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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?

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    $\begingroup$ 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. $\endgroup$
    – ttnphns
    Nov 6, 2014 at 13:57
  • $\begingroup$ In your situation I'd probably use common mean and std for both samples, for example those of the combined sample. $\endgroup$
    – ttnphns
    Nov 6, 2014 at 14:12
  • $\begingroup$ @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? – $\endgroup$
    – dr.green
    Nov 6, 2014 at 14:28

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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.

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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.

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