# Standardization of index components without mean centering

There are many questions and answers (see here for example) related to standandardization of variables, carried out by taking a value, subtracting the mean (centering) and dividing by standard deviation.

I do this in an index that has 3 components (x, y, z), to adjust their scale and make the sum possible. See here: $$R_{i} = \frac{x_{i}-\mu_{x}}{SD_{x}} + \frac{y_{i}-\mu_{y}}{SD_{y}} + \frac{z_{i}-\mu_{z}}{SD_{z}}$$

My variables are all greater or equal to zero.

Of course when values are below the mean I get a negative sum, which is a problem. So I am looking for a way to keep the final index always positive (or zero), as the original variables. But I don't want to use a min-max normalization.

If I avoid centering, but keep dividing by the SD, do I bias my index? Is the sum still correct? I mean that the variables have very different scales, so I am wondering if dividing them by SD (without substracting the mean) is enough to make them comparable (and summable). The result would be like this:

$$R_{i}^{'} = \frac{x_{i}}{SD_{x}} + \frac{y_{i}}{SD_{y}} + \frac{z_{i}}{SD_{z}}$$

MY REASONING TO SAY THAT $$R_{i}^{'}$$ IS STILL OK

I think that $$R_{i}^{'}$$ keeps the ranking of observations as $$R_{i}$$ and doesn't bias the original index formulation, because mean and SD are constant values, so I would have that the two indices only differ by a constant. See my reasoning here: $$k_{x} = \frac{\mu_{x}}{SD_{x}}, k_{y} = \frac{\mu_{y}}{SD_{y}}, k_{z} = \frac{\mu_{z}}{SD_{z}}$$ so that $$R_{i} = \frac{x_{i}}{SD_{x}} - k_{x} + \frac{y_{i}}{SD_{y}} - k_{y} + \frac{z_{i}}{SD_{z}} - k_{z}$$ and $$R_{i} = R_{i}^{'} - (k_{x}+k_{y}+k_{z})$$ so I should not have problems using $$R_{i}^{'}$$ instead of $$R_{i}$$, as they only differ by a constant.

HOWEVER I THINK I AM WRONG: as in this way I am still summing variables on different scales (since I did not subtract the mean, see comments here), right? Could a solution be to simply divide by the mean?

• You are not scaling $x_i+y_i+z_i$ by a constant, so may get different results. What is $R_i$ supposed to represent? May 27, 2021 at 15:01
• Thanks, I am not sure I get exactly the answer. Numbers will be different, but for example ranking of different observations (i) still concordant? You can imagine the index as an intelligence score, measured along 3 intelligence dimensions. May 27, 2021 at 15:05