A recent question here on the forum was about iolite's error propagation: what is it, and how is it calculated?

The purpose of error propagation is to ensure that you have a reasonable estimate of the uncertainty of your calculated result(s). Depending on what you consider "reasonable" this might be: what range of values would I get if I measured this sample repeatedly in different sessions? This is normally impossible with samples as they might be consumed during their analysis. So we have to rely on repeat measurements of reference materials to estimate this uncertainty.

This estimate should take into account some of the decisions you've made in your data processing. For example, it should account for how you've modeled your baselines using splines, and similarly what spline types you used for your reference materials.

You might be thinking: but iolite already provides an uncertainty for each result. This is the "internal uncertainty" because it's the variation of the points within your selection for your chosen channel. If we take a result for Pb206/U238 final as an example, for a selection you'll get a mean/median value and some uncertainty. This is calculated for the points within the selection for that channel. However, this uncertainty doesn't take into account things such as the effect of splining your baselines, your choice of downhole fit etc. So we find that these internal uncertainties are usually lower than we'd expect if we'd measured the same reference material over and over again, and this suggests that there are other sources of uncertainty not captured just by reporting the uncertainty of the points within your selection.

So Chad Paton (and some other advisors) came up with the idea of looking at your primary reference material results to look at how much the results vary in comparison with the internal uncertainty.
To do this, iolite uses a metric called the "Mean Squared Weighted Deviation" or MSWD for short. This metric looks at a group of measurements, each with their own uncertainty, and determines whether the spread in results is realistic given the uncertainties on each of the measurements. We can use this to identify when our internal uncertainties are perhaps too small given our spread in means.

In the early days when this was first introduced, there were not a lot of secondary reference materials measured along with our samples, so instead of using the secondary reference materials, the error propagation routine uses the primary reference material results. Now we can't use them just as they are because they've been used in the calculation of the final results, and that would be circular reasoning. Instead what the error propagation routine does is pulls each primary RM measurement out, and treats it like it was an unknown. When the measurement is removed, the splines are recalculated and the result for that measurement is recorded. After this is done for all the measurements of the primary RM*, we have a pool of results that we can examine with our MSWD statistic to see if the internal uncertainties match up with the spread in results. This pool of results, where the primary RM is pulled out one measurement at a time and treated as an unknown, is called the "pool of pseudo-secondary reference materials results". It's pseudo-secondary because it's actually our primary RM, but we're treating it as an unknown.

One property of the MSWD is that it will be greater than 1 if the spread in measurements is larger than that predicted by the internal uncertainties. If this is the case for the pool of pseudo-secondary RM results, it suggests that the internal uncertainties are not taking into account all the uncertainty in our experiment. So iolite will add a little bit of uncertainty to the results and recalculate the MSWD. This little bit of extra uncertainty is called the "excess uncertainty" because it's added onto the internal uncertainty. iolite keeps adjusting this excess uncertainty until it gets an MSWD of roughly 1 for the pool of pseudo-secondary results. It then records this excess uncertainty as a percentage so that it can be applied to all the other results by adding it in quadrature.

The "propagated uncertainties" you see in iolite are the internal uncertainty with the excess uncertainty added to it.

If this was to work perfectly, you could look at your secondary reference materials, and with their propagated uncertainty, they should have an MSWD of 1 because the uncertainty of each measurement matches the spread in measurements.

Sometimes iolite doesn't need to add any excess uncertainty at all because the pool of pseudo secondary results has an MSWD of 1 to start with.

Sometimes you might see a message that iolite did not calculate propagated uncertainties because you haven't measured the primary RM enough times. To get a reasonable estimate of the MSWD, you need to have measured the primary RM about 15 times. If we calculated the MSWD on fewer results, would could be adding unnecessary excess uncertainty to our results just because the MSWD is faulty, which iolite would rather not do, so it just tells you why it didn't propagate the errors.

Note that all of this may not be necessary if you collect your data into iolite's Database feature and calculate uncertainties from that. You would still need to add any excess uncertainty you might calculate to each of your results, but there is a script that can do that for you.

I hope that clears up some of these concepts. Please post below if you have any questions.

-Bence

We don't actually use all the results of the primary RM because the spline might go funny (unconfined) if we took out the first and last measurements. So we can only, at most, use the total number of measurements - 2 to create our pool of results to test the MSWD.