Since binary variables are 0 or 1 to begin with, it doesn't make sense to normalize them.

On the other hand, we normalize continuous variables which usually results in values greater than 0 and less than 1.

There seems to be an asymmetry here. The continuous variables are normalized, but the binary variables are not. The magnitude of the binary variables is either as high or as low as it can possibly be, not because of any information in the variable itself but because that is our convention. On the other hand, the continuous variable can take on any value in the range 0 or 1. Doesn't this make it inappropriate to use both continuous variables and binary variables as predictors in a model? Am I overthinking this?

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    $\begingroup$ Hi John, could you elaborate on your logic, going from your explanation, to the last sentence? I can understand your explanation, but not why you draw that conclusion from it. Just as a first guess.. Don't you think that the character of the variable (binary or continuous), is less what we choose, but more a characteristic of the underlying information? $\endgroup$ – Tom Apr 28 at 17:48
  • $\begingroup$ @Tom I understand if my explanation is unclear because I am not even clear on this myself. I'm confused about why the convention of making binary variables 0 or 1 is reasonable. It is to some extent arbitrary; if we chose the convention to be 0.3 and 0.6 instead, then the character of the variable would be preserved but the variable would have less of an effect on the results of the model. Of course, not using 0 and 1 may cause mathematical inconvenience depending on the model. But my point is that unlike with continuous variables, we are choosing 0 and 1 for the binary variable. $\endgroup$ – John S Apr 28 at 17:52
  • $\begingroup$ I'm starting to understand what you are getting at. But I think you are indeed really over thinking this ;). Although the value of 1 might be arbitrary, the value of zero is not. Have you checked out Wooldridge Introductory Econometrics, chapter seven I believe on Dummy variables? $\endgroup$ – Tom Apr 28 at 18:00
  • $\begingroup$ If you would change the 1 values of binary independent variable, from 1 to 2. The estimate changes because you changed the measurement. But nothing else will change.. Have you tried messing around with this in a statistical program to see what happens? $\endgroup$ – Tom Apr 28 at 18:05
  • $\begingroup$ @Tom Funny you say that, I'm in the middle of playing around in MATLAB trying exactly that. Your comment about how 1 is arbitrary but 0 is not was eye opening. I will checkout the textbook chapter you mentioned, assuming I can find the text. $\endgroup$ – John S Apr 28 at 18:13

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