Statistics in Quantitative X-ray Maps

[AndrewLocock]

From discussion with John:

With regard to standard error of the mean, it may be useful to clarify an underlying implicit assumption: homogeneity.
By taking an average of some data set, the underlying assumption is that the data all belong together (represent a single composition). That is, they are implicitly homogeneous.

And by taking the standard error of the mean, further replication reduces the perceived uncertainty. (Further replication continues to reduce random error, but not systematic error - which will eventually set a limit on such replication).

However, in a natural sample of wide compositional variation, the data are clearly heterogeneous (and should not be averaged together). In this case, the standard deviation of the entire data set is a proxy for the range of the data. And the standard deviation of an individual point represents the X-ray counting statistics that generated that point.

Cheers, Andrew

[Probeman]

And by taking the standard error of the mean, further replication reduces the perceived uncertainty. (Further replication continues to reduce random error, but not systematic error - which will eventually set a limit on such replication).

Andrew brings up a good point here.  Clearly one should restrict pixel averaging statistics to single phase domains which appear (at least visually) to be homogeneous.

Unless of course one is attempting to calculate the average composition of a heterogeneous material. In that case one should most definitely average pixels that are already quantified!  See here for more on this topic on why "defocused beam" analysis yields inaccurate results:

https://probesoftware.com/smf/index.php?topic=44.0

Meanwhile to see the averaging effects for heterogeneous samples on the standard error statistics I extracted pixels a heterogeneous sample and trying to select the (roughly) same area while increasing the number of pixels in the pixel extraction as seen here:



Here are the averaging results using standard error statistics:

Shape extraction number: 1
Pixels shape extracted/filtered: 100

  Fe WT%,  79.3341 +/-  .706792
  Mo WT%,  13.2246 +/-  .669886
  Cr WT%,  8.86299 +/-  .062921
  Ni WT%,  .021593 +/-  .012833
   O WT%,  -.30588 +/-  .027846
   Total,  101.137 +/-  .275705

Shape extraction number: 2
Pixels shape extracted/filtered: 225

  Fe WT%,  78.5930 +/-  .476070
  Mo WT%,  13.9332 +/-  .453728
  Cr WT%,  8.86584 +/-  .037695
  Ni WT%,  -.00166 +/-  .009303
   O WT%,  -.31037 +/-  .022373
   Total,  101.080 +/-  .168631

Shape extraction number: 3
Pixels shape extracted/filtered: 400

  Fe WT%,  78.5473 +/-  .376199
  Mo WT%,  14.0504 +/-  .358766
  Cr WT%,  8.87389 +/-  .029576
  Ni WT%,  .016975 +/-  .007130
   O WT%,  -.30123 +/-  .015141
   Total,  101.187 +/-  .131458

Shape extraction number: 4
Pixels shape extracted/filtered: 625

  Fe WT%,  81.5242 +/-  .282746
  Mo WT%,  10.9013 +/-  .267420
  Cr WT%,  8.80694 +/-  .023563
  Ni WT%,  .000539 +/-  .005664
   O WT%,  -.28093 +/-  .011598
   Total,  100.952 +/-  .106799

Shape extraction number: 5
Pixels shape extracted/filtered: 900

  Fe WT%,  80.3186 +/-  .234372
  Mo WT%,  12.3108 +/-  .224741
  Cr WT%,  8.76934 +/-  .019332
  Ni WT%,  .000602 +/-  .004625
   O WT%,  -.28657 +/-  .010154
   Total,  101.113 +/-  .090955

Shape extraction number: 6
Pixels shape extracted/filtered: 1225

  Fe WT%,  80.2674 +/-  .207730
  Mo WT%,  12.1888 +/-  .198964
  Cr WT%,  8.83089 +/-  .016758
  Ni WT%,  .007073 +/-  .003975
   O WT%,  -.30282 +/-  .008717
   Total,  100.991 +/-  .076728

Seems like this might be worth taking a closer look at?  Any comments?