4.3.3. Homogeneity Index
The precision due to counting statistics is the same for one long count as for several shorter ones done for the same total time. The latter method is useful for determining whether counting statistics are the limit to attainable precision. If counting statistics are the principle cause of X-ray count variation for a homogeneous sample, then comparisons of observed and calculated parameters can be used to evaluate sample homogeneity. Consider analyses of a number of different grains (or different points on one grain). Are the observed variations the result of counting statistics or do they demonstrate true compositional variations? Define,
and the homogeneity index (H.I.) as:
The value of F would equal 1 if the observed deviation was the same as the variation expected just from counting X-rays produced from a homogeneous sample. An "F test" of the assumption of homogeneity can be used to evaluate the probability of equal variance. The test uses (n - 1) "degrees of freedom" of the numerator of the F-ratio (number of analyses). The number of degrees of freedom of the denominator is taken to be infinite. Tables of F-ratios and probabilities for equal variance (i.e., homogeneity) are available in many sources. Some sample values are given below:
|
(n-1) |
H.I. |
F |
Probability of |
|
1 |
2.5 |
6.26 |
2.5 % |
|
1 |
3.0 |
9.00 |
0.5 % |
|
3 |
1.5 |
2.25 |
8.0 % |
|
3 |
1.8 |
3.24 |
2.0 % |
|
3 |
1.9 |
3.78 |
1.0 % |
|
4 |
1.5 |
2.25 |
6.0 % |
|
4 |
2.0 |
4.00 |
0.5 % |
|
5 |
1.5 |
2.25 |
5.0 % |
|
5 |
1.7 |
3.02 |
1.0 % |
As an example, if 6 measurements were made and H.I. = 1.7, then the probability of homogeneity is 1.0%. Figure 4.3.3 presents the relationship between H.I., the number of analyses, and the probability of homogeneity. Rigorous statistical tests of homogeneity have not been applied much in the geological literature, but H.I. is often reported.
A value of for H.I. greater than 3 has been cited as proof of a sample's inhomogeneity; however, if counting statistics were the only cause of variation, then much smaller values of H.I. effectively would demonstrate inhomogeneity. Since counting statistics are only one source of variability, the statistical test must be applied with care. Often, correlated variations (e.g., Ca decreases when Na increases) can reveal inhomogeneity.
| Figure 4.3.3. Probability of homogeneity as a function of homogeneity index (H.I.) and the number of analyses. |
|
