I split my dataset into 2 parts: 75% of it is the training set, 25% of it is the test set. Then I estimated the logistic regression parameters in the training set and I compute the Area Under the ROC Curve (AUC) (of the model estimated in the training set) from the test set. Since the test set is formed by 2500 observations on 20 variables, I was expected to get 2500 AUC, one each row. Why I just get one AUC?
Area under the ROC curve is a summary statistics for the performance on your entire test set. It quantifies your model's performance on all 2500 observations, yielding a single number.