Russell measure (RM) and enhanced Russell measure (ERM) are popular non-radial measures for efficiency assessment of decision-making units (DMUs) in data envelopment analysis (DEA). Input and output data of both original RM and ERM are assumed to be deterministic. However, this assumption may not be valid in some situations because of data uncertainty arising from measurement errors, data staleness, and multiple repeated measurements. Interval DEA (IDEA) has been proposed to measure the interval efficiencies from the optimistic and pessimistic viewpoints while the robustness of the assessment is questionable. This paper draws on a class of robust optimisation models to surmount uncertainty with a high degree of robustness in the RM and ERM models. The contribution of this paper is fivefold; (1) we develop new robust non-radial DEA models to measure the robust efficiency of DMUs under data uncertainty, which are adjustable based upon conservatism levels, (2) we use Monte-Carlo simulation in an attempt to identify an appropriate range for the budget of uncertainty in terms of the highest conformity of ranking results, (3) we introduce the concept of the price of robustness to scrutinise the effectiveness and robustness of the proposed models, (4) we compare the developed robust models in this paper with other existing approaches, both radial and non-radial models, and (5) we explore an application to assess the efficiency of the Master of Business Administration (MBA) programmes where data uncertainties influence the quality and reliability of results.