Normalization of power-law distributed variables. Z-scores or Min-Max?

I need to make a composite index from the sum of three power-law distributed variables, which vary on different scales and have different variances. For each variable there are many observations with very low scores and few observations with high scores.

I need to normalize the variables to obtain a common scale, before summing them to obtain a single score of the final index. I'm considering two possibilities:

Min-Max Normalization

(Xi - min (X)) / (max(X) - min(X))

Standardization (Z-scores)

(Xi - mean(X)) / std(X)

Which solution is appropriate, given the power-law distribution of the three variables? Or are they both wrong? Why?

EDIT Please have a look to an example of the distribution I am referring to:

I have three variables distributed like X and I need to normalized them before making a sum of the three.

• Why do you think the variables follow a power law when they are the sums of three exponential variates, which do not follow a power law? Have you left something out? Commented Dec 26, 2017 at 23:23
• "many observations with very low scores and few observations with high scores" would suggest skewed, with a possibly unimodal, perhaps even monotonic density, but none of those things are sufficient for a power law, which says something more specific about the shape. If your variates were ordinary one-parameter exponential and you wanted to render them of comparable scale, dividing by the mean would be the most obvious thing to do. Commented Dec 27, 2017 at 4:17
• I think the distribution is a power law. So I edited the question and added a graph. Moreover I never meant that the final index has a power-law distribution. I meant that the variables that I need to sum follow a power-law distribution. They change on different scales, and have different SD. Before summing them I want to normalize them, choosing the best way given their distribution. Commented Dec 27, 2017 at 8:51
• Whether one way of standardizing is superior to another one will likely depend on what you plan on doing with your sum score afterwards. Commented Dec 27, 2017 at 9:40
• @Stephan thank you. Could you explain more the difference and what you mean? I'm constructing an index for comparison among observations and ultimately for forecasting purposes. Commented Dec 27, 2017 at 11:24