In a regression equation applied to data in which the X and Y variables are both standardized (i.e., transformed into z scores), what is the value of the intercept?

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BlackTom AI · Accepted Answer

First, restate the scenario: we have a regression equation where both X and Y have been standardized to z-scores.

Next, recall the standardization effect: when a variable is transformed into a z-score, its mean becomes 0 and its standard deviation becomes 1.

In a simple linear regression of Z_Y on Z_X, the model has the form: Z_Y = b0 + b1 * Z_X, where Z_Y and Z_X are the standardized variables.

Consider the intercept b0 in this context. By definition of the regression equation, the expected value of Z_Y given Z_X is E[Z_Y | Z_X] = b0 + b1 * Z_X. The intercept b0 is the predicted value of Z_Y when Z_X = 0.

But since Z_X is a standardized variable with mean 0, the point Z_X = 0 corresponds to the mean of the original X transformed into a z-score. At this point, the expected value of Z_Y must equal the mean of Z_Y, which is 0 (because Z_Y is standardized to have mean 0). Therefore, E[Z_Y] = b0 + b1 * E[Z_X] = b0 + b1 * 0 = b0.

Since E[Z_Y] = 0, it follows that b0 = 0.

Intuitively, when both variables are centered and scaled, the regression line passes through the origin (0,0) in the standardized space, so the intercept is 0.

For completeness, note that the slope b1 in this standardized regression equals the correlation coefficient r between X and Y, but the key point here is that the intercept is determined by the centering to be zero, irrespective of the slope value.

If only one variable were standardized, the intercept would not necessarily be zero; the zero-intercept property arises specifically from standardizing both variables so that their means are zero.

PSYCH 3321 SP2025 (28481) Homework 9

In a regression equation applied to data in which the X and Y variables are both standardized (i.e., transformed into z scores), what is the value of the intercept?

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