- 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 of Filler 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 data for Laboratories B, O and R at all three levels. In the case of Laboratory B (Number code 32), their data give outlying between-test-result ranges in Levels 1 and 2, and a high between-test-result range in Level 3. With Laboratory O (Number code 252), their data give outliers in either the between-specimen standard deviations or between-test-result ranges at all three levels. Laboratory R obtained outlying laboratory averages at all three levels. The data for Laboratory N (Number code 251) for Test result 1 in Level 1 were excluded because they give an outlying between-test-specimen standard deviation. The data for Laboratory L (Number code 234) in Levels 2 and 3 were excluded because they give outlying between-test-result ranges. The data for Laboratory S (Number code 170) in Level 1 were excluded because they give an outlying between-test-result range.

In Figure 1 it can be seen that the three materials give similar repeatability and reproducibility standard deviations, so that it is possible to summarise the results using the simple averages of these standard deviations.

From the densities reported it appears that Laboratories B, M, N, P and S used water as the liquid in the test. Although water is permitted by the CEN method, it is possible that it is not as suitable as the organic liquids because it is polar. Inspection of the results, however, does not indicate that the laboratories that used water obtained results that differed from those of the other laboratories in a consistent manner.

*Subsequent to the publication of the report on the experiment, it was observed that there is a relationship between the density of the liquid and the results obtained by different laboratories. Thus standardising the liquid should improve the precision of the method.*

Particle densities are not used to check compliance of a filler with limits in specifications, so the "*8% criterion*" used in other reports in this project is not applicable here.

6 of participants reported data that were classed as outliers. This is a high proportion of the 19 participants that took part in the experiment. Many of the outliers occurred in the between-test-result ranges or between-test-specimen standard deviations, so the precision of the test will be improved if laboratories use the critical value for the range of three determinations established in this experiment to carry out repeatability checks. Note that the average value over the three levels in Table is W_{c} = 0.025 Mg/m^{3}.

However, once the outliers have been removed, the data give estimates of reproducibility standard deviations that are much higher than the estimates of repeatability standard deviations. The determination of density is a simple procedure involving only a number of weighings which are required to be carried out to an accuracy of 0.01 g, which is to 0.1 % of the test specimen mass. It is rather surprising that the reproducibility is as high as found in this experiment. Sensitivity analysis is used in the following sections to try to identify possible reasons for this.

By varying the values used in the calculations it can be seen that the result of the determination of the particle density of a filler is very sensitive to variations in the density of the liquid used in the test. This can be illustrated by an example. Laboratory A (31) reported the following values for their Test Specimen *a* in Level 1:

- Volume of pyknometer V = 53.32 mL
- Mass of pyknometer m
_{0}= 53.00 g - Mass of pyknometer + filler m
_{1}= 63.56 g - Mass of test specimen m
_{t}= 10.56 g - Mass of pyknometer + filler + liquid m
_{2}= 105.47g - Liquid density p
_{1}= 0.865 Mg/m^{3} - Volume of test specimen V
_{t}= 4.87 mL - Filler density p
_{f}= 2.168 Mg/m^{3}

If the value for the liquid density is changed by just 0.001 Mg/m^{3} to 0.866 Mg/m^{3} one obtains instead:

- Volume of test specimen V
_{t}= 4.93 mL - Filler density p
_{f}= 2.142 Mg/m^{3}

so that the particle density of the filler is changed by the much larger amount of 0.026 Mg/m^{3}.

Laboratory O (252) determined the liquid density afresh for each test result (i.e. for each set of three determinations of the particle density of the filler), and for Level 1 obtained p_{1} = 0.8644 Mg/m^{3} for their first test result, and p_{1} = 0.8619 Mg/m^{3} for their second test result. The difference of 0.0025 Mg/m^{3} between these two liquid densities explains why their two test results for Level 1 differed by 0.11 Mg/m^{3} (and were classed as outliers).

The reason why the particle density is so sensitive to the liquid density may be seen by applying a little calculus. The particle density is calculated using the formula:

p_{f} = (m_{1} - m_{0})/(V - (m_{2} - m_{1})/p_{l})

so the partial differential of p_{f} with respect to p_{l} is:

dp_{f}/dp_{l} = (p_{f}/p_{l})(1 - V/V_{t})

This partial differential may be used as a multiplier to convert an error in the determination of the liquid density e(p_{l}) to the error caused in the determination of the particle density of a filler e(p_{f}) , using:

e(p_{f}) = (dp_{f}/dp_{l}).e(p_{l})

With the CEN method, the volume of the pyknometer is V = 50 mL and the volume of the test specimen is V_{t} = 4.5 mL, so the ratio of volumes in the expression for the partial differential is large (about 11). The ratio of densities is also greater than one (about 3 usually), so both factors combine to make the test result very sensitive to errors in the determination of the density of the liquid. With the data for Laboratory A given above one obtains:

p_{f}/p_{l} = 2.5

and

V/V_{t} = 10.9

so that

e(p_{f}) = - 25 e(p_{l})

i.e. an error in the determination of the liquid density causes an error 25 times larger in the determination of the particle density of a filler, as shown in the numerical example for Laboratory A above.

One can replace errors in the above formulae with standard deviations, or repeatability or reproducibility limits, to obtain, for example, an equation that shows how much the reproducibility of the determination of the density of the liquid R(p_{l}) contributes to the reproducibility of the determination of the particle density of filler R_{1}(p_{f}) :

R_{1}( p_{f}) = 25 R( p_{l})

Thus if the reproducibility of the determination of the density of the liquid is R(p_{l}) = 0.004 Mg/m^{3} , it would be enough to explain the reproducibility found for the determination of the particle density of filler of R_{1}(p_{f}) = 0.1 Mg/m^{3}. It would be of interest to carry out an experiment to find out the reproducibilities of the determination of the densities of the liquids used in the test.

The sensitivity could be reduced simply by increasing the test specimen mass. If the mass is increased to about 60 g so that the pyknometer is half-filled, giving V_{t} = 25 mL, the partial differential is reduced from 25 to 2.5 - a very substantial improvement. Obviously there is an upper limit to the size of the specimen that can be used - it may be difficult to expel all the air if the test specimen is too large. However, it is recommended that consideration should be given to increasing the test specimen mass to the maximum that can be used without causing such practical problems.

Also, it does not make sense, as in the proposed CEN method, to require triplicate determinations on filler specimens, but only a single determination of the liquid density, when the result of the test is so sensitive to the latter. The method should require, instead, several replicate determinations of the liquid density.

If particle densities of fillers are to be quoted to the nearest 0.1 Mg/m^{3} , as required by the CEN method, then, for this to be justified, the density of the liquid should be determined to the nearest 0.1 ÷ 25 = 0.004 Mg/m^{3}. Annex B in the CEN method requires the density of the liquid to be recorded to the nearest 0.001 Mg/m^{3}, so the Standard is correct on this point. However, the densities of the liquids permitted by the CEN method change by about 0.0002 Mg/m^{3} per °C . In view of this, it is surprising that the CEN method requires the temperature of the water bath to be maintained within the very narrow range of 25.0 ± 0.1°C. A wider tolerance of ± 0.5°C may well be adequate.

Laboratories 32 and 252 used pyknometers with volumes of 100 mL or 128 mL. This increases the partial differential from 25 to about 60, so it makes the sensitivity of the test much worse. In view this, the use of pyknometers larger than that prescribed in the proposed CEN method, with 10 g test specimens, should be discouraged.

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

dp_{f}/dV = -p_{f}/V_{t}

Hence increasing the mass of the test specimen will increase its volume V_{t}, which will reduce the size of the partial differential, and so make the result of the test less sensitive to the determination of the volume of the pyknometer.

With the data for Laboratory A given at the beginning of the previous section one obtains:

dp_{f}/dV = -0.45

so that an error e(V) in the determination of the pyknometer volume will cause an error e(p_{f}) in the determination of the particle density of a filler, where:

e(p_{f}) = -0.45 e(p_{V})

Again, if particle densities of fillers are to be quoted to the nearest 0.1 Mg/m^{3}, then, for this to be justified, the pyknometer volume should be determined to the nearest 0.1 ÷ 0.45 = 0.2 mL. Annex A in the draft CEN method requires the pyknometer volume to be recorded to the nearest 0.001 m^{3}, which is obviously a mistake. The Standard requires the weighings to be made to 0.01 g, so that it is justifiable to record the volume of the pyknometer to the nearest 0.01mL. This is an order of magnitude better than is really required according to the sensitivity analysis. Hence, according to this analysis, the calibration of the pyknometer does not appear to be a major source of errors in the determination of the particle density of filler.