Range of standardized beta in linear regression In regression, the beta value represents the increase in $y$ if $x$ changes one unit. The standardized beta gives the same information, but for increase in standard deviations. But why can't $y$ increase by 2 SDs if $x$ increases by 1 SD, for example? I mean, if the influence is heavy enough?
Also, I assumed the correlation coefficient is just a measure for how much two variables vary together, independent of the slope. But in regression, the slope is dependent on the correlation coefficient. Are these two different information?
 A: Simple linear regression provides a simple way of addressing the idea behind your question.  Recall that 
$\hat{\beta_1}=rS_y/S_x$ and
$\hat{\beta_0} = \bar{Y}-\hat{\beta_1}\bar{x}$, which leads to the linear regression equation
$\hat{Y} = \hat{\beta_0} + \hat{\beta_1}x$ which can be rewritten as 
$\hat{Y} = (\bar{Y}-r\dfrac{S_y}{S_x}\bar{x}) + r\dfrac{S_y}{S_x}x$.
After a little algebra, the previous equation can be written as 
$\dfrac{\hat{Y} -\bar{Y}}{S_y} = r(\dfrac{x -\bar{x}}{S_x})$
Now, notice that if $x$ is is one sd above it's mean, then $y$ is predicted to be $r$ sd's above or below it's mean depending on whether the correlation $r$ is positive or negative, and $r$ is of course bounded between $-1$ and $1$  This is known as the regression effect, or regressing to the mean, since $Y$ is predicted to be closer to it's mean than $x$ is to its mean.  
Analogously, let $\hat{Y_1}$ and $\hat{Y_2}$ correspond to the predicted values for $x_1$ and $x_2$ respectively.  It can then be shown after a little algebra that  
$\dfrac{\hat{Y_2} -\hat{Y_1}}{S_y} = r(\dfrac{x_2 -x_1}{S_x})$, which implies that a 1 sd change in $x$ can be associated with no more than an $r$ sd predicted change in $Y$.
