Reverse scaling

Question

I had to standardize a range of numbers to a scale from 0 to 1. The end result is interpreted in the new range as higher the number in new scale is positive (0.9 is better than 0.5). How is it possible to rearrange the formula so that the scale can be reversed, that is lower number in the scale is positive?

Answer 1

Let the old numbers be $x_1, x_2, x_3 \dots x_n$. If you want to scale them so that the smallest number becomes $0$ and the largest becomes $1$, the new numbers are $y_1, y_2, \dots y_n$, where $$ y_k = \frac{x_k - x_{\text{min}}}{x_{\text{max}} - x_{\text{min}}}, \qquad k = 1, 2, 3 \dots ,n $$ where $x_{\text{min}}$ is the smallest number in the original set and $x_{\text{max}}$ is the largest number in the original set. You can see easily that now $0 \leq y_k \leq 1 $. If you want to turn it upside down, so that the smallest number in the original set corresponds to $1$ and the largest corresponds to $0$, you just have to take one minus the previous result, or $$ y_k = 1 - \frac{x_k - x_{\text{min}}}{x_{\text{max}}- x_{\text{min}}} $$