# Testing of industrial products - Aggregates for construction

## Formulae

### Cochran's test, and standardised ranges (k-statistics), for ranges of two test results, or ranges of two determinations.

Cochran's test statistic = W^{2} / Sum W^{2}

k - statistic = W/W_{pooled}

where

W = the range for a laboratory

Sum W^{2} = the sum of squared ranges for a level

W_{pooled} = the pooled range for a level

### Cochran's test for standard deviations of three determinations

and standardised standard deviations of three determinations.

Cochran's test statistic = S^{2} ÷ (Sum S^{2})

Standardised standard deviation = S ÷ { Sum S^{2}/(2N) }^{½}

where

2N = the number of standard deviations

S = the standard deviation of three determinations

Sum S^{2} = the sum of squared standard deviations for the level

### Grubbs' test, and standardised laboratory biasses (h-statistics).

Grubbs' test statistic = h-statistic = (X - X_{ave}) / X_{sd}

where

X = a laboratory average

X_{ave},X_{sd} = the average and standard deviation

Grubbs' test for a pair of outliers both exceeding the average:

test statistic = (S_{p-1,p})^{2} / (S_{0})^{2}

where

(S_{p-1,p})^{2} = the corrected sum of squares of the p-2 values that do not include the two suspect values;

(S_{0})^{2} = the corrected sum of squares of all p values.

Note: The use of Cochran's and Grubbs' tests for outliers is described in the standard on precision (ISO 5725, 1994).

### Repeatability and reproducibility standard deviations.

These are calculated using formulae for a uniform-level precision experiment with an unequal number of replicates per cell, given in the standard on precision (ISO 5725, 1994).

### Repeatability and reproducibility limits.

R_{1} = 2.8 S_{R1}

r_{1} = 2.8 S_{r1}

When two determinations are made on a test portion:

W_{c} = 2.8 S_{spec}

When three determinations are made on a test portion:

W_{c} = 3.3 S_{spec}

where S_{spec} is the between-specimen standard deviation

### Functional relationships.

These are calculated using formulae for weighted regression analysis given in the standard on precision (ISO 5725, 1994).

### Coefficients of variation.

C_{r1} = 100 S_{r1} / X_{ave}

C_{R1} = 100 S_{R1} / X_{ave}

### Coefficients of variation for the Sand Equivalent test, Percentage of Crushed Particles, Percentage of Totally Crushed Particles.

C_{r1} = 100 S_{r1} / (100 - X_{ave})

C_{R1} = 100 S_{R1} / (100 - X_{ave})

### Sensitivity ratios.

The sensitivity ratio for "Level 2 - Level 1" and repeatability is calculated as:

Sensitivity ratio = (X_{ave,1} - X_{ave,2}) / sqrt{ (S_{r1,1})^{2} + (S_{r1,2})^{2} }

where

X_{ave,1} and X_{ave,2} are the averages for Levels 1 and 2

S_{r1,1} and S_{r1,2} are the repeatability standard deviations for Levels 1 and 2

The other sensitivity ratios are calculated similarly.