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Prior proposals for cumulative statistics suggest making tiny randomperturbations to the scores (independent variables in a regression) in order toensure the scores' uniqueness. Uniqueness means that no score for any member ofthe population or subpopulation being analyzed is exactly equal to any othermember's score. It turns out to be possible to construct from the original dataa weighted data set that modifies the scores, weights, and responses (dependentvariables in the regression) such that the new scores are unique and (togetherwith the new weights and responses) yield the desired cumulative statistics forthe original data. This reduces the problem of analyzing data with scores thatmay not be unique to the problem of analyzing a weighted data set with scoresthat are unique by construction. Recent proposals for cumulative statisticshave already detailed how to process weighted samples whose scores are unique.

Ties in ranking scores can be treated as weighted samples