evaluation) is nearly always done by qualified humans, called assessors, by means of counting and/or scoring on bounded scales related to quality criteria that are specific for 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 available (though for different reasons, we consider psycho-metric tests and learning analytics as non-viable, though in principle available). 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 device or software (exceptions, of course, are the simplest forms of assessment like multiple-choice tests and short essays).

Thus, human assessment per se is not an issue in educational assessment practice. The real issue 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 on bounded scales that would satisfy a long list of technical and practical requirements.

First, it should enable assessors to add, multiply, and average bounded scores measured on any number of quality criteria. Second, 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. Third, there should be a systematic way of standardizing bounded scales, so that scores can be interpreted independently from the original measurement scale. Fourth, it should be easy to aggregate, or aver-age, scores from some bounded scale to get a single bounded score called quasi-arithmetic mean, that represents the given bounded scores in the same way that the arithmetic mean does for a set of boundless decimal numbers.

The above list is not complete, but it highlights the key features of the bounded scale concept/construct. 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. Using those module operations, it appears straightforward to define the concept of quasi-arithmetic mean. As modules are a generalization of linear spaces, we may still talk about scores and their manipulations as if we were talking about decimal numbers and number operations, but when it comes to concrete calculations, we must apply the operations defined for bounded scales.

We will demonstrate the construction of the percentage scale, i.e., the standard bounded scale of numbers between 0 (0%) and 1 (100%). For neutral scores of 0%, 50%, or 100%, there are simple formulae for score addition, multiplication, and averaging. However, for arbitrary neutral scores, we must introduce a so-called generating function with two auxiliary parameters (called α and β, both dependent on the neutral score) to get the correct definitions for the score operations. We will also show how to get a complete characterization of the signed percentage scale (with scores between -100% and +100%) using the same approach but slightly different formulae and auxiliary parameters.

Keywords:

Assessment, scoring, bounded scale, percentage scale, signed percentage scale, quasi-arithmetic, higher education.