Ranking different variables by standardization I have seven numeric variables and 12 cases. All of them are measures for how well 5 different casinos are performing. The higher the variable, the better the performance. Four of the variables are: attendance figures, no. of new members, visit frequency count, and spend per head. I need to combine these four variables into single performance score which will allow me to successfully rank in order of performance.
My idea is very simple: Standardize all four variables, sum them up and whatever you get is the score which should be rank-order. Do you see any problem with applying this approach? Is there any other (better) approach that you would recommend?
 A: What method of combining is best will depend on what you want to achieve.
If you standardize and sum, as you suggest, then you are weighting each variable equally on the basis of its own standard deviation - that is, you are saying that a 1 SD increase in one variable is just as good as a 1 SD increase in another.  Perhaps that is what you want.  Perhaps not.
You might want to simply assure that each has the same mean, by multiplying each by a constant so that they do. Then increases are equally good if they are of the same extent in raw figures, adjusted only for scale.
You can use substantive knowledge to suggest different weights. 
But I am curious - you write that you have 7 variables and 12 cases for 5 casinos. I would guess that these 12 cases might be months. If so, there might be some more complex methods to be used. 
A: You could do standardization:
$$x' = \frac{x- \mu}{\sigma}$$
But as Peter pointed out, the representation of this procedure is somehow strange in your application.
In this case, I think normalization to range 0 and 1,
$$x' = \frac{x-\text{min}}{\text{max}-\text{min}},$$
would be a better choice, as its representation is more interpretable and straightforward: After normalization, the weight you apply on a variable directly indicates the importance of that variable. Since each variable in your application has the monotone property, i.e., the more the better, all the weights will be positive values.
As there is no training data available here for choosing the weights in a data-driven way, I would suggest you pick up the weights based on your domain knowledge as a first step.
