# Constructing an index: is it necessary to standardize? Which is the best solution?

I'm constructing an index made by the sum of three variables, which all vary in the range [0,1] (as they have been divided by their theoretical maximum).

Depending on the sample analyzed, it is not always assured that the three variables have the same variances (and st. deviations). In order to attribute them the same importance, one solution, before making the sum, would be to standardize them (subtracting the mean and dividing by the standard deviation). In this case the index would correspond to the sum of standardized scores.

My questions are: - is standardization necessarily required? does it have other drawbacks (apart from changing the scale)? I cannot always assume that the variables follow a specific distribution. - Would you reccommend standardization or is there another more appropriate solution? - Could the sum of unstandardized scores be OK?

EDIT: I'm adding a data example after Placidia's answer, to provide a better description of my case and make the question more generalizable. Please consider an example for the three variables I want to sum in a final index and their transformations as in the following table. I have ten observations to rank.

The final index can be obtained either by: (1) the sum of raw values divided by the theoretical maximum; (2) or the sum of the rescaled values in [0,1]; (3) or the sum of the standardized values (subtract mean and divide by SD); (4) or the sum of percentile scores. Please see the table and the graph: Different choices could produce different rankings for the 10 observations. Which solution would be more appropriate? Is there a totally wrong one?

• There are many ways to normalize variables before combining in one index. We don't know your specific data and situation. I would recommend to lie on a sofa and meditate over variants and their implications for you. Two topics I might point to as especially important: (i) do you really nead standardization based on subtraction of the mean? (ii) do you really want to standardize with sample values statistics? – ttnphns Dec 22 '17 at 14:27
• Thanks, I would love to receive your help to better understand the implications of these choices. – Forinstance Dec 22 '17 at 14:34
• your question is ambiguous. Terms like normalisation, standardisation and standard score (z) or standard normal (0,1) distribution. have different themes and have contextual meanings e.g. measurement theory and mathematical-statistics. You are citing additive model without going into issues of linearity or non-linearity of variables . Rescaling ? for what reasons ? – Subhash C. Davar Dec 23 '17 at 9:15
• It is exactly to address these issues that I need your help. Apart from the mathematical solution, what are the things that I need to consider? And what would be appropriate for the data in the example? – Forinstance Dec 23 '17 at 12:44
• Rescaling was mostly a test. I think the real choice to make is between dividing by the theoretical maximum (given the sample) or subtracting the mean and dividing by the standard deviation. – Forinstance Dec 23 '17 at 12:49