Until now, our emphasis has been on grouping uncertainty sources as either systematic or random. The ISO in its guide does not recommend grouping uncertainty sources or errors that way. Instead, it suggests grouping them as either Type A, where there are data to calculate a standard deviation, or Type B, where there aren’t. This approach seems in conflict with the commonly applied terminology of systematic and random uncertainty sources.

Emerging approach
However, a new uncertainty model that combines the best features of both methods is now coming into vogue. It handles the ISO recommendations of using Type A and Type B classifications and still allows the engineer to quote uncertainties in the more physically understandable vocabulary of systematic and random. How can this be? What compromises were reached? Tune in next time for the rest of this story and the answers.
For now, though, let’s review the basic principles of each model.
We’ll start with the ISO model. For Type A uncertainties, data can be used to calculate standard deviation. Type B uncertainties must be estimated in some other way. The total uncertainty, which ISO calls “expanded uncertainty,” is then calculated by root-sum-square of the two types of uncertainties. But first, all the elemental Type A and Type B uncertainties are combined by root-sum-square. That is, we first calculate:
<graphic>
and then:
<graphic> .
Note here that the <ital>U<subscript>B,i<end subscript, end ital> need an assumed distribution and degrees of freedom. The new U. S. National Standard published by the ASME recommends that you assume the <ital>U<subscript>B,i<end subscript, end ital> are normally distributed and have infinite degrees of freedom.
It is also important to recognize that all the <ital>U<subscript>A,i<end subscript, end ital> and <ital>U<subscript>B,i<end subscript, end ital> uncertainties are standard deviations of the average for that uncertainty source <em dash>—<em dash> that is, they all represent one <graphic> .
We then need to combine the <ital>U<subscript>A<end subscript, end ital> and <ital>U<subscript>B<end subscript, end ital> uncertainties into the total or expanded uncertainty. That expanded uncertainty is:
<graphic> .
The constant out front, <ital>K<end ital>, is used to provide the confidence desired. The most common choice for that constant is Student’s <ital>t<end ital><subscript>95<end subscript>, which provides an uncertainty with 95% confidence. This ISO expanded uncertainty would then be written:
<graphic> .

A question of degree
Before the Student’s <ital>t<end ital><subscript>95<end subscript> can be determined, there is one more important step. Do you know what it is? The degrees of freedom for <ital>U<subscript>ISO<end subscript, end ital> are needed. How do we get that? Well, each standard deviation of the average we’ve used in the <ital>U<subscript>A<end subscript, end ital> and <ital>U<subscript>B<end subscript, end ital> equations above has its associated degrees of freedom. For the <ital>U<subscript>A<end subscript, end ital>, the degrees of freedom come directly from the data that are used to calculate the standard deviations of the average, that is,
<graphic>
where <graphic>  represents degrees of freedom, sometimes abbreviated as <ital>d.f.<end ital>
These degrees of freedom are for all the <ital>U<subscript>A,i<end subscript, end ital> where <ital>N<subscript>i<end subscript, end ital> is the number of data points used to calculate the standard deviations of the average.
For the <ital>U<subscript>B,i<end subscript, end ital>, the degrees of freedom are assumed to be infinite.
The degrees of freedom, <lower case Greek nu>, for the <ital>U<subscript>ISO<end subscript, end ital> is computed for the total uncertainty with the Welch-Satterthwaite approximation:
<graphic>
This formula is a real pain. Hand calculations are very frustrating here. So, program the formula on your computer. One simplifying aspect is that one term in the denominator, <graphic> , is zero when the <graphic>  is infinity.
Now, with the degrees of freedom, the Student’s <ital>t<end ital><subscript>95<end subscript> can be found in a table in any statistics text. Not a problem.
If 99% or some other confidence is desired, just use the proper Student’s <ital>t<end ital>.
Well, there you have it. Now, we need to consider the U. S. Uncertainty Standard and how to calculate that uncertainty. What are its major components? Hint, they are associated with the impact of uncertainties on the test result. Second hint, these groupings are familiar to engineers the world over. Do you know what they are? Next time....
Until then, remember, “use numbers not adjectives.”

Ronald H. Dieck is the principal of Ron Dieck Associates, Inc., Palm Beach Gardens, FL. E-mail him at rondieck@aol.com.