Estimates of repeatability and reproducibility are given in Table 1. The standard deviations are also shown in Figure 1. These estimates have been calculated by excluding the following data.
Percentage of crushed particles (and the percentage of rounded particles).
All the data for Laboratories B and E at all three levels were excluded. This was because they gave low results at all three levels, and the laboratory averages for Laboratory B in Levels 1 and 3, and the laboratory average for Laboratory E in Level 1, were outliers.
The data for Laboratory O in Level 1 were excluded because their between-test-portion range was an outlier.
Percentage of totally rounded particles.
The data for Laboratory P in Level 1 were excluded because their between-test-portion range was an outlier.
Figure 1 shows lines that represent functional relations for the repeatability and reproducibility standard deviations.
In the case of repeatability, the statistical theory presented below allows the functional relation to be derived solely from theoretical considerations, so the broken lines for Sr1 shown in Figure 1 are derived using this theory - they are not obtained by fitting an equation to the results of the experiment. However, the broken lines can be seen to fit the experimental results satisfactorily, and the same formula can be used to calculate the repeatability of determinations of the percentage of crushed particles, the percentage of rounded particles, the percentage of totally crushed particles, or the percentage of totally rounded particles. The formula includes the aggregate size and the test portion mass as parameters, so it could be used to calculate the repeatability for aggregate sizes or test portion masses not included in the experiment.
In the case of reproducibility, inspection of Figure 1 shows that the points for the reproducibility standard deviation follow similar patterns to those for the repeatability standard deviation, so the same form of relation may be used, but with the coefficient (0.0018) derived by fitting the relation to the results of the experiment. It should be noted though that, in the case of reproducibility, there is no theoretical justification for the inclusion of the term D3 / M , so that one cannot be confident that the functional relation will correctly predict the reproducibility for aggregate sizes or test portion masses different to those used in this experiment.
Inspection of Figure 1 also shows that the reproducibility of the determination of the percentage of totally crushed particles is very much worse than that of the other properties determined in the test. The implication is that the operators do not agree on how "totally crushed particles" should be distinguished from "crushed particles". If the determination of the percentage of totally crushed particles is to be retained in the test method then it will be necessary to add guidance to the method description to help operators carry out the test consistently. The points shown in Figure 1 for the determination of the percentage of totally crushed particles were not included in the calculation of the coefficient of the functional relation for the reproducibility standard deviation.
It has been argued (J»rck, Sym and Powell, 1994. A study of mechanical tests of aggregates. Green Land Reclamation Ltd Report GLR 3036/03a.) that the reproducibility standard deviation of a mechanical test, when expressed as a coefficient of variation, should be no more than about 8 %, if the test method is to be used to assess the compliance of aggregates with specifications. The justification of this rule applies equally well to the determination of the percentages of crushed and broken surfaces, provided that proper allowance is made for specifications that apply lower limits to the results of the test.
The results in Table 3 show that the reproducibility of the test fails to meet this criterion by a wide margin. Hence if the test is to be called up in specifications its reproducibility must be improved. The Mandel plots provide strong evidence of correlations between results for the three aggregates, showing that, generally, each operator is consistent in the way he sorts the aggregates into classes, but the operators have different ideas on how this should be done. Consideration should therefore be given to amplifying the definitions of the classes in the test method description. For example, at present it is not clear if a hemisphere should be classed as a "crushed particle" or a "rounded particle". It would also be helpful if the test method contained drawings or photographs illustrating how particles should be classified in marginal cases.
It is also worth noting that the correlations between the results for the three materials demonstrate that the between-laboratory variation was not caused by variations between the samples, so that the method used for sample preparation produced samples that were sufficiently homogeneous for the purposes of this experiment.
The test divides the particles in a test portion into two classes: crushed particles and rounded particles. These classes can then be sorted again to determine the percentages of totally crushed and totally rounded particles. If the operators carry out the sorting consistently, the repeatability variation should arise only during the random process of selecting the particles that are included in the test portion, so it should be possible to predict the repeatability of the test using statistical theory. The statistical theory developed by Gy (Gy, 1982. Sampling of particulate materials, theory and practice. Elsevier, Oxford.) was studied in a BRE report (Sym, 1992. An investigation into the relationship between test portion mass and repeatability for quantitative petrographic examination of aggregates. Building Research Establishment Note N 194/92.), and shown to give predictions of the repeatability of sieving and sorting tests on aggregates that agreed with experimental results.
According to this theory, the repeatability of the test should be:
(1) r1,pred = 2.8 ( X (100 - X) F l f gD3 ÷ M )½ %
The symbols in this equation have the following meanings. Note that the units of F, M and D must be consistent, e.g. kg/m3, kg and m3.
Substituting the values for F, l, f and g in the above equation gives:
(2) r1,pred = 87.4 ( X (100 - X) D3 ÷ M )½ %
If the aggregate size is expressed in mm this equation becomes:
(3) r1,pred = 0.0028 ( X (100 - X) D3 ÷ M )½ %
(4) Sr1,pred = 0.0010 ( X (100 - X) 3 ÷ M )½ %
The values of repeatability predicted using Equation (4) agree very well with the values obtained from the experiment (see Figure 1). Some useful conclusions may be drawn from this. Thus in Figure 1, the repeatability standard deviations do not follow a straight line. This can be seen to be a consequence of the combinations of values of D and X that occurred in the experiment. However, because the predicted repeatabilities agree with those derived from the experiment, the equation for the predicted repeatability could be used as a statement of the repeatability of the test method.
It was noted above that the reproducibility coefficient of variation of the test does not meet the 8 % criterion. The reproducibility coefficient of variation cannot be smaller than the repeatability coefficient of variation, so when the repeatability coefficient of variation is greater than 8 %, as it is in many of the cases in Table 3, it is impossible for the reproducibility coefficient of variation to meet the 8 % criterion. Equation (4) can be re-arranged to allow one to calculate the test portion mass that is needed for the test method to give a specified level of repeatability:
(5) M = 0.00001 ( X (100 - X) D3 } ÷ (Sr1,specified)2 kg
This equation could be used to derive a table of test portion masses that would give a satisfactory, low, level of repeatability, and hence help the test give better reproducibility. An example of this approach is given in the report on the cross-testing experiment on the determination of the flakiness index.
The precision of a test method is critical only when a material gives results close to its specification limit. With an aggregate that gives results well within its specification, it may be acceptable for the repeatability and reproducibility coefficients of variation to be greater than 8 %. It would thus make sense for the test method to allow the user a choice of test portion masses, so that smaller masses can be used with aggregates that easily meet their specifications, and larger masses can be used with aggregates that may give results close to specification limits. Equation (5) could be used to calculate test portion masses appropriate to these situations.
According to Table A in the description of the experiment, the percentage of totally rounded particles in a "Category B" aggregate should not exceed 3 %. Very large test portion masses are required to determine small percentages precisely, so producers will find that they can meet this requirement consistently only if their aggregates contain a much smaller percentage of totally rounded particles than 3 % , or if they test very large test portions.
Equations (4) and (5) apply to aggregates, such as those used in this experiment, for which D <= 2d . It may be necessary, on occasion, to apply the test method to aggregates for which D > 2d . In such cases the test portion has to be sieved into size fractions so that each size fraction can be sorted into classes. In practice, some size fraction can contain very small or very large numbers of particles, and the method has to tell the operator what to do when these situations arise. The statistical theory used to derive Equations (4) and (5) can be extended to cover this more complicated problem. A method was derived during the development of the British procedure for quantitative petrographic examination of aggregates (BS 812 Part 104, 1994). That Standard gives a table of test portion masses, and a simple way of calculating how much of each size fraction to test. The method is illustrated below in Table B. The principle on which the method is based is that it minimises the total number of particles that have to be examined to achieve a specified level of repeatability. (I.e. it minimises the amount of work that has to be done by the operator.) The approach is applicable to other tests that involve sorting aggregates into classes, such as the test method that is the subject of this report.
|Size fraction||Mass found in test portion||Reduction factor||Mass of size fraction to test|
|di / Di mm||Mi kg||(Di / D1)3||mi kg|