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Mean signed deviation

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Source updated November 25, 2024

In statistics, the mean signed difference (MSD), also known as mean signed deviation, mean signed error, or mean bias error is a sample statistic that summarizes how well a set of estimates ${\hat {\theta }}_{i}$ 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 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 $\theta _{i}$ would be the i-th out-of-sample value of the dependent variable, and ${\hat {\theta }}_{i}$ would be its predicted value. The mean signed deviation is the average value of ${\hat {\theta }}_{i}-\theta _{i}.$

01Definition

The mean signed difference is derived from a set of n pairs, $({\hat {\theta }}_{i},\theta _{i})$ , where ${\hat {\theta }}_{i}$ 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 in a time series analysis context, a forecasting procedure might be evaluated using the mean signed difference, with ${\hat {\theta }}_{i}$ being the predicted value of a series at a given lead time and $\theta _{i}$ being the value of the series eventually observed for that time-point. The mean signed difference is defined to be

$\operatorname {MSD} ({\hat {\theta }})={\frac {1}{n}}\sum _{i=1}^{n}{\hat {\theta _{i}}}-\theta _{i}.$

02Use Cases

The mean signed difference is often useful when the estimations ${\hat {\theta _{i}}}$ are biased from the true values $\theta _{i}$ in a certain direction. If the estimator that produces the ${\hat {\theta _{i}}}$ values is unbiased, then $\operatorname {MSD} ({\hat {\theta _{i}}})=0$ . However, if the estimations ${\hat {\theta _{i}}}$ are produced by a biased estimator, then the mean signed difference is a useful tool to understand the direction of the estimator's bias.

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Sources and credits

This article is adapted from the Wikipedia article “Mean signed deviation”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Mean signed deviation

01Definition

02Use Cases

Sources and credits

Related topics

Bias of an estimator

Deviation (statistics)

Mean absolute difference