Why is the absolute mean z-score value of two groups always exactly the same? I have 50 data points each with 7 variables.
              v1  v2  v3  v4  v5  v6  v7   
data point 1
data point 2
...
data point 50

These 50 data points are divided into two groups, with label a or b. Each group has 25 data points. I need to compute a final score using the 7 variables for each data point.
Since these 7 variables are not initially on the same scale, I standardize them all using z-score. So for each variable and using all 50 data points, I separately compute the mean and standard deviation and then compute the z-score. I then compute the final score in the following way:
score for data point 1 = v1 + v2 + (v3 - v4) - v5 - v6 - v7 (all v* are z-scores)

So here is my problem. After computing the scores for all data points, I compute the mean score for the two groups (I want to know the difference between the two groups). The two mean values are always exactly the same but in opposite directions. For example:
[group=a] mean of scores: 2.16
[group=b] mean of scores: -2.16

I'm pretty sure I'm missing something since it's unusual that the two groups always have the same absolute values in opposite directions. So I don't know why this happens or what I'm missing.

The variable v* all have zero mean already (because of the standardization). So a linear combination of them will still have a zero-mean. That's why the means of the two (equal size) groups add to 0.
