Testing of industrial products - Aggregates for construction

Formulae

• Return to List of test methods
• Return to Home Page

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

Cochran's test statistic = W² / Sum W²

k - statistic = W/W_pooled

where

W = the range for a laboratory

Sum W² = 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 = S² ÷ (Sum S²)

Standardised standard deviation = S ÷ { Sum S²/(2N) }^½

where

2N = the number of standard deviations

S = the standard deviation of three determinations

Sum S² = 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 / (S₀)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;

(S₀)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.

R₁ = 2.8 S_R1

r₁ = 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.