MEASURE FOR MEASURE
Applying t-distribution to measurement uncertainty estimation
by Dilip Shah
While assisting laboratories with their measurement uncertainty estimation for ISO 17025 accreditation, I am frequently asked the question of when to use Student’s t-distribution. Generally, the laboratory technicians and managers are proficient in their technical capabilities but struggle with the statistical part of the measurement uncertainty analysis.
So just as a refresher, let’s describe the use of t-statistics in layman’s terms and show some examples of the measurement uncertainty application. First, a little background on Student’s t-distribution, which was discovered by William S. Gosset in 1908. Gosset was a statistician employed by the Guinness & Co., which did not allow his discovery to be published under his own name. He therefore wrote under the pen name "Student."
These distributions can be applied to the following situation: Suppose you have a simple random sample of size n drawn from a normal population with mean µ and standard deviation σ. Let x denote the sample mean and s the sample standard deviation. In this case, the quantity
has a t-distribution with n – 1 degrees of freedom.
Note there is a different t-distribution for each sample size—in other words, it’s a class of distributions. When you speak of a specific t-distribution, you need to specify the degrees of freedom. The degree of freedom for this t-statistic comes from the sample standard deviation s in the denominator of the previous equation.
The t-density curves are symmetric and bell-shaped—similar to the the normal (Gaussian) distribution—and have their peak at 0. But their spread is more than that of the standard normal distribution. In the equation, the denominator is s rather than µ. Because s is a random quantity varying with various samples, there’s more variability in t, resulting in a larger spread.
The larger the degrees of freedom, the closer the t-distribution is to the normal (Gaussian) distribution. This reflects the fact that the standard deviation s approaches µ for large sample size n. At 29 degrees of freedom (sample size 30) or higher, the t-distribution begins to approximate the normal (Gaussian) distribution. This is illustrated in Figures 1-3 and the t-distribution in Table 1.
The k coverage factor for the normal distribution for 95% confidence interval is 1.96. In the measurement uncertainty estimation, you would round it to k = 2 at approximately 95% confidence interval.
Time to apply
After this brief overview of t-distribution, let’s focus on its application in measurement uncertainty. When estimating measurement uncertainty, there is at least one contributor that is Type A uncertainty (repeatability). ISO 17025 accrediting bodies mandate that repeatability must be considered in the measurement uncertainty estimation.
Because repeated measurements take time—and time is money—laboratories take about 10 repeated measurements instead of 30, approximating the normal distribution. With 10 measurements, there are nine degrees of freedom (n – 1).
In addition, accrediting bodies also recommend that laboratories consider reproducibility (another Type A uncertainty contributor, with even smaller degrees of freedom—for example, three operators or instruments with 10 measurements performed by each) as a contributor.
Other uncertainty contributors in the measurement uncertainty budget may be Type B—those derived by means other than statistical analysis. Examples of Type B uncertainty contributors are specification, resolution and uncertainty reported in calibration reports for metrological traceability. Type B uncertainty contributors are considered to have infinite degrees of freedom after you convert them to standard uncertainty.
Thus, the uncertainty budget has Type A and B contributors that are added together—using the root sum square (RSS) method—to report combined uncertainty. The combined uncertainty is then reported as expanded uncertainty with a coverage factor k = 2 at approximately 95% confidence interval.
But this k factor is assumed for normal distribution with infinite degrees of freedom. This is assuming all of the uncertainty contributors have infinite degrees of freedom—either all Type B data or both Type A and B data with each Type A contributor having 30 degrees of freedom or more.
Now it’s possible to refine the original question to: When should you use coverage factor k = 2 or when should coverage factor k be used based on 95% confidence interval based on the t-distribution table?
In the measurement uncertainty budget, you can possibly have Type A contributors with varying degrees of freedom and Type B contributors with infinite degrees of freedom—in the spreadsheet examples, a value 100 (greater than 30) is used. ISO Guide 98-3:2008 describes a method to calculate effective degrees of freedom using the Welch-Satterthwaite formula:
In that equation, uc is combined uncertainty, ui is individual uncertainty, and vi is individual uncertainty’s degrees of freedom.
It’s always a good idea to design a template for measurement uncertainty. The template can be a paper form filled out manually or one designed using a spreadsheet. The spreadsheet template’s advantage is that it will perform all calculations automatically, thus reducing the chance of any calculation error. Still, any spreadsheet template should always be validated.
If you calculate the effective degrees of freedom in the measurement uncertainty budget in this measurement uncertainty template, it will help you determine when you can get away with using k = 2 and when you need to use the t-distribution.
An important part of our measurement uncertainty spreadsheet template—or paper form—should be the percentage contribution column, which should be calculated using the variance of each uncertainty contributor divided by the total variance.
A good measure to ensure it is calculated correctly is to make sure the total contribution equals 100%. If the total contribution does not equal 100% and is greater than 100%, it’s likely the individual uncertainties are divided by the total combined uncertainty (see Table 2). This is a common mistake performed by technicians and managers alike.
Online Table 1 shows several examples with varying magnitudes of Type A uncertainty and its percentage contribution to illustrate the effective degrees of freedom.
In those examples, you can see that if the Type A data do not contribute significantly to the overall measurement uncertainty, you can use k = 2 for the coverage factor for expanded uncertainty (the effective degrees of freedom is greater than 30).
If Type A measurement uncertainty contributes significantly to the overall measurement uncertainty, the effective degrees of freedom (less than 30) will reflect that, and the t-distribution table will be referred to obtain the correct k-factor multiplier.
Performing hands-on analysis by using a spreadsheet or manually calculating the measurement uncertainty helps determine how the correct, appropriate coverage factor k is used to report expanded measurement uncertainty at approximately 95% confidence interval.
Another advantage of using this method is the data and justification are documented to defend the use of correct statistics in the measurement uncertainty estimation during the ISO 17025 assessment audit or customer inquiry.
- 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.
- International Organization for Standardization, ISO/IEC Guide 98-3: 2008—Guide to the expression of uncertainty in measurement (GUM:1995).
- Shah, Dilip, "Balanced Budget," Quality Progress, May 2009, pp. 54-55.
- Shah, Dilip, "In No Uncertain Terms," Quality Progress, January 2009, pp. 52-53.
- Shah, Dilip, "Standard Definition," Quality Progress, March 2009, pp. 52-53.
Dilip Shah is president of E = mc3 Solutions in Medina, OH. He is a past chair of ASQ’s Measurement Quality Division and Akron-Canton Section 0810, and is co-author of The Metrology Handbook (ASQ Quality Press, 2004). Shah is an ASQ&-certified quality engineer and calibration technician, and a senior member of ASQ.