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In statistics, the mean signed difference (MSD), also known as mean signed deviation and mean signed error, is a sample
statistic A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hy ...
that summarises how well a set of estimates \hat_i match the quantities \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 In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is ...
model has been estimated over a sample of data, and is then used to extrapolate predictions of the
dependent variable Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or dema ...
out of sample after the out-of-sample data points have become available. Then \theta_i would be the ''i''-th out-of-sample value of the dependent variable, and \hat_i would be its predicted value. The mean signed deviation is the average value of \hat_i-\theta_i.


Definition

The mean signed difference is derived from a set of ''n'' pairs, ( \hat_i,\theta_i), where \hat_i is an estimate of the parameter \theta in a case where it is known that \theta=\theta_i. In many applications, all the quantities \theta_i will share a common value. When applied to
forecasting Forecasting is the process of making predictions based on past and present data. Later these can be compared (resolved) against what happens. For example, a company might estimate their revenue in the next year, then compare it against the actual ...
in a
time series analysis In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
context, a forecasting procedure might be evaluated using the mean signed difference, with \hat_i being the predicted value of a series at a given
lead time A lead time is the latency between the initiation and completion of a process. For example, the lead time between the placement of an order and delivery of new cars by a given manufacturer might be between 2 weeks and 6 months, depending on vari ...
and \theta_i being the value of the series eventually observed for that time-point. The mean signed difference is defined to be :\operatorname(\hat) = \frac\sum^_ \hat - \theta_ .


See also

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Bias of an estimator In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In st ...
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Deviation (statistics) In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some other value, often that variable's mean. The sign of the deviation reports the direction of that difference (the deviation is po ...
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Mean absolute difference The mean absolute difference (univariate) is a Statistical dispersion#Measures of statistical dispersion, measure of statistical dispersion equal to the average absolute difference of two independent values drawn from a probability distribution. ...
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Mean absolute error In statistics, mean absolute error (MAE) is a measure of errors between paired observations expressing the same phenomenon. Examples of ''Y'' versus ''X'' include comparisons of predicted versus observed, subsequent time versus initial time, and ...
Summary statistics Means Distance {{statistics-stub