Almost always, the cause of too good a chi-square fit is that the experimenter, in a “fit” of conservativism, has overestimated his or her measurement errors. Very rarely, too good a chi-square signals actual fraud, data that has been “fudged” to fit the model. | Fitting Data to a Straight Line 661 as a distribution can be. Almost always the cause of too good a chi-square fit is that the experimenter in a fit of conservativism has overestimated his or her measurement errors. Very rarely too good a chi-square signals actual fraud data that has been fudged to fit the model. A rule of thumb is that a typical value of x2 for a moderately good fit is X2 v. More precise is the statement that the x2 statistic has a mean v and a standard deviation V v and asymptotically for large v becomes normally distributed. In some cases the uncertainties associated with a set of measurements are not known in advance and considerations related to x2 fitting are used to derive a value for a. If we assume that all measurements have the same standard deviation ai a and that the model does fit well then we can proceed by first assigning an arbitrary constant a to all points next fitting for the model parameters by minimizing x2 and finally recomputing X y - X tfHN - M i 1 Obviously this approach prohibits an independent assessment of goodness-of-fit a fact occasionally missed by its adherents. When however the measurement error is not known this approach at least allows some kind of error bar to be assigned to the points. If we take the derivative of equation with respect to the parameters ak we obtain equations that must hold at the chi-square minimum n _ X A - y xi f dy xj .ak . . . M 02 A @ak J i 1 i k 1 . M Equation is in general a set of M nonlinear equations for the M unknown ak. Various of the procedures described subsequently in this chapter derive from and its specializations. CITED REFERENCES AND FURTHER READING Bevington . 1969 Data Reduction and Error Analysis for the Physical Sciences New York McGraw-Hill Chapters 1-4. von Mises R. 1964 Mathematical Theory of Probability and Statistics New York Academic Press . 1 Fitting Data to a Straight Line Sample page from NUMERICAL RECIPES IN C
