# Scaling data, keeping relative numerical dispersion

I have values two vectors that are spread across ranges of [-5, -1] and [1, 5] respectively. They can be decimals. Here is a toy example:

x <- c(-5, -4.5, -4.4, -3, -1.95, -1)
y <- c(1, 2, 3, 3, 3.5, 3.7)


I concatenate these to deal with one vector: z <- c(x, y) So we can see that ydoes not actually have the full range of possible values, i.e. it's largest value is 3.7, not 5.

There is nothing between -1 and 1 apart from some zeros, so nothing like 0.7.

I want to scale all the data to a [-1, 1] range, but if I use the common method of normalisation with my own function:

common_scaler <- function(input_data, new_min, new_max) {

## Define parameters for scaling
old_min <- min(input_data)
old_max <- max(input_data)
a <- (new_max - new_min)/(old_max - old_min)
b <- new_max - a * old_max

## Scale the input_data
output_data <- a * input_data + b
return(output_data)
}


The output is all nicely between -1 and 1, however there is a value of +1 included.

> common_scaler(z, -1, 1)
>  -1.00000 -0.88506 -0.86207 -0.54023 -0.29885 -0.08046  0.37931  0.60920  0.83908  0.83908  0.95402  1.00000


We know there was no +5 in the original data, so this leads me to believe we have fundamentally altered the properties of the data set. If I take the mean before and after, the interpretation will be different. For example:

> mean(z)
 -0.3042
> mean(scaler2(z, -1, 1))
 0.0795


This changes the whole outcome for my research, i.e. the results being positive or negative.

I thought "All I want to do is scale the data down to my chosen range, so I need a factor to divide through by". I took the maximum value of the data set (so from x and y) and divide through by that, meaning my largest value should now be -1, while on the positive side, I have 0.74 (3.7/5). Here is the function, with the rsults and the comparison of means:

my_scaler <- function(x) {x / max(sqrt(x*x))}

> my_scaler(z)
 -1.00 -0.90 -0.88 -0.60 -0.39 -0.20  0.20  0.40  0.60  0.60  0.70  0.74

> mean(z)
 -0.3042
> mean(my_scaler(z))
 -0.06083


This seems more reasonable to me, however I don't know if I am breaking some fundamental rules of scaling/transforming data. Have I changed the distribution in some detrimental way?

I actually have a data set with 1000 observations of 70 variables, each with their own scales. The desired result is that I have can take perform rowMeans over all the data, without getting any kind of distortion on the results. Maybe I could take the mean first, then scale all the data with my_scaler?

I think your my_scaler function in the last snippet will fail for the same motive the function common_scaler. What you want (depending what you want) is a linear mapping between two scales, for example $$T:[-5,5] \to [-1,1]$$ This being linear, it's derivative is a constant, so every other linear function (mean, variance, etc.) will be scaled with the same constant.