# 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?

• The general answer is: depends on your purpose and context. Zscored variable X is (X-mean)/std. Usually we take both mean and std from the same sample as X data itself. But not always. Often, for example, when we add new data cases and recalculate predictions with a model already parameterized, we do not recalculate mean or std (given new cases) but standardize the new cases with "old" values of mean and std. Another example is when X is the sample but mean or std are taken from literature. Etc Etc. So, you decide - what makes sense for you. Nov 6, 2014 at 13:57
• In your situation I'd probably use common mean and std for both samples, for example those of the combined sample. Nov 6, 2014 at 14:12
• @ttnphns. I highly appreciate your explanation. The purpose of my questions 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 substact x from the mean of the second group then divide by standard deviation for the second group and vice versa? – Nov 6, 2014 at 14:28