(Portions of this text were contributed by João Caldeira.)

What to discuss and conclude

After making sure that the fit to the data is “good”, obtaining uncertainties to all measurements, and calculating everything that needs to be calculated… what can we discuss and conclude? Let’s use the example of the  Gamma Cross Sections experiment since we are all familiar with it and the necessary analysis.

For that experiment, we asked you to compare your obtained linear attenuation coefficients to a provided set of literature values, and comment on which process was predominant in the attenuation at each energy. When comparing the results of your experiment to a known value, your main tool is the uncertainty calculated or obtained from the fit; without it, any comparison is meaningless. The same measured value can be in fine agreement with the literature or be an indication that something went very wrong, and the uncertainty is the tool to decide this.

Note that we can’t simply say our obtained values agree well (or not so well) to the literature without providing evidence. In a case like the gamma cross sections experiment – where we wanted to compare several values – something like a table or a plot would be a good way to start. For example, a table from one submitted report is shown in Table 1.

Photon Energy (MeV) Measured $\mu\;\mathrm{(cm^{-1})}$ Literature $\mu\;\mathrm{(cm^{-1}})$
0.0813 3.37 ± 0.75 4.80 ± 0.48
0.122 1.62 ± 0.06 3.0 ± 0.3
0.360 0.67 ± 0.02 0.70 ± 0.07
0.511 0.50 ± 0.02 0.50 ± 0.05
0.662 0.51 ± 0.02 0.50 ± 0.05
1.270 0.36 ± 0.01 0.40 ± 0.04

Table 1: Table showing measured and literature values of $\mu$.

Now we can actually compare. The bottom four rows all agree within uncertainties, that is, the intervals defined by the uncertainties for measured and literature values intersect each other. Note that this doesn’t mean we should end all comments here. One thing that should always be included is some discussion of what the major sources of error were in our procedure and how the precision of the measurement could be improved.

For the first row, the intervals are disjoint. This doesn’t always have to mean something went wrong. We generally assume, bar any systematic errors, that many such measurements of this physical quantity will have some distribution centred on the actual value, and the uncertainty here can be thought of as estimate of the standard deviation of that distribution. For a normal distribution, 68% of the values are within one standard deviation of the mean.

Strictly, however, to assume a normal distribution we need to use the central limit theorem, which applies only to averages of a large number of measurements: even $A_1, \ldots, A_N$ does not follow a normal distribution, the random variable $S_N = (A_1+\ldots+A_N)$ $N$ is approximately normal for large $N$. Since it is often impractical or impossible to repeat all measurements ten times, we often use a measurement uncertainty, which is not the same as a standard deviation.

So, you may be asking, is a value within two uncertainties of the literature value okay or not? Well, usually yes, but you should check for errors that you may have committed in your measurements and analysis more carefully than you would check if there was an agreement.

If the deviation is close to or larger than twice the uncertainty assigned to the measurement, however - such as in the second row of the table above - this is a strong indication that something went wrong. You should go back and look at the analysis critically, possibly correcting something. Perhaps you missed a factor, or something could be improved. In this case, the fit of the exponential decay was made without a constant offset, but the measured values converged to a non-zero intensity due to some background. The fit should be repeated, including the missing piece.

If you cannot find the problem, maybe one of your measurements is at fault. Think back to the experiment and discuss what could have failed, and how it can be improved. A detailed lab notebook is very useful in this task.

Of course, if nothing is found, it is always possible that your measurement and calculation of uncertainty has been done correctly, but yielded a statistically unlikely result. For more than two standard deviations this should be a conclusion taken only after double-checking all the steps, and for more than four or five it is unlikely enough that you should think of it as impossible.

Finally, after thoroughly discussing the obtained results, you should describe what could be learned with this experiment, and how any insight ties in with what you knew before. For the gamma cross section experiment, you should compare the results obtained to the given plot and conclude which process was predominant. Note, of course, that the predominant process changes in the tested range, so it is not the same for all data points. Here you could also relate your findings to the question in the lab manual about pair production. Why did we not reach energies where it dominates?