Residuals

Residual is a term with several closely related meanings that occur in the fields of business, finance, optimization, and statistics. Here, it will be only considered as a statistical term. In statistics, residuals are differences between observed values and values predicted on the basis of a statistical model.

This entry presents what residuals are in the examples of continuous (regression analysis) and [Page 1415]categorical (contingency tables analysis) data. It is important to understand what residuals stand for, and what their properties are, as they are part of every statistical model used in educational research. Moreover, a wide repertoire of statistical models has specific assumptions about distribution of residuals that has to be met, so that the model represents unbiased relationships between variables. Besides the diagnostic of a statistical model, residuals can be used to identify unusual observations (anomalies) or to indicate which category occurs less often or more often than expected.

Residuals should not be confused with statistical error, which is the amount by which observations are different from their expected value based on the whole population (this quantity cannot be observed directly). Residuals refer to the amount by which observations are different from the sample mean. Therefore, residuals usually are treated as the estimates of statistical error.