- Tables of repeatability and reproducibility values.
- Figure showing functional relations for the precision values.
- Histograms of the data from the precision experiment.
- Mandel plots of the data from the precision experiment.
- Return to Flakiness Index test Summary Page.
- Return to Home Page.

Estimates of repeatability and reproducibility are given in Tables 1 and 5. The standard deviations are also shown in Figure 1. These estimates have been calculated by excluding the data for Laboratories G and L in the European experiment. It is clear from inspection of the data that Laboratory G has not carried out the test correctly. In the case of Laboratory L, their data were excluded because they gave consistently high biasses and ranges in the Mandel plots.

For the European experiment it is reasonable to summarise the reproducibility standard deviations using a linear functional relationship. Functional relations for the repeatability standard deviations are considered in detail below.

In the case of the French experiment, the reproducibility standard deviations do not lie on a straight line. Examination of Figure 2 shows that this is because Laboratories c and i obtained widely-differing laboratory averages in Level 2. In view of the small number of laboratories that took part in the French experiment, it is proposed that only the relationships for the European experiment should be quoted in the European Standard test method as a statement of the precision of the Flakiness Index test.

The test method gives estimates of the repeatability and reproducibility of the Flakiness Index test as r = 2.8 FI% and R = 5 FI%, for levels of Flakiness Index between 8 and 20. These estimates are consistent with those obtained in the European experiment for Levels 1 and 2.

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. Because specifications for flakiness index apply upper limits, as with mechanical tests, the criterion may also be applied to the Flakiness Index test. The results in Tables 3 and 6 show that the reproducibility of the Flakiness Index test does not meet this criterion.

The Flakiness Index test divides the particles in a test portion into just two classes: flaky and non-flaky. Because of this, 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 Flakiness Index test should be:

Equation (1): r_{1, pred} = 2.8 sqrt{ FI (100 - FI) G L f g D^{3} / M }

The symbols in this equation have the following meanings. Note that the units of G, M and D must be consistent, e.g. kg/m^{3}, kg and m^{3}.

- FI
- is the percentage of flaky particles in the test portion.
- G
- is the density of the aggregate (assumed to be 2600 kg/ m
^{3}). - L
- is the liberation factor (1.0 for flakiness).
- f
- is the particle shape factor (assumed to be 0.5 for most aggregates).
- g
- is the size range factor (0.75 for aggregates for which D/d = 2).
- D
- is the upper size of the aggregate (the sieve through which about 95% of the aggregate will pass).
- M
- is the test portion mass.

Substituting the values for G, L, f and g in the above equation gives:

Equation (2): r_{1, pred} = 87.4 sqrt{ FI (100 - FI) D^{3} / M } %

If the aggregate size is expressed in mm this equation becomes:

Equation (3): r_{1, pred} = 0.0028 sqrt{ FI (100 - FI) D^{3} / M } %

or

Equation (4): S_{r1, pred} = 0.0010 sqrt{ FI (100 - FI) D^{3} / M } %

Table X gives the results of applying this equation to the three levels in this experiment.

The predicted values of repeatability in Table X agree very well with the values obtained from the experiment (see Table 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 now be seen to be a consequence of the combinations of values of FI, D and M 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.

Level 1 | Level 2 | Level 3 | |
---|---|---|---|

Aggregate size D mm | 20 | 14 | 10 |

Test portion mass M kg | 12 | 3 | 3 |

Flakiness Index FI % | 8.8 | 29.2 | 50.7 |

Predicted repeatability standard deviation S_{r1, pred} % |
0.73 | 1.38 | 0.91 |

Predicted repeatability r_{1, pred} % |
2.0 | 3.8 | 2.5 |

It was noted above that the reproducibility coefficient of variation of the flakiness index 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 7.7%, as shown in Table 3 for Level 1, it is very difficult for the reproducibility coefficient of variation to meet the 8% criterion. The repeatability coefficient of variation is 4.7% at Level 2. Although it is possible for the reproducibility coefficient of variation to meet the 8% criterion in this case, it would be easier if the repeatability was better.

If a target for the repeatability coefficient of variation is set at one-half of 8%, i.e. at 4%:

Equation (5): 100 x S_{r1, pred} / FI = 4 %

so that

Equation (6): r_{1, pred} = 2.8 x 0.04 x FI = 0.11 x FI %

this may be substituted into Equation (3) to calculate the test portion mass required to give this chosen level of repeatability. The following equation for the test portion mass M (kg) is obtained:

Equation (7): M = 0.0006 (100 - FI) D^{3} / FI kg

This shows how the test portion mass should depend on the flakiness index FI (%) and the size D (mm) of the aggregate.

Often, the Flakiness Index test is applied to single-sized aggregates (i.e. an aggregate contained within a size range d/D for which D/d = 2). In this case, it is possible to calculate a formula, equivalent to Equation (7), that gives instead the approximate number of particles that should be tested. In Equation (1), G f D^{3} is an estimate of the mass of one particle, so M / (G f D^{3}) is an estimate of N, the number of particles in the test portion. With the target for repeatability as in Equation (5), the required number of particles is found to be:

Equation (8): N = 469 x (100 - FI) / FI

Thus if a single-sized aggregate is being tested, and one is interested in measuring flakiness indices around 50%, and a repeatability coefficient of variation of 4% is to be achieved, then the test portion should contain 469 particles (approximately). Note that the same number of particles is required whatever the upper aggregate size. If lower levels of flakiness are of interest, then larger test portions are required.

Some values derived from the above equations for M and N are given in Tables Y and Z. Note that these apply to aggregates with a size range D/d = 2. For aggregates with a wider size range, the masses given in Table Y will give a repeatability coefficient of variation less than (i.e. better than) 4%. It can be seen in Table Y that the required test portion mass increases very rapidly as the aggregate size increases, so that the masses required for large-sized aggregates may well be considered to be impractical. The table does not show masses above 50kg to avoid alarming any laboratory technicians who might read this report.

Upper aggregate size | FI = 15 % | FI = 20 % | FI = 35 % | FI = 50 % |
---|---|---|---|---|

D = 10 mm | 3.4 kg | 2.4 kg | 1.1 kg | 0.6 kg |

D = 16 mm | 13.9 kg | 9.8 kg | 4.6 kg | 2.5 kg |

D = 20 mm | 27.2 kg | 19.2 kg | 8.9 kg | 4.8 kg |

D = 32 mm | (>50.0 kg) | (>50.0 kg) | 36.5 kg | 19.7 kg |

Upper aggregate size | FI = 15 % | FI = 20 % | FI =35 % | FI = 50 % |
---|---|---|---|---|

Any | 2700 | 1900 | 870 | 4703 |

The precision of a test method is critical only when a material gives results close to its specification limit. With an aggregate that give results well within its specification, a lower standard of precision than that given by Equation (5) will be acceptable. In this situation, the theory could be used to calculate the test portion masses that give some chosen level of repeatability above 4%, and these masses will be smaller than those shown in Table Y. The theory given above could thus be used to find a reasonable balance between the costs of testing and the risks of making incorrect decisions.

Large test portions are unavoidable if the Flakiness Index of large-sized aggregates must be measured with the level of precision implied by Equation (5). In these circumstances it would be better to use, instead of the Flakiness Index test, some other test method in which the three dimensions of each particle are measured and used to calculate a shape coefficient for each particle. Such a test would give more information per particle than the Flakiness Index test, and so would allow a satisfactory level of precision to be achieved with smaller test portions.