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

#### Comfort Eagle

##### New Member
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?

#### hlsmith

##### 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?

#### Buckeye

##### 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.

#### Miner

##### 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.