Methods for standardizing / normalizing different rank scales

Question

I know there is the normal subtract the mean and divide by the standard deviation for standardizing your data, but I'm interested to know if there are more appropriate methods for this kind of discrete data. Consider the following case.

I have 5 items that have been ranked by customers. First 2 items were ranked on a 1-10 scale. Others are 1-100 and 1-5. To transform everything to a 1 to 10 scale, is there another method better suited for this case?

If the data has a central tendency, then the standard would work fine, but what about when you have more of a halo effect, or some more exponential distribution?

Answer 1

For item-ratings type of data with the restriction that an item's rating should be between 1 and 10 after transformation, I would suggest using a simple re-scaling, such that the item's transformed rating $x_t$ is given by:

$$x_t = 9\left(\frac{x_i - x_{min}}{x_{max} - x_{min}}\right) + 1$$

where $x_{min}$ and $x_{max}$ are the minimum and maximum possible rating in the specific scale for the item, and $x_i$ is the item rating.

In the case of the above scaling, the transformation applied is independent of the data. However, in the normalization, the transformation applied is dependent on the data (through mean and standard deviation), and might change as more data becomes available.

Section 4.3 on page 30 of this document shows other ways of normalizing in which your restriction (transforming to the same absolute scale) might not be preserved.

Answer 2

There is another interesting technique in this paper called the decoupling normalization method. I've used it and found that the results are good. This finds the affinity of user to a particular item and then you can scale it to a scale of 5 or 10, whichever you want. Hope it helps.