Create a histogram. Do the data follow a normal (Gaussian) distribution?
ANSWER
No, the distribution is not Gaussian. The histogram, and a part of the Excel spreadsheet is shown below.
| Channel | Tabulated Energy (keV) | source |
| 379.1 +/- 3.9 | 510.9996 | 22Na |
| 870.0 +/- 5.5 | 1274.5450 | 22Na |
| 25.8 +/- 1.4 | 32.1000 | Ba X-ray |
| 481.8 +/- 3.3 | 661.6610 | 137Cs |
| 638.9 +/- 2.4 | 898.0460 | 88Y |
| 814.8 +/- 4.9 | 1173.2380 | 60Co |
| 911.4 +/- 4.1 | 1332.5130 | 60Co |
| 24.3 +/- 1.3 | 26.3450 | 241Am |
| 50.6 +/- 2.8 | 59.5370 | 241Am |
The channel numbers and their uncertainties are given in column 1 of the table. The g-ray energies (given in column 2) have been measured very precisely and for this purpose can be considered to have negligible uncertainty.
Perform a linear fit to this data to determine the calibration, Energy=a*(channel) + b. Judging by the reduced c2 of the fit, how good is the fit? Can you determine the uncertainties of a and b?
Make as plot of the residuals "measured - fit" (it is up to you to define what is "measured" and what is "fit"). Do you notice a trend in this plot that might suggest some way to improve your calibration?
ANSWER
First, you have to recognize that in this problem, the quantities are "inverted". The formula suggests that the independent variable is the Channel number, but the Tabulated energy has no error, and the Channel number has. Therefore we first express the channel number as Channelfit = (1/a)(Energy) + b/a. We will consider Energy as the x variable, and make a plot of Channel vs. Energy, with error bars on the y axis. The result of the fit is shown here:
Although the fit looks OK, it gives a reduced c2=14.9
(the number is in cell "I6"). Something must be wrong!
The plot of the residuals (Channel number - fit) indicates that there is a systematic
deviation from the fit, that is much larger than the error bars. Therefore trying to figure out the error of parameters a and
b is quite meaningless. An increase of the (non-reduced) c2=104.5 by 1
(the usual test for determining the error of a parameter, as described by Bevington) does not really change the fit by much - it remains
equally terrible. (By the way, if nothing else works, you may argue that a possible way to get at least
some idea about the error of the fitting paramaters is to look for the condition of c2=c2best
+ reduced c2 = (1+1/(N-P))c2best. )
We can improve the fit by adding a third parameter. For example:
Channelfit = (1/a)E) + b/a + g*E2
works great! The residual plot looks OK, and we get reduced c2 = 0.69
(the number of parameters is P=3).
At this point we may start looking for the error of the fitting parameters in a meaningful way.
But let us keep in mind, that this was a calibration: it has to be done accurately, but it is only the
first step
of the measurement. The real question is NOT the error of parameters a, b and c.
The question is, how much error
will the calibration cause in the actual measurement. For most practical purposes
the error of the calibration (if done properly, like in the second fit) can be deduced by
looking at the residuals plot: it will be represented by a
+/-1 channel number error, that has to added to the error of the actual measurement.
| Angle(deg) | Counts/10sec |
| 34 | 183 |
| 34.1 | 192 |
| 34.2 | 238 |
| 34.3 | 372 |
| 34.35 | 594 |
| 34.4 | 907 |
| 34.45 | 771 |
| 34.5 | 574 |
| 34.55 | 355 |
| 34.6 | 288 |
| 34.7 | 232 |
| 34.8 | 189 |
| 34.9 | 173 |
- Assume that the detector counts follow a Poisson distribution. Fit a Lorentzian, y = A (G/2)2/((x-x0)2+ (G/2)2), to the data by the c2 method, adjusting parameters A, x0 and G. Report the best fit results.
- Fix (to the best value) parameters A and G. Determine the error of the peak position by mapping c2 vs. x0 and looking for the c2min+1 points.
- Fix the parameter A, and determine if there is any correlation between the other two parameters. For this, you have to make a contour plot of c2 vs. x0 and G.
