# How to interpret beta coefficient while regressing a normalized value on a dummy variable?

My regression is of the form:

$$y = \beta_{0} + \beta_{1} Treat + \beta_{2} X + u$$

The $$y$$ variable is normalized. The $$Treat$$ variable is a binary dummy.

How should I interpret $$\beta_{1}$$ or the effect of the treatment? Is it something like, "getting the treatment changes $$y$$ by t standard deviations" where $$t=\beta_{1} / \sigma_{y}$$?

Thanks!

The model predicts that the treatment group scores $$\beta_1$$ standard deviations higher on the (unstandardized) $$y$$ variable than the non-treatment group, controlling for the $$X$$ variable. No need to divide $$\beta_1$$ by $$\sigma_y$$ if $$y$$ is already standardized and no need to divide $$\beta_1$$ by $$\sigma_{treat}$$ if $$treat$$ is a dummy code.