# Re-scaling standardized values

I am constructing an index composed of around 100 indicators aggregated in sub-index and these in an Index, among the indicators different types can be found: categorical, ordinal, interval and ratio.

• Categorical indicator are normalized with a categorical approach (i.e. assign numbers to values).
• Ordinal, Interval and ratio are normalized by Min-Max approach.

The scale of the index, will be 0-1 and the weights for indicators will be determined by "expert opinion" approach (i.e. theoretical).

The indicators of the index and its classification in sub-indexes have been developed to represent a theoretical model, and in a next step Confirmatory Factorial analysis would be applied for proving the theoretical model with the empirical data.

The final objective of the index is not only to rank the units of analysis (communities), but to reflect the distance between them. It also aims to be reproducible and applicable to different contexts (countries), allowing to compare unit of analysis across the contexts.

During the normalization of the indicator was observed that indicators referred to % over total population (e.g. % of children and % of elders) did not vary among the different units of analysis.

Though the indicators are already normalized on a scale 0-1, as being %, if taken the raw % each unit of analysis (e.g. communities within a country) obtain a very similar value (e.g. A=0.2; B=0.21; C=0.19), thus not contributing to the variability of the index (moving across the whole range 0-1) when aggregated.

I started then a process of standardization of the obtained % (basic Z-score) aiming at bringing it again in a scale 0-1 with the max-min rescaling over the Z-score, so its variability increased, as it is relative to the sample distribution.

Could anyone suggest if this is a reasonable approach or if other approaches would be recommended?. Bibliography references are always welcome.

PS: I refer to normalize when all indicators are in the same scale of measurement; I refer to standardization, when I speak about the formula (X-MEAN/SD) also know as Z-scoring.

• Why do you think this is a good idea at all? You start with intelligible units, e.g. it could be interesting and useful to compare proportions of children and older people. So why throw away the differences? (Watch out: normalize has various different meanings, even within statistical contexts.) Jun 25, 2018 at 13:16
• The purpose is to create an index, thus, despite I know stadardizing would change the meaning of the value (i.e. not anymore a proportion, but a deviation from the average) it might help, or at least is why I thought of this, to ensure more variability across the 0-1 range, once it is normalized again into the 0-1 range again. But maybe is not the best approach...? Jun 26, 2018 at 14:58
• Creating an index -- to me not a goal in itself, at best a step towards something else. Jun 26, 2018 at 15:34
• X - MEAN/SD should usually be (X - MEAN) / SD. In fact the former is not only what you don't usually want, it only makes sense if X is dimensionless and unit-free. (if this seems obvious, fine, but if you type the equivalent in your software it won't guess what you intend.) Jan 24, 2021 at 10:30

## 1 Answer

If I understand this correctly, you have a bunch of indicators that range from 0 to 1 in theory, but some have very restricted ranges in your data set while others have wider ranges and you want to know whether to standardize them before adding them.

That would depend on whether you want each indicator to have equal importance in your index, regardless of its actual range. Without knowing a lot more about your variables and your goal for this index, it's impossible for us to tell which is better.

And there is another possibility: You could do exploratory factor analysis.

• thank you very much for your constructive answer. I tried to explain more in detail the characteristics of the index and its objectives, nonetheless i would remain available if more clarifications are needed. Jun 29, 2018 at 15:43