# Testing of industrial products - Aggregates for construction

## Results of the cross-testing experiment on the Shape Index test

### Repeatability and reproducibility

One source of variation in the test results must be sought in the principles of the test method. The test answers either with "yes" or "no" the question: is this particle of "cubical" shape? When the test is used to estimate the percentage of non-cubical particles in a bulk sample (by counting the number of non-cubical particles, not by weighing them), the repeatability standard deviation for test portions of n particles can be expressed by the equation:

(1) Sr = {c × nc ÷ n}½

where

Sr = repeatability standard deviation;

n = number of particles in the test portion;

c = number of cubical particles in the bulk sample, expressed as a percentage of the total number of particles in the bulk sample;

nc = number of non-cubical particles in the bulk sample, expressed as a percentage of the total number of particles in the bulk sample;

This means that, even under ideal conditions, the repeatability standard deviation for test portions of 300 particles out of a bulk sample with 50 per cent cubical particles, will be:

(2) Sr = {50 × 50 ÷ 300}½

which is 2.9 %, equivalent to about 9 particles. In terms of weight this would mean a standard deviation of 2 % by mass, if we assume that the non-cubical particles weigh half as much as the cubical ones.

From Equation (1) it can be seen that 50 % of non-cubical particles is the pessimum and for size fractions with a higher or lower percentage of poorly shaped particles, the standard deviation becomes smaller. In the practical test, of course, there are additional factors adding to the variation of results.

The reproducibility standard deviation in the German Standard (DIN52114) is given as:

(3) SR = {0.01 × C × NC}½

when the test is used to estimate the percentage of non-cubical particles in a bulk sample (by weighing the non-cubical particles, not by counting them), and:

SR = reproducibility standard deviation (in per cent by mass);

C = percentage (by mass) of cubical particles in the sample;

NC = percentage (by mass) of non-cubical particles in the sample.

This formula results, for 50 % of non-cubical particles, in a reproducibility standard deviation of 5 % by mass. The corresponding repeatability standard deviation is given in the German Standard as:

(4) Sr = {0.002 × C × NC}½

The graphs of functional relations compare the results obtained in the cross-testing experiment with the values calculated using the formulae from the German Standard.