Uncertainty estimation from InHouse Validation Studies. Determination of Organophosphorus Pesticides in Bread
This is the example A4 of the EURACHEM / CITAC Guide "Quantifying Uncertainty in Analytical Measurement", Second Edition.
Model Equation:
{calculation of the level of pesticide in the sample: P_{op} = (I_{op} * c_{ref} * V_{op})/(I_{ref} * Rec * m_{sample}) * F_{hom} * 1e6 but not used to calculate uncertainty}
{calculation of uncertainty}
P_{op} = P_{op}_{ }_{nominal} * f_{repeatability} * f_{bias} * f_{other};

List of Quantities:
Quantity  Unit  Definition 

P_{op}  mg/kg  Level of pesticide in the sample 
P_{op}_{ }_{nominal}  mg/kg  Nominal level of pesticide in the sample 
f_{repeatability}  Uncertainty contribution covering repeatability  
f_{bias}  Uncertainty contribution due to bias (recovery)  
f_{other}  Other uncertainty contributions 
P_{op}_{ }_{nominal}: 
Constant Value: =1/0.9 
The nominal value of the level of pesticide in the sample is not associated with any uncertainty. A value of 1/0.9 is chosen so that P_{op} will be 1, the the combined uncertainty of the result can be used directly as a relative uncertainty.
f_{repeatability}: 
Type B normal distribution Value: 1 Expanded Uncertainty: =0.382/sqrt(2) Coverage Factor: 1 
Repeatability has been investigated in a precision study. A number of duplicate tests (same homogenised sample, complete extraction/determination procedure) were performed for typical organophosphorous pesticides found in different bread samples (for data see Table A4.2 in the EURACHEM / CITAC Guide). The normalised difference data (the difference between the duplicate results divided by the mean) provides a measure of the overall run to run variability. To obtain the estimated relative standard uncertainty for single determinations, the standard deviation of the normalised differences is taken and divided by sqrt(2) to correct from a standard deviation for pairwise differences to the standard uncertainty for the single values. This gives an uncertainty of 0.382/sqrt(2) = 0.27.
f_{bias}: 
Type B normal distribution Value: 0.9 Expanded Uncertainty: =0.28/sqrt(42) Coverage Factor: 1 
Data from an inhouse validation study was used to obtain data on recovery. On 42 spiked bread samples, the mean recovery was 90% with a standard deviation of 28%. The standard uncertainty of the recovery is calculated as the standard deviation of the mean, 0.28/sqrt(42).
f_{other}: 
Type B normal distribution Value: 1 Expanded Uncertainty: 0.2 Coverage Factor: 1 
This factor includes all other sources of uncertainty not adequately covered by f_{repeatability} and f_{bias}. Due to the design of the precision study, all uncertainty contributions arising from balances and volumetric measurement devices are already covered by f_{repeatability}. This leaves only the purity of the reference standard, the possible nonlinearity of the GC response and the sample homogeneity as possible sources of uncertainty. The purity of the reference standard is biven by the manufacturer as 99.53% ±0.06%. Assuming a rectangular distribution, this would be an additional uncertainty of 0.0006/sqrt(3) = 0.00035. This uncertaintiy contribution is so small compared to e.g. the contribution of f_{repeatability,} that it can be safely neglected. Linearity of the response of the GC within the given concentration range is established for the substances under investigation during the validation studies. No allowance is made for nonlinearity. Homogeneity of the bread samples has been estimated on the basis of the sampling method used. A number of feasible pesticide residue distribution scenarios were considered, and a simple binominal statistical distribution used to caluclate the standard uncertainty for the total included in the analysed sample. The scenarios, and the calculated relative standard uncertainties in the amount of pesticide in the final sample, were: Residue distributed on the top surface only: 0.58 (This scenario would arise in the case of decorative additions of whole grains to the surface of the bread) Residue distributed evenly over the surface only: 0.2 Residue distributed evenly throgh the sample, but reduced in concentration by evaporative loss or decomposition close to the surface: 0.05  0.10 The second scenario was chosen as the most likely worst case. Therefore an uncertainty of 0.2 was chosen.
Uncertainty Budgets:
P_{op}: Level of pesticide in the sample
Quantity  Value 
Standard Uncertainty 
Distribution 
Sensitivity Coefficient 
Uncertainty Contribution 
Index 

P_{op}_{ }_{nominal}  1.1111111111111 mg/kg  
f_{repeatability}  1.000  0.270  normal  1.0  0.27 mg/kg  63.3 % 
f_{bias}  0.9000  0.0432  normal  1.1  0.048 mg/kg  2.0 % 
f_{other}  1.000  0.200  normal  1.0  0.20 mg/kg  34.7 % 
P_{op}  1.000 mg/kg  0.340 mg/kg 
Results:
Quantity  Value 
Expanded Uncertainty 
Coverage factor 
Coverage 

P_{op}  1.00 mg/kg  0.68 mg/kg  2.00  95% (ttable 95.45%) 