- 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 Particle Density and Water Absorption test Summary Page.
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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.

"*pre-dried*" test:

- Laboratory A (31), all data for Level 1 (outlying laboratory average)
- Laboratory J (170), all data for Level 3 (outlying between-test-result ranges)
- Laboratory T (310), all data for Levels 1 and 3 (outlying between-test-result ranges)

"*SSD*" test:

- Laboratory O (252), all data for Levels 1, 2 and 3 (outlying laboratory averages)

Water absorption test:

- Laboratory N (249), all data for Level 3 (outlying between-test-result range)
- Laboratory O (252), all data for Levels 1, 2 and 3 (outlying laboratory averages)
- Laboratory P (253), all data for Level 3 (outlying laboratory average)
- Laboratory R (273), all data for Level 3 (outlying laboratory average)

Figure 1 shows the repeatability and reproducibility standard deviations plotted against the water absorptions of the three materials. This figure may be used to see how the precision values may be summarised by functional relations. With the "*pre-dried*" particle density test, the reproducibility is higher in Level 3 (water absorption 3.1 %) than in the other two levels. However, as the method is intended only for non-porous aggregates, it is appropriate to summarise its reproducibility by the average reproducibility for Levels 1 and 2.

In the case of the "*SSD*" particle density test, in Levels 2 and 3 the data from Laboratories E and M give rise to laboratory averages which are stragglers, and which inflate the estimates of reproducibility obtained at these two levels. In view of this, it is not appropriate to fit a relationship to the reproducibility standard deviations. The reproducibilities may instead be summarised by the average value over the three levels.

Results of tests of particle density are usually rounded to the nearest 0.01 Mg/m^{3}: the repeatability standard deviations for both tests of particle density are low compared with this, so again it is appropriate to summarise them by simple averages.

The repeatability and reproducibility standard deviations for the water absorption test fall reasonably close to straight lines, so they may be summarised by linear relations.

Table 2 gives the functional relations that have been obtained.

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. Water absorptions are used to decide if there is a need to subject an aggregate to a weathering test (e.g. a freeze/thaw test, or a magnesium sulfate test), and a moisture content of 1 % has been tentatively put forward as a critical value. Hence the above criterion may be applied to the water absorption test.

In Level 1 (average water absorption 1.0 %) the coefficient of variation for the reproducibility of determinations of water absorption (see Table 3) is 10.0 %, and in Level 3 (average water absorption 3.1 %) the coefficient of variation for reproducibility is 7.1 % . Hence the test method fails to meet the "*8 % criterion*", marginally.

Particle densities are not used to check compliance of a coarse aggregate with limits in specifications, so the "*8 % criterion*" is not applicable to such tests. Particle densities are used in concrete mix design to calculate the yield (in m^{3} of concrete) from the batch weights and the densities of the constituents:

Y = Sum (X_{i}/p(_{i})

where

Y = the yield (m^{3})

X_{i} = the mass batched of constituent i (Mg)

p_{i} = the density of constituent i (Mg/m^{3})

This equation may be used to assess the reproducibility of the density tests as follows.

Using calculus one obtains:

(dY/dp(_{i}) = -V_{i}/p(_{i})

where

V_{i} = the volume batched of constituent i (m^{3})

A typical concrete mix design would have:

Constituent | Density Mg/m^{3}
| Mass Mg | Volume m^{3} |
---|---|---|---|

Cement | 3.12 | 0.310 | 0.099 |

Coarse aggregate | 2.60 | 1.033 | 0.397 |

Sand | 2.60 | 0.826 | 0.318 |

Water | 1.00 | 0.186 | 0.186 |

Total | 2.352 | 1.000 |

giving a water/cement ratio of 0.60 and an aggregate/cement ratio of 6.0. For this mix,

(dY/dp_{p}) = -0.397/2.60 = 0.15

Using this value it can be seen that it would require an error in the determination of the particle density of the coarse aggregate of e(p_{p}) = 0.067 Mg/m^{3} to cause an error of e(Y) = 0.01 m^{3} (or 1% of the total volume) in the calculated yield, as:

e(Y) = (dY/dp_{p}) × e(p_{p}) = 0.067 × 0.15 = 0.01 m^{3}

Both the "*pre-dried*" and "*SSD*" tests give reproducibilities R_{1} that are substantially smaller than 0.067 Mg/m^{3} , so that (on the assumption that errors in yield of 1 % are acceptable) they both appear to have adequate precision for the purpose of concrete mix design.

With both the "*pre-dried*" and "*SSD*" tests for particle density, and the determination of water absorption, the repeatability limits are much smaller than the reproducibility limits, so within-laboratory variability contributes only a small part of the reproducibility of the test methods. However, six of the participants reported data that were classed as outliers. This is a high proportion of the twenty participants that took part in the experiment. In view of this, it might be considered desirable to require duplicate tests in the "*SSD*" method for determining particle density and water absorption, so that repeatability checks can be made.

In the "*pre-dried*" method, outliers occurred in the between-test-result ranges but not in the between-determination ranges. This suggests that these tests were not carried out under repeatability conditions, or that the sample division was not carried out sufficiently carefully. If repeatability checks are to be effective, laboratories need to take care on these points.

The influence of the density of the water on the result of a determination of particle density may be seen as follows. In the "*pre-dried*" test, the particle density is calculated using the formula:

p_{p} = (M_{2} - M_{1})/(V - (M_{3} - M_{2})/ p_{w})

where

p_{p} = the "*pre-dried*" particle density of the aggregate

p_{w} = the density of water at the test temperature

M_{1} = mass of pyknometer + funnel

M_{2} = mass of pyknometer + funnel + test specimen

M_{3} = mass of pyknometer + funnel + test specimen + water

V = volume of pyknometer + funnel

so the partial differential of p_{p} with respect to p_{w} is:

(dp_{p}/dp_{w}) = (p_{p}/ p_{w}) × (1 - V/V_{t})

where

V_{t} = volume of the test specimen

This partial differential may be used as a multiplier to convert a change in the water density e(p_{w}) to the error caused in the determination of the "*pre-dried*" particle density of an aggregate e(p_{p}), using:

e(p_{p}) = (d p_{p}/d p_{w}) × e(p_{w})

Typical values that occur in the test are:

p_{p} = 2.60 Mg/m^{3}

p_{w} = 0.9970 Mg/m^{3}

V = 1000 mL

V_{t} = 250 mL

giving:

p_{p}/p_{w} = 2.6

and

V/ V_{t} = 4.0

so that

e(p_{p}) = -7.8 e(p_{w})

If the density of water is taken to be 1.0000 Mg/m^{3} (the value that it assumes at 5°C), instead of the correct value of 0.9970 Mg/m^{3} at the test temperature of 25°C, the difference of e(p_{w}) = 0.0030 Mg/m^{3} affects the particle density of the aggregate by:

e(p_{p}) = -7.8 × 0.0030 = 0.023 Mg/m^{3}

This is a small but perceptible difference, and it is about the same size as the reproducibility limit R_{1} for the test, so the use of the correct density of water at the test temperature does appear to be justified.

However, small variations in temperature about the nominal test temperature will have only a very small influence on the particle density. This may be seen as follows. A change of 1°C in the temperature of the water changes its density by e(p_{w}) = 0.00026 Mg/m^{3}, and this will change the particle density by only:

e(p_{p}) = -7.8 × 0.00026 = 0.002 Mg/m^{3}

This is small compared with the repeatability limit of the test, so, on the basis of this calculation, it would appear to be reasonable to allow variations in temperature of several degrees centigrade about the nominal temperature in the "*pre-dried*" test.

Referring back to the remarks made about the inconsistencies between the "*pre-dried*" and "*SSD*" tests with respect to their requirements on temperature, these calculations indicate that:

- it would be better to calculate the particle density in the "
*SSD*" test using the correct density of water for the test temperature (as specified in the "*pre-dried*" test); - it should be acceptable in the "
*pre-dried*" test to allow variations of ±3°C about the nominal temperature during the test (as specified in the "*SSD*" test).

However, calculations can only predict what might happen in practice, so if there is a need to harmonise these two test methods, these proposals should be properly tested by practical trials.

The relationship between errors in the determination of the pyknometer volume and the errors they cause in the determination of the "*pre-dried*" particle density may also be studied using calculus. The partial differential in this case is:

(dp_{p}/dV) = -(p_{p}/V_{t})

With the typical values given in the previous section one obtains:

(dp_{p}/dV) = -0.01

so that an error e(V) in the determination of the pyknometer volume will cause an error e(p_{p}) in the determination of the "*pre-dried*" particle density of a coarse aggregate, where:

e(p_{p}) = -0.01 e(V)

If particle densities are to be quoted to the nearest 0.01 Mg/m^{3} , then, for this to be justified, the pyknometer volume should be determined to the nearest 0.01 ÷ 0.01 = 1.0 mL . Annex A in the draft CEN method requires the pyknometer volume to be recorded "in millilitres", without specifying the degree of rounding. Further, it requires a balance weighing to 0.1 % of the test specimen mass: with test specimens of about 0.7 kg this is to 0.7 g , which is barely adequate. On the other hand, participants recorded the volumes of their pyknometers to the nearest 0.1 mL , and recorded masses to the nearest 0.1 g , both of which appear to be appropriate degrees of rounding. It might be better if the method required masses to be determined to 0.01 % of the capacity of the pyknometer, and the volume of the pyknometer to be determined to 0.01 % too.