I have a set of several test scores for a large group of students.

The goal I have is to quantify each students' overall performance across each of these tests, as it relates to his or her population of peers.

The distribution of the scores differs, such that the range, median, and std deviation are not constant.

Each of the scores has a certain priority or weight. For example, score 1 might represent 12%, score 2: 29%, score 3: 33%, etc., with all of the weights adding up to 100 in order to create a weighted average.

My first thought was to standardize each score on a range of 0 to 1, create the weighted average based on those values, and create a percentile rank from that, but in doing so, each score's placement within its respective distribution is not adequately recognized.

If I take the percentile rank of each score and multiply it by it's respective weight, will I end up with a statistically valid representation of total value, as it relates to the aggregated population of scores/peers? I realize that this in itself will not be a percentile.

Would standardizing via z score be a more sound method? If so, why? And what would the weighted aggregation be?

Any comments or alternative methods are welcomed.

The only requirements are:

  1. Standardized score for each test based on relevant placement in population
  2. Logical aggregation of scores (weighted) that adequately represents the sum of its parts, such that the number can be interpreted as overall performance respective to the greater population

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