How are constant and heterocedastic variances different from each other? Homocedasticity is assumed for Ordinary Least Square but for Weighted Least Square, the variance is different given $x_i$. What do we mean by a constant variance? and also what do we mean by a heterocedastic variance?
 A: Constant variance means that over all possible values of your independent variables, the variance of your dependent variable (DV) is about the same. If you made a scatter plot of the DV's variance, the slope of the scatter plot would be approximately zero regardless of where it crosses the y-axis. 
Heteroskedastic variance is different from this. Heteroskedastic variance is when the variability of your dependent variable changes across different values of the independent variable(s). When you make a scatter plot of the DV's variance, there will be a discernible pattern. It might be shaped like a cone, parabola, etc. Since the slope of this scatter plot cannot be expressed in terms of a constant (i.e. y=3, or y=1, or ANY constant), the slope is not zero and therefore it is not constant.
In the context of your question, Weighted Least Squares circumvents the OLS assumption of constant variance. In OLS, all independent variables are (basically) weighted equally and when homoskedasticity is violated, OLS will give misleading/erroneous results...which is no bueno.
Does this make sense?
