MEASURE FOR MEASURE
Into the Unknown
A guide to estimating Type A measurement uncertainty data
by Dilip Shah
For estimation of measurement uncertainty,1-4 both Type A and B uncertainty contributors may need to be considered, depending on the parameter that is being estimated. ISO/IEC Guide 99:2007—International vocabulary of metrology—Basic and general concepts and associated terms defines Type A evaluation of measurement uncertainty as: "Evaluation of a component of measurement uncertainty by a statistical analysis of measured quantity values obtained under defined measurement conditions."5
One of the quickest ways of estimating Type A measurement uncertainty data is by taking a series of measurements and calculating its appropriate standard deviation. For example, if you took 10 measurements, as shown in Table 1, its standard deviation would be calculated by using the sample standard deviation formula. This is also classified as repeatability data in the measurement uncertainty budgets.
One of the reasons ISO/IEC 170256 accrediting bodies require accredited laboratories to estimate Type A contributors to their measurement uncertainty budget is to ensure that the laboratory is capable of making measurements with the equipment that they list in their scope of accreditation. This validates the competence of the laboratory and the personnel making the measurements.
The repeatability of measurements is usually the minimum requirement for listing Type A uncertainty contribution to the measurement uncertainty budget. However, any uncertainty contributor that is estimated by statistical means should be listed as Type A data with its appropriate degrees of freedom. Examples include:
- Actual, statistical analysis of monitored data of environmental conditions (temperature and relative humidity).
- Long-term drift of equipment that is characterized statistically.
When collecting Type A data for analysis, ensure that the data collected takes into account the instrument resolution.7 If the instrument reads to a 0.001 resolution, ensure that the data collected also reports to a 0.001 resolution. For example, a 0.500 reading on display is reported as 0.500. There should not be any premature rounding of data. When data get collected by automated data acquisition systems, ensure that it is capable of collecting and reporting to the appropriate resolution.
In most cases, a laboratory has more than one technician that is capable of making the measurements for that parameter. Depending on the availability of the technician, either one may be assigned to the task of making the measurement. In that case, it is important to know that measurements made by either of the technicians are statistically insignificant (reproducible) or if they are significant, the reproducibility of the technicians need to be taken into account and listed as the second Type A uncertainty contributor with its degrees of freedom.
Conducting such a study may require each of the technicians to make 10 or more measurements on a single artifact, as shown in Table 2, and employing appropriate statistical analysis. One-way analysis of variance (ANOVA) is one technique that can be used. ANOVA is available in popular spreadsheets and statistical software packages. It uses the F-distribution to determine if there is a statistically significant difference in the between groups (reproducibility) and within groups (repeatability) data. It determines this by looking at the F-critical value (2.86626) from the F-distribution table (degrees of freedom numerator of three and degrees of freedom denominator of 36 at 95% confidence interval) and comparing it with the F-calculated value, which is the between groups mean square (MS)-value divided by within groups MS-value (see Online Figure 1 and Online Table 1 on this column’s webpage at www.qualityprogress.com for more on the F-distribution). The MS-value is derived by dividing the sum of squares value by degrees of freedom. It also is the variance value.
The decision rules are:
- If the F-value is less than the F-critical value, then the data are considered statistically insignificant. In this case, reproducibility may be ignored if it does not make a significant overall contribution to measurement uncertainty.
- If the F-value is greater than the F-critical value, then the data are considered statistically significant. The reproducibility should be included in the measurement uncertainty budget.
- If the F-value is very close to the F-critical value, the data should be considered statistically significant. The reproducibility should be included in the measurement uncertainty budget. See Rule No. 4 for p-value (0.6195 = 61.95%).
- Also consider the p-value (0.6195 = 61.95%). It shows the probability of significance. The higher it is, the better the confidence. This should also be factored into the decision-making process.
To calculate reproducibility from the data in Table 2, divide the square root of the MS-value for between groups by the square root of 10 (or the appropriate sample size for the user’s data).
To calculate repeatability from the data in Table 2, calculate the square root of the MS-value for within groups.
Ensure that the degrees of freedom associated with Type A uncertainty are included in the measurement uncertainty budget. Depending on the percent contribution to the overall uncertainty, the k coverage factor may be dependent on the effective degrees of freedom using the Welch Satterthwaite formula.8, 9
A future column will look at common Type B uncertainty contributors, and how they are estimated for the measurement uncertainty budget.
- International Organization for Standardization, ISO/IEC Guide 98-3:2008—Guide to the expression of uncertainty in measurement.
- Dilip Shah, "In No Uncertain Terms," Quality Progress, January 2009, pp. 52-53.
- Dilip Shah, "Standard Definition," Quality Progress, March 2009, pp. 52-53.
- Dilip Shah, "Balanced Budget," Quality Progress, May 2009, pp. 54-55.
- International Organization for Standardization, ISO/IEC Guide 99:2007—International vocabulary of metrology—Basic and general concepts and associated terms.
- International Organization for Standardization, ISO/IEC 17025:2005—General requirements for the competence of testing and calibration laboratories.
- Dilip Shah, "Keep Your Resolution," Quality Progress, March 2011, pp. 56-58.
- International Organization for Standardization, ISO/IEC Guide 98-3:2008—Guide to the expression of uncertainty in measurement, see reference 1.
- Dilip Shah, "Student Teaching," Quality Progress, January 2012, pp. 47-49.
Dilip Shah is president of E = mc3 Solutions in Medina, OH. He is chair of ASQ’s Measurement Quality Division and past chair of Akron-Canton Section 0810. Shah is also co-author of The Metrology Handbook (ASQ Quality Press, 2012). Shah is an ASQ-certified quality engineer, quality auditor, calibration technician and an ASQ fellow.