# Suppose your adopted values of σ m are off by a

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Suppose your adopted values of σ m are off by a common scale factor, i.e. if σ m,adopted = m,true . Then hatwider χ 2 f - 2 instead of hatwider χ 2 1. And to obtain the parameter errors from δχ 2 , you must find the offset δx such that Δ χ 2 = f - 2 hatwider χ 2 . You can correct for this erroneous common factor f by dividing your adopted values of σ m by f . Of course, you don’t know what this factor f is until you do the chi square fit. Dividing them by f is equivalent to multiplying them by hatwide χ . And, of course, the same as multiplying σ 2 m by hatwider χ 2 . 8.5.2. When you’re using equation 8.9. . . To be kosher, after having run through the problem once with the adopted σ m , calculate the hatwider χ 2 ; multiply all σ m by hatwide χ ; and redo the problem so that the new hatwider χ 2 = 1. Then the derived variance

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– 30 – s a is also correct. You can obtain it either as the corresponding diagonal to the covariance matrix (equations 8.9 and 8.10, which are identical in this case) or by finding what departure from x 0 is necessary to make Δ χ 2 = 1. 4 This redoing the fit may seem like unnecessary work, but when we deal with multiparameter error estimation in § 9 it’s the best way to go to keep yourself from getting confused. 8.5.3. Think about your results! In the case Δ χ 2 1 (and hatwider χ 2 1) the dispersions of the observed points s m are equal to the intrinsic dispersions of the datapoints σ m and the mathematical model embodied in the least- squares fit is perfect. That, at least, is the theoretical conclusion. In practice, however, your obtaining such a low, good value for hatwider χ 2 might mean instead that you are using too large values for σ m : you are ascribing more error to your datapoints than they really have, perhaps by not putting enough faith in your instrument. But there is another way you can get artificially small values for hatwider χ 2 . This will occur if your measurements are correlated . Suppose, for example, that by mistake you include the same mea- surements several times in your fit. Then your measurements are no longer independent. Cowan discusses this possibility in his § 7.6. High values of hatwider χ 2 indicate that the model is not perfect and could be improved by the use of a different model, such as the addition of more parameters—or, alternatively, that you think your equipment works better than it really does and you are ascribing less error to your datapoints than they really have. And in this case, using equation 8.10 instead of 8.9 is disastrous. Think about your results. 8.5.4. When your measurements are correlated. . . One more point, a rather subtle one. There are circumstances in which your datapoints are not independent. Then the formulation of chi-square fitting (and least-squares fitting, for that matter) is more complicated. You need to calculate the covariance matrix for the measured values y m ; call this covariance matrix V . If this matrix is not unitary, then χ 2 is no longer given by equation 8.8i.
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