# PHYCS 3730/6720 Lab Exercise

#### Exercise 1 Learning to guess chi square.

The file guess.pdf contains a figure showing a straight line fit to data.

Estimate the value of chi square for this fit to within a few tenths. Do this "by eye". No need for a calculator or program. State the number of degrees of freedom. Explain how you arrived at this estimate.

For the remaining exercises we experiment with linear least squares fitting. The data set we work with is
  x  y   sigma
------------
4  3   1
5  5   2
6  8   1


#### Exercise 2. Getting the best slope and intercept

For this dataset, find the matrix $$M$$ and vector $$c$$ in Eqs (10) and (11) of the handout.

Find the best slope $$m<$$ and intercept $$b$$ from Eqs (15) and (16).

#### Exercise 3. Chi square surface

Find the minimum chisquare from Eq (2) of the handout.

Write the equation for the chisquare surface as a function of the deviation of the parameters from the best fit values. That is, insert the numerical values of the coefficients of Eq (19). Keep it in terms of $$u = \Delta m$$ and $$v = \Delta b$$.

#### Exercise 4. Plotting the error ellipses

The error ellipses are the contour levels of the chi square surface. Construct the contour plot for $$\chi^2 - \chi^2_0$$. That is, the elevation of chi square above the minimum. (So zero elevation is at the center of the ellipses.) You may use either gnuplot or Pyplot.

For gnuplot, use splot to do it. We want to plot the contours as a function of the deviations $$u = \Delta m$$ and $$v = \Delta b$$. The following gnuplot commands may also help with splot:

  set view 0,0
set contour
set cntrparam levels discrete 1,2

For Pyplot, see the course notes linked above.

Find the standard deviations in $$m$$ and $$b$$ by using the maximum and minimum values on the $$\chi^2 - \chi^2_0 = 1$$ contour.