What is the correct method to calculate the Z score for the mean of a variable between two groups I have two groups, 23 Patients and 27 controls with a list for their age and the head circumference measurements. I want to do scatter plot for the Z score of age by the Z score of head circumference. 
I will simplify the matter by the following table for 4 controls and 5 patients:
GROUP       age    head circumference   
Patient     5         20 cm
Patient     7         19 cm
Patient     4         22 cm
Patient     8         17 cm
Patient     6         18 cm
Control     9         17 cm
Control     3         23 cm
Control     4         21 cm
Control     6         24 cm

My question is about the correct method in this case to calculate the Z score for age and head circumference between the groups (I will mention the possible ways that I am thinking about regarding how to calculate Z score for every age point for example)
1. Z score patients = { age - mean age (patients)}/STD(patients) and Z score controls = { age - mean age (controls)}/STD(controls)
2. Z score patients = { age - mean age (controls)}/STD(controls) and Z score controls = { age - mean age (patients)}/STD(patients)
3. Z score patients = { age - mean age (controls)}/STD(controls) and Z score controls = { age - mean age (controls)}/STD(controls)

The purpose of my question is to inquire that if I have the means for two groups and I want to compare the z score for the means between the groups using scatter plot for the z scores. In order to calculate the Z score for the first group is it correct to subtract the age (in my example) from the age mean of the second group then divide by the standard deviation of the age in the second group and vice versa?
 A: Actually there is no difference between the three methods that you mentioned before and the results. 
Simply try to do scatter plot for the Z score by doing calculations with any of the previous methods and you will have the same results.
A: Your question needs to be reformulated because is seems a little ambiguous as stated in my opinion.


*

*first off there is no such thing as a between sample z-score as far as I know. Z scores are single population or sample relative measures. They might help compare the relative positions of two otherwise incomparable raw values within their respective samples or populations. But never does it compare directly values between two samples or populations

*Second,  as pointed out by ttnphns, not all ratios (X-a)/b qualify as z-score. This is precisely why the subtrahend (a) has to be the mean of a sample or population containing X and the divisor (b), its standard deviation. Otherwise it is simply a rescaled translation.

*Finally, note that rescaled translations keep the sample or population order. No matter what you divide by or what you subtract, the within sample order stays the same.
