### Author Topic: Statistics in Quantitative X-ray Maps  (Read 1358 times)

#### AndrewLocock

• Professor
• Posts: 104
##### Re: Statistics in Quantitative X-ray Maps
« Reply #15 on: March 14, 2024, 09:52:52 AM »
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
« Last Edit: March 14, 2024, 10:57:53 AM by John Donovan »

#### Probeman

• Emeritus
• Posts: 2842
• Never sleeps...
##### Re: Statistics in Quantitative X-ray Maps
« Reply #16 on: March 14, 2024, 11:16:31 AM »
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?
« Last Edit: March 14, 2024, 11:52:06 AM by Probeman »
The only stupid question is the one not asked!