This paper analyzes nitrogen (N) flows on organic and conventional dairy farms in Sweden, and compares three indicators for the N pollution associated with the milk: (1) the farm-gate N surplus, (2) the chain N surplus, and (3) the N footprint. We find that, compared to indicators based on N surplus, the N footprint is a more understandable indicator for the N pollution associated with a product. However, the N footprint is not a replacement for the often-used farmgate N surplus per unit area, since the two indicators give different information. An uncertainty analysis shows that, despite the large dataset, 1566 conventional and 283 organic farms, there is substantial uncertainty in the indicator values, of which a large part is due to possible bias in estimates of biological N fixation (BNF). Hence, although the best estimate is that conventional milk has 10-20% higher indicator values than organic, it is conceivable that improved estimates of BNF will change that conclusion. All three indicators simplify reality by aggregating N flows over time and space, and of different chemical forms. Thus, they hide many complexities with environmental relevance, which means that they can be misleading for decision-makers. This motivates further research on the relation between N surpluses and N footprints, and actual environmental damages.