`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?