Bell Curve algorithm to adjust set of scores

Question

I am faced with a challenge whereby the business user would like a "Bell curve" applied to their scoring.

This system scores people on a 1-5 point scale. The requirement is that most people score too generously, and they would like for the scores within a group of people to be adjusted down (or up) based on a bell curve.

I would assume then that they are trying to make the majority of people sit at the median level i.e. 3 in this case. I am not sure that the client is correct in their terminology wrt Bell Curve but the requirement is that the scores are leveled out to the 3 level.

What would be the best algorithm to achieve this?

For example, in one group they might have a 3,4,4,3,5 group of scores. in this case the scoring is on average higher than 3.What would be a fair way to adjust all these scores so that the "bell curve" is applied?

Answer 1

The bell curve is the Probability Distribution Function (PDF) of the normal distribution, so that's your goal.

The key to this transformation is the Cumulative Distribution Function (CDF). In words, "y% of the values are less or equal to x". You can easily table the CDF that you have in your input. The CDF of the normal distribution is also known (integral of the bell curve).

Together, this gives you: "y% of the scores are less than x, but according to the normal distribution, y% of the scores should be less than x', therefore the correction is x -> x' "

Mathematically, this is done via the probit function.

Answer 2

You usually assume that your data fit a distribution instead of transforming your data into a given distribution.

If your input data fit a normal distribution ("bell curve"), then you can adjust by simply add/remove the same value from all the sample.

The distribution will be preserved, only the mean will change.

If you want to center your distribution on a given mean, just add the difference between your target mean and the actual one.