1. The chi-squared statistic  2 N Measures “goodness of fit” Used for model fitting and hypothesis testing e.g. fitting a function C(p 1,p 2,...p M ; x) to a set of data pairs (x i,y i ) where the y i have associated uncertainties  i : Define statistic: If C has M fitting parameters, expect  2 ~ N - M

2.  2 fitting approach Consider a set of data points X i with a common mean and individual errors  i We’ve already seen that the weighted average: Alternatively use goodness of fit: Find the value of A that minimises  

3. Parameter fitting by minimizing  2 Set derivative of   w.r.t. A to zero and solve: In other words, the optimally weighted average also minimizes  .

4. Using  2 to estimate parameter uncertainties Variance of optimally weighted average: What is   for Use Taylor series: Now So Hence: A 22  2 min

5. Error bars from  2 curvature We’ve just seen that: Hence  2 ≤1 encloses 68% of probability for A. We use  2 ≤1 to get “1  ” error bars on the value of a single parameter fitted to data. Use the second derivative (curvature): For the case where We get

6. Scaling a profile by  2 minimization As before: –X i = data, known. –  i = error bars, known. –p i = profile, known. –A p i = profile scaled by factor A. Goodness of fit:

7. Error bar on scale factor Use the  2 curvature method. Second derivative: Use  2 = 1: A 22  2 min