How to linearly transform variables before t-test if their range is unknown

I want to conduct a paired t-test to estimate the difference between two variables x1 and x2.

For sake of simplicity, let's assume that the two variables measure how well participants in a study performed on two cognitive reasoning tests. In the first test (x1, float), participants receive a score between 1 and 5. The range of the score in the second test (x2, float) is theoretically between -1 and 1. However, I cannot know if these extreme scores are practically achievable. The empirical range in my data set for x2 is between -.41 and .65. Note that my sample size is quite small (N = 140).

Now, usually, my approach would be to transform/rescale x2 to the same range as x1, so to take on values between 1 and 5 in order to be able to compare them via t-test. To do so I need to know from what range x2 should be scaled. Choosing the theoretical range (-1 to 1) doesn't seem right, since I cannot be sure if a human could ever achieve these values. Choosing the empirical range (-.41 and .65) surely won't be correct, since I cannot be sure that my small sample exhausted the most extreme scores.

Long story short: Is there a sensible way of linearly transforming x2 to the same range of x1 to make these two variables comparable by a t-test?


Less is more. Stay pure. Stay poor.
Can you provide a sample of your data - it can be made up. Can you have any value within these two ranges?


Active Member
I'm a bit confused as I thought the dependent variables should be the same in a paired t-test. For example, a before and after measure of heart rate. Suppose after an intervention. I don't see how we can compute a paired t-test with different measures/scales.


TS Contributor
This does seem more like a relationship between x1 and x2 rather than a difference. Plot the paired points with x1 on one axis and x2 on the other axis to see if there is a relationship. If there is, you can try a regression.


Ambassador to the humans
Maybe you can tell us what you're researching and what your hypothesis is. It's not clear to me why you want to do what you're describing but I think it might just be lost in translating your question into math.


Well-Known Member
The problem is that you can transform any variable to have any mean and SD you wish. If you try to find a suitable transformation, you need some criterion to decide if that transformation is in some sense the "best" one. I suspect that if you work out some scheme of doing this (maximum likelihood for example) you will find the means and SDs for both variables will be the same.
Coming from a different tack, can you invent some data where it is obvious to the eye that there is a difference in the means after scaling? I can't.