Analytical Measurement Uncertainty and Method Validation

Question 1:

In each question below, identify the most appropriate type of distribution that is being referred to. Then, calculate the standard uncertainty.

(a) Specification from the manufacturer for a 10 mL volumetric flask is 10 ± 0.025 mL. What is the standard uncertainty in the volume of the flask?

(b) A method specifies that a test portion of 1.000 ± 0.010 g is weighed for analysis. In subsequent calculations, the mass is treated as 1.000 g. What is the standard uncertainty attributable to this assumption?

(c) The calibration certificate for an analytical balance states that the measurement uncertainty is ± 0.000035 g @ 20 g load. The uncertainty is based on a confidence interval of 95%. What is the standard uncertainty of weighing?

(d) A dieldrin reference standard is described by the supplier as "99% min". Estimate the standard uncertainty of the purity.

(e) A calibration mass is certified as 0.05003 g ± 0.02 mg. What is the standard uncertainty of the mass in grams and in mg?

(f) The standard deviation of repeat weighings of a 50 mg mass is 0.033 mg. What is the standard uncertainty of a single weighing?

(g) Five absorbance readings were 0.123, 0.125, 0.128, 0.120 and 0.122.
(i) What is the standard uncertainty of a single absorbance reading?
(ii) What is the standard uncertainty of the mean absorbance?

Question 2:

(a) Calculate the combined standard uncertainty for the volume of iso-octane in a 100 mL volumetric flask from the following information.
- The results of 10 fill and weigh experiments on a 100 mL Class A volumetric flask had a standard deviation of 0.017 mL.
- Specification from the manufacturer for the flask is ± 0.1 mL.
- The coefficient of volume expansion for organic solvents, α, is 1x10^-3 ºC^-1.
- The mean laboratory temperature was 21.5ºC. The standard deviation of its variations was 2.3ºC.
NOTE: Temperature has an effect on the amount of the solvent contained in the flask, because of high coefficient of thermal expansion of organic liquids.

(b) A 5 mL pipette is used to dispense an organic solvent. Calculate the standard uncertainty in the volume of liquid delivered by the pipette.

- 10 repeat dispensings of liquid had a standard deviation of 0.005 mL.
- Tolerance stated by the manufacturer for the pipette is 5 ± 0.015 mL
- The coefficient of volume expansion for organic solvents, α, is 1x10^-3ºC^-1
- The mean laboratory temperature was 21.5ºC. The standard deviation was 2.3ºC.

(c) A method requires 50 mg of a dieldrin standard to be weighed out on an analytical balance. What is the standard uncertainty in the mass of the compound?
- 10 replicate weighings of a 50.03 ± 0.02 mg certified mass had a mean 50.039 mg and standard deviation of 0.033 mg.

(d) A primary stock solution is prepared by dissolving approx. 50 mg of dieldrin in iso- octane and making up to 100 mL in a volumetric flask.

I. Calculate the concentration of the solution in mg mL^-1.

II. What is the standard uncertainty in the solution concentration?
- 50.3 mg of dieldrin was weighted out. The standard uncertainty due to mass as calculated in Question 2(c).
- The purity of the standard was quoted by the supplier as being 99% minimum. The standard uncertainty for this was calculated in Question 1(d).
- The standard uncertainty due to the 100 mL of iso-octane liquid was calculated in Question 2(a).

Question 3:

A standard solution of cadmium was prepared by dissolving cadmium metal in a nitric acid solution. The certificate of analysis for the cadmium metal stated that the purity was "0.9999±0.0001". The report also stated that, "the reported uncertainty is an expanded uncertainty with a coverage factor of 2, which gives a level of confidence of approximately 95%". In an analysis, 100.28 mg of the cadmium was weighed out, and the standard uncertainty on this mass was determined to be 0.05 mg, the associated degrees of freedom with this determination was 19. A calibrated 100 mL volumetric flask was used to prepare the solution to volume; the standard uncertainty on this flask was determined to be 0.07 mL. The associated degrees of freedom with this determination were 9. Using this information,

(a) Determine the concentration of cadmium in the standard solution.

(b) Determine the standard uncertainty of the concentration of this solution.

(c) Use the Welch-Satterthwaite equation to estimate the effective degrees of freedom associated with this standard uncertainty (note, you will need to estimate the degrees of freedom associated with the purity of the cadmium).

(d) Determine the coverage factor to be used to convert the standard uncertainty into an expanded uncertainty (assume a 95% confidence level) and report the result and its expanded uncertainty.

(e) The Eurachem guide on measurement uncertainty recommends using a coverage factor of 2 to calculate an expanded uncertainty with 95% level of confidence. Comment on this assumption compared to the coverage factor that you determined in (c) and (d).

Question 4:

In a laboratory, a standard solution must be prepared based on an aqueous solution of CH₃COOH of concentration 4% (w/w) using the following procedure.

Forty millimetres of a specified stock solution of 100% (v/v) (standard deviation 0.5% (v/v)) is pipetted into a 1 L volumetric flask using a class A 20 mL pipette and the flask is filled with water. The difference between the laboratory temperature and the calibration temperature of the pipette and volumetric flask is not more than ±2°C.

The manufacturer's calibration data of the volumetric flask and the pipette and the standard deviation of the manual operations, obtained by earlier tests in the laboratory, are tabulated
below. The coefficient of volume expansion of water is 2.1x10^-4°C^-1.

Calibration data at 20°C

1 L volumetric flask ±4 mL
20 mL class A pipette ±0.03 mL

Standard deviation of the manual operations
1 L volumetric flask 1.5 mL
20 mL class A pipette 0.016 mL

Calculate the expanded uncertainty at the 95% confidence level. Include a cause and effect diagram in your analysis.

Question 5:

(a) To re-calibrate a 10 mL pipette, the volume of water (0.998207 g mL-1) was measured by 10 replicates giving the following results in grams.

What are the mean value and the standard uncertainty for a single pipetting step?

(b) The data set for the calibration of the photometric determination of nitrite is given below.

The measured absorbance values for a sample are: 0.4892; 0.4886; 0.4895. Calculate the predicted average [nitrite] and the standard uncertainty of the sample.

(c) The specification for a 10 mL burette is quoted by the manufacturer as ±0.02 mL. Calculate the standard uncertainty under the conditions of a rectangular and a triangular distribution.

(d) According to the calibration certificate for a balance, the measurement uncertainty is
±0.0005 g at a 95% confidence level. Calculate the standard uncertainty.

(e) The standard deviation of repeated weighing of 0.1 g is calculated to be 0.00015 g. Calculate the standard uncertainty.

(f) The results obtained in a proficiency test of the determination of benzo[a]pyrene in drinking water according to a standard method are tabulated below. Calculate the standard uncertainty.

ctrue/mg L^-1      caverage/mg L^-1    RSD / %   sbetween-lab/mg L    swithin-lab/mg L^-1

24                     19.05                  79.4                   4.92                    2.34