Standardized Euclidean distance ($d_S(x,y)=\sqrt{{\sum }_j(x_j-y_j{)}^2/s_j^2}$)

With fixed positive scale estimates $s_j$, this is Euclidean distance in standardized coordinates. It removes marginal units but ignores correlations. Small estimated variances can greatly amplify measurement noise; a zero scale must be handled separately.