Mean signed deviation

In statistics, the mean signed difference (MSD),^[1] also known as mean signed deviation, mean signed error, or mean bias error^[2] is a sample statistic that summarizes how well a set of estimates ${\displaystyle {\hat {\theta }}_{i}}$ match the quantities ${\displaystyle \theta _{i}}$ that they are supposed to estimate. It is one of a number of statistics that can be used to assess an estimation procedure, and it would often be used in conjunction with a sample version of the mean square error.

For example, suppose a linear regression model has been estimated over a sample of data, and is then used to extrapolate predictions of the dependent variable out of sample after the out-of-sample data points have become available. Then ${\displaystyle \theta _{i}}$ would be the i-th out-of-sample value of the dependent variable, and ${\displaystyle {\hat {\theta }}_{i}}$ would be its predicted value. The mean signed deviation is the average value of ${\displaystyle {\hat {\theta }}_{i}-\theta _{i}.}$