Notes from “Measurement Uncertainty: Methods and Applications”
Overview
These are notes from a book called Measurement Uncertainty: Methods and Applications by Ronald H. Dieck. Instrument Society of America 1992. This is an “independent learning module from the Instrument Society of America. [1] The book has ”units“ rather than chapters.
Unit 1: Introduction and Overview
A description of how this independent learning module (ILM) works.
Unit 2: Fundamentals of measurement uncertainty analysis
Two sources of error in this book:
- precision error
- random error
- bias error
- systematic error
- blunders
- assumed not to exist
- (no term)
- Purpose of measurement is to numerically characterize the state or performance of a physical process
The pressure to use test data for decision making is often so great that there is a tendency to ABM the data is correct, even to the point of almost never reporting an estimate of the measurement uncertainty with its test result. (p. 9)
It is important to note that every measurement ever made by every engineer or scientist has been in error and will be so in the future. There has never been and never will be a case when a person measured a variable and obtained the true value. Error is the difference between the measurement and the true value. (p.9)
- Rather than “good data,” “bad data,” “nominal data,” etc. we need objective criteria for reporting the quality of test data.
The knowledge of the measurement uncertainty of a test result is as important as the result itself in characterizing the state or performance of a Prof. Test results should NEVER be reported without also reporting their measurement uncertainty. No manager or process owner should take action based on test results with an undefined measurement uncertainty.
Precision error
This source of error adds a (positive or negative) unknown component to the measurement; each successive measurement has a random component added to it. The errors are related in that they come from the same distribution but are uncorrelated in the sense that knowing one error doesn’t tell you about another. (This is a nice way to describe uncorrelated errors.)
The author assumes (maybe just for now) Gaussian errors. Errors are then said to follow the rule of 68% within \(1\sigma\), 95% within \(2\sigma\), and 99.7% within \(3\sigma\). This roughly holds for large samples substituting \( s \) for σ except when \(n < 31\). With small samples, we use \( t \). I forget the \( t_{\alpha/2} \) critical values so here they are from R:
n <- c(5, 10, 15, 20, 25, 30)
cbind(n, round(qt(0.975, n), 2))
| 5 | 2.57 |
| 10 | 2.23 |
| 15 | 2.13 |
| 20 | 2.09 |
| 25 | 2.06 |
| 30 | 2.04 |
I never thought of this: if using \( t \) critical values compensates for \( s \) underestimating \( \sigma \) why not adjust \( s \) via \( n-2 \) or something?
There is some discussion of “degrees of freedom” and the author says it is the “freedom left in a data set for error or variability” (p. 16).
The author gives an example of 20 measurements where the true value is a known 100°F and errors are assumed Gaussian. He suggests using Bartlett’s test to check for normality which is not something I do. In the example 19/20 data points are within the t-based 95% CI. The interval \( \pm t s \) is “sometimes called the precision error for a particular data set.”
Having now understood the calculation of the standard deviation and the utility of the value \( t s \), it is important to note that usually a measurement is taken to understand the average of the set of data, not the spread or scatter in the data. What is usually needed or wanted is a statement about the error present in the average \( \bar{x} \).
That desire may be expressed by asking, “How good is the average?” Since the term “good” is poorly defined conceptually and undefined numerically, it would be more proper to ask, “Where the experiment to be repeated numerous times, thus securing many averages, how much scatter would be expected in those averages? (p. 21).
I love how he put that. And then the miracle is that we can use one sample to inform us about the scatter in the many averages thanks to the central limit theorem. using \( s_\bar{x} = \sigma / \sqrt{n} \) and \( t \s_\bar{x} \), this last quantity being the “precision of the average.”
There is a side note about pooling different estimates of \( s \) if they pass Bartlett’s homogeneity test.
There is another side note about using two identical instruments to take measurements at the same time and using the differences between the readings to learn about \( s \) for one instrument. I didn’t understand this because if you do multiple readouts of the pair of instruments you can get \( s \) for each instrument. This is what the notation suggested but I could imagine one readout on each instrument at one point in time would be the use case.