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The statistical errors are then =, with expected values of zero, [4] whereas the residuals are = ¯. The sum of squares of the statistical errors, divided by σ 2, has a chi-squared distribution with n degrees of freedom:
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For a confidence level, there is a corresponding confidence interval about the mean , that is, the interval [, +] within which values of should fall with probability . ...
When either randomness or uncertainty modeled by probability theory is attributed to such errors, they are "errors" in the sense in which that term is used in statistics; see errors and residuals in statistics.
In statistics, sampling errors are incurred when the statistical characteristics of a population are estimated from a subset, or sample, of that population. Since the sample does not include all members of the population, statistics of the sample (often known as estimators ), such as means and quartiles, generally differ from the statistics of ...
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Because actual rather than absolute values of the forecast errors are used in the formula, positive and negative forecast errors can offset each other; as a result, the formula can be used as a measure of the bias in the forecasts. A disadvantage of this measure is that it is undefined whenever a single actual value is zero.