3.17 FAQ-242 Why is my Reduced Chi-Sqr value very different from 1?
Last Update: 8/8/2018
The Nonlinear Curve Fitter tool in Origin computes and reports as one of the measures of goodness of fit. If a weight is included in the fitting process and the Reduced Chi-Sqr is very different from 1, please examine if an improper weighting method is chosen. If the Reduced Chi-Sqr value is much smaller than 1, it may indicate a too large weight. Vice versa.
Why do you need check the weight method?
Typically Reduced Chi-Sqr value is closer to 1, better a fit we get. If weight is involved during fitting, Reduced Chi-Sqr close to 1 also indicates that the difference between observed data and fitted data has a similar magnitude of weight. When fitting with weight, Reduced Chi-Sqr is calculated as follow:
\[Reduced\chi ^2 = \frac{1}{n-p}\sum_{i=1}^n \frac{(y_i-f_i)^2}{\sigma_i^2}\]
where n is the number of data, p the degree of freedom, \(y_i\) the ith y data, and \(\sigma_i\) the ith error. When the number of data is large enough, n - p approaches n, and Reduced Chi-Sqr is determined by the difference between source data and fitted data and the weight.
Therefore, if Reduced Chi-Sqr is very different from 1, it may indicate an improper weighting method. Origin provides several weighting methods, please refer to for formula of each method. If the Reduced Chi-Sqr value is much smaller than 1, it may indicate a too large weight. Vice versa.
Quick Example
If you perform a nonlinear curve fit with the Statistical weighting method on a set of data and it generates a fitting result of Reduced Chi-Sqr close to 1, it indicates the fit result is good. Then you scale the y data by multiplied a factor of 10. Note that the Statistical weighting method takes ~y magnitude as error variation, the weight is also scaled by 10. However, for a random variable \(y\) with weight \(\sigma^2\), the proper weight for \(ny\) should be
- \(E(nX-E(nX))^2=n^2E(X-E(X))^2=n^2\sigma^2\ \).
So the Statistical cannot calculate a proper weight for the scaled data. The Redcued Chi-Sqr value which is scaled much larger than 1 also indicates this improper. To make Redcued Chi-Sqr close to 1 again, you should choose Instrumental or Variance ~ y^2 method, which take ~y^2 magnitude as weight.
| Note:
In addition to Reduced Chi-Sqr, Origin outputs many other quantities such as R-Square and Adjusted R-Square which can also be used to estimate goodness of fit. |
Keywords:goodness, chi-square, residual sum of squares, fit, fitting, nonlinear, variance