The performance of a school system can be evaluated through the learning levels of the pupils, usually summarized by school mean scores. The variability of the mean scores among schools is rarely studied in detail, though it is a crucial issue especially in primary schools: in fact, a high variability among schools raises doubts on the capacity of the system to guarantee equal educational opportunities. To investigate the patterns of variability in Italy, we analyse data from INVALSI, the Italian national institute for the evaluation of the school system, which regularly carries out standardized tests to assess the learning levels of the pupils at various grades. We consider the mathematics test administered to fifth-grade pupils at the end of the 2008/2009 year, along with a pupil's questionnaire for measuring socio-economic factors. The analysis is performed using a random intercept linear model on the Rasch score of the mathematics test, with pupil-level errors depending on gender and school-level errors depending on the geographical area. The model includes several demographic and socio-economic explanatory variables and some compositional variables obtained as school means of pupil variables. The results show a considerable increase in the residual variance among schools when going from North to South, pointing out a serious issue of fairness in Southern Italy. The situation is mitigated by the finding that a substantial part of the residual variance among schools is due to a few schools with exceptionally positive results. (Contains 6 tables and 3 figures.)