Relative scores such as Local Outlying Factor and mass ratio have been shown to be better scores than global scores in detecting anomalies. While this is true, our analysis reveals for the first time that these relative scores have a key shortcoming: anomalies have greatly different relative scores if they are located in different regions where the curvatures of the density surface are very different. As a result, the low-score anomalies could be ranked lower than some normal points. This revelation motivates (i) a new score called Neighbourhood Contrast (NC) which produces approximately the same high scores for all anomalies, regardless of varying curvatures of the density surface in different regions; and (ii) an anomaly detection method based on NC. Our experiments show that the proposed method which employs the new score significantly outperforms methods using the aforementioned relative scores on benchmark datasets.