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As a rule of thumb, when the variance of the measurement error is known ''a priori'', a <math>\chi_\nu^2 \gg 1</math> indicates a poor model fit. A <math>\chi_\nu^2 > 1</math> indicates that the fit has not fully captured the data (or that the error variance has been underestimated). In principle, a value of <math>\chi_\nu^2</math> around <math>1</math> indicates that the extent of the match between observations and estimates is in accord with the error variance. A <math>\chi_\nu^2 < 1</math> indicates that the model is "over-fitting" the data: either the model is improperly fitting noise, or the error variance has been overestimated.<ref>{{citation |first=Philip R. |last=Bevington |title=Data Reduction and Error Analysis for the Physical Sciences |___location=New York |publisher=McGraw-Hill |date=1969 |page=89 |quote=For {{math|χ<sup>2</sup>}} tests, {{math|χ<sub>ν</sub><sup>2</sup>}} should be approximately equal to one.}}</ref>
When the variance of the measurement error is only partially known, [[Weighted arithmetic mean#Correcting for over- or under-dispersion|the reduced chi-squared may serve as a correction estimated ''a posteriori''
== Applications ==
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