Abstract:

Various investigations describe errors and difficulties concerning central position measures, focusing mainly on the arithmetic mean. The tasks involved data measured on an interval scale, where equal numerical differences correspond to the same quantity differences in the underlying magnitude. Currently, the median and other statistics summaries used in graphic representations, such as boxplots, receive great emphasis.
Though these statistics are also suitable for data measured on an interval scale, they become meaningful, especially with ordinal data, which represent the order in a certain magnitude; however, equal distances in the variable may not correspond to the same differences in the amount of magnitude. Ordinal data are relevant in many applications of social sciences and play a fundamental role in exploratory data analysis. In particular, we can use the median to summarize ordinal information and compare different data distributions.
In this paper, we analyse the difficulties of a group of Spanish primary school prospective teachers (PTs) when comparing two sets of ordinal data. We gave the scores obtained in a topic by two different school groups to a sample of 60 PTs who were finishing their second year of university studies in the Faculty of Educational Sciences. They studied the subject “Mathematical Bases for Primary Education”, which has a duration of 90 teaching hours, including a block of descriptive statistics (type of data, frequency tables, graphs, central position measurements and spread) the previous year.
We asked these PTs to compare two ordinal data sets to decide which group performed better and describe the procedure used in detail. The data corresponded to an ordinal scale commonly used in Spanish schools to evaluate the students: I= insufficient (scores 0.0-4.9), S= sufficient (5.0-6.9), N=notable (7.0-8.9) and S= excellent (9.0 to 10.0). The participants should use this scale in their future professional work.
Results suggest a poor understanding of ordinal data by the PTs, few of which used the median to compare the data sets. A high proportion transformed the data to an interval scale and used the mean to compare the data, and their solution depended on the transformation used. About 20% did not use averages to solve the problem, comparing isolated values of the variable instead. These results point to the need to reinforce the PTs' knowledge of ordinal data and their properties, using examples from their future professional work, like the situation used in this work.

Keywords:

Prospective teachers, understanding, ordinal data, averages.