In statistics, the mean signed difference, deviation, or error (MSD) is a sample statistic that summarises how well a set of estimates θ ^ i match the quantities θ 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 θ i would be the i-th out-of-sample value of the dependent variable, and θ ^ i would be its predicted value. The mean signed deviation is the average value of θ ^ i − θ i .
The mean signed difference is derived from a set of n pairs, ( θ ^ i , θ i ) , where θ ^ i is an estimate of the parameter θ in a case where it is known that θ = θ i . In many applications, all the quantities θ i will share a common value. When applied to forecasting in a time series analysis context, a forecasting procedure might be evaluated using the mean signed difference, with θ ^ i being the predicted value of a series at a given lead time and θ i being the value of the series eventually observed for that time-point. The mean signed difference is defined to be
MSD ( θ ^ ) = ∑ i = 1 n θ i ^ − θ i n . 