Does BIAS equal to MEAN ERROR Bias is defined as an average of all errors (without abs) and this is, IMO, what I want.
However, I have been asked to give MEAN ERROR. Is this the same than bias and is it wrong to call bias as mean error?
Just in case I’m messing these definitions totally, description what I try to do:
Positions are forecasted and compared to absolute values. To give the error, RMSE and BIAS are calculated.
 A: Very briefly, the MSE is the second moment of the bias. 
Let $\hat{\theta}$ be an estimator for the true quantity $\theta$. Then we have that
$Bias_\theta(\hat{\theta})=E(\hat{\theta})-\theta=E(\hat{\theta}-\theta)$
while
$MSE_\theta(\hat{\theta)} = E((\hat{\theta}-\theta)^2)$ and $RMSE \equiv \sqrt(MSE)$.
A: In plain English, in a model a Mean Error is typically equal to zero.  Otherwise, you have a really bad model.  Let's face it the simplest model is a "naive" model where you simply take the average of all your values.  Such a simple model would already have a Mean Error = 0.  If you have constructed a more complex model, and its Mean Error is different than zero, than your model clearly has a "bias."  If your model's Mean Error is positive it has an upward bias (it overestimates the actual values).  If your model's Mean Error is negative it has a downward bias (it underestimates the actual values).  This is not to be confused with the Mean Absolute Error (MAE) of your model and the Root Mean Squared Error (RMSE) also called Standard Error of your estimate.  The vast majority of models do have an MAE and an RMSE, but most often do not have a Mean Error (or bias).    
