1. MARLAP Chapter 19 Measurement Uncertainty
Keith McCroan
Bioassay, Analytical & Environmental Radiochemistry Conference 2004

2. Outline What you should know
The Guide to the Expression of Uncertainty in Measurement (the “GUM”)
Uncertainty in the radiochemistry lab
Summary of recommendations

3. What You Should Know
Chapter 19 of MARLAP is the measurement uncertainty chapter
It’s big and it has lots of equations
What do you really need to know about it?

4. What You Should Know It has more than one target audience
The first 3 sections present concepts and terms, with no math
They are intended for readers who want to know what uncertainty means or who want to learn the terminology and notation
The remaining sections contain the mathematical details for lab personnel who need to evaluate and report measurement uncertainty

5. What You Should Know
At the end of the 3rd section, we summarize our major recommendations
If you don’t like math, you can stop reading after the recommendations
But the fun begins in Section 4

6. Top Recommendations
Use the terminology, notation, and methodology of the GUM
Report all results – even if zero or negative – unless it is believed for some reason that they are invalid
Report the uncertainty of every result and explain what it is (e.g., 1σ, 2σ ?)
Consider all sources of uncertainty and evaluate and propagate all that are believed to be potentially significant in the final result

7. All the Rest The chapter summarizes the GUM
General information in Section 3
Mathematical details in Section 4
Section 5 discusses the evaluation of uncertainty for radiochemical measurements

8. Questions So Far?

9. Part 1
The GUM

10. What is the GUM?
It is a guide, published by ISO and available from ANSI
It presents terminology, notation, and methodology for evaluating and expressing measurement uncertainty
It tries to get everyone speaking and writing the same language about uncertainty

11. The GUM
Published in 1993 by ISO in the name of 7 international organizations
Revised and corrected in 1995
Accepted by NIST and other national standards bodies
Endorsed by MARLAP
Gradually being adopted by ANSI & ASTM

12. Don’t Fight It
The GUM approach is no harder than what you’ve done before
More than anything else, you need to learn its terms and symbols
You’re going to see it more and more (e.g., in ASTM documents)
Resistance is futile

13. The GUM Approach
What follows is an oversimplified summary of the terminology, notation, and methodology of the GUM

14. The Measurand
Metrologists define the measurand for any measurement to be the “particular quantity subject to measurement”
For example, if you’re measuring the specific activity of 137Cs in a sample of soil, the measurand is the specific activity of 137Cs in that sample

15. Uncertainty of Measurement
The GUM defines uncertainty of measurement as a
parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand
An uncertainty could be (for example) a standard deviation, a multiple of a standard deviation, or the half-width of an interval having a stated level of confidence

16. Error of Measurement
Statisticians and metrologists disagree about the meaning of the word “error”
Statisticians use error to mean uncertainty, as in the “standard error” of an estimator
To a metrologist, the error of a measurement is the difference between the result and the true value
Metrological error is a theoretical concept – You can never know what its value is

17. Mathematical Model of Measurement
Before one ever makes a measurement, one makes a mathematical model of the measurement
Typically the value of the measurand is not measured directly but is calculated from other quantities (input quantities) that are measured
The model is an equation or set of equations that determine how the value of the measurand, Y, is to be calculated from the values of the input quantities X1,X2,…,XN

18. Mathematical Model of Measurement
When we talk about the model, we may also refer to the measurand Y as the output quantity
Although the model may consist of one or more equations, we’ll denote it here abstractly as a single equation
Y = f(X1,X2,…,XN)

19. Making a Measurement
To make a measurement, one determines values for the input quantities and plugs them into the model to calculate a value for the output quantity, Y
The values determined for the input quantities in a particular instance of the measurement are called input estimates and may be denoted by x1,x2,…,xN
The value calculated for the output quantity is called the output estimate and may be denoted by y

20. Uncertainty Propagation
Each input estimate has an uncertainty, and the uncertainties of the input estimates combine to produce an uncertainty in the output estimate
The operation of mathematically combining the uncertainties of the input estimates to obtain the uncertainty of the output estimate is called propagation of uncertainty

21. Steps in Uncertainty Propagation
Determine values for the input quantities (the input estimates) and calculate the value of the output quantity (the output estimate)
y = f(x1,x2,…,xN)
Evaluate the uncertainty of each input estimate and the covariance of each pair of correlated input estimates
Propagate the uncertainties and covariances of the input estimates to calculate the uncertainty of the output estimate

22. Standard Uncertainty
Before uncertainties can be propagated, they must be expressed in comparable forms
The standard uncertainty of any measured value is the uncertainty expressed as an estimated standard deviation – i.e., the “one-sigma” uncertainty
The standard uncertainty of an input estimate, xi, is denoted by u(xi)
We express all the uncertainties as standard uncertainties when we propagate them

23. Evaluating Uncertainties
There are many ways to evaluate the standard uncertainty of an input estimate, xi
For example, one might average the results of several observations and calculate the standard error of the mean, or take the square root of the number of counts observed in a single radiation counting measurement
Or one might make a wild (educated) guess about the maximum possible error in a value and divide it by sqrt(3) or sqrt(6)

24. Type A and Type B Evaluations
The GUM groups all evaluations of uncertainty into two categories: Type A and Type B
A Type A evaluation of uncertainty is a statistical evaluation based on series of observations
A Type B evaluation of uncertainty is anything else

25. Type A and Type B
An uncertainty evaluated by a Type A method used to be called a “random uncertainty” (but not anymore!)
An uncertainty evaluated by a Type B method used to be called a “systematic uncertainty” (but not anymore!)

26. Type A Evaluation
Statistical evaluation of uncertainty involving series of observations
Example:
Make a series of observations of a quantity, then calculate the mean and the “standard error of the mean,” or what metrologists call the “experimental standard deviation of the mean”

27. Type B Evaluation
Any evaluation that is not a Type A evaluation is a Type B evaluation
Examples:
Calculate Poisson counting uncertainty as the square root of the observed count
Use professional judgment to estimate the maximum possible error in the value, then divide by sqrt(3) or some other constant

28. Covariance
Correlations between input estimates affect the uncertainty of the output estimate
The estimated covariance of two input estimates, xi and xj, is denoted by u(xi,xj)
The estimated correlation coefficient is denoted by r(xi,xj)
See the MARLAP text, the GUM, or Ken Inn for more information

29. Uncertainty Propagation
Recall that the output estimate, y, is given by
y = f(x1,x2,…,xN)
The following equation shows how the standard uncertainties and covariances of input estimates are propagated to produce the standard uncertainty of the output estimate

30. Combined Standard Uncertainty
The standard uncertainty of y obtained by uncertainty propagation is called the combined standard uncertainty
Notice that it is denoted here by uc(y), not u(y)
The subscript “c” means “combined”

31. Sensitivity Coefficients
Each partial derivative is called a sensitivity coefficient
It equals the partial derivative of the function f(X1,X2,…,XN) with respect to Xi , evaluated at X1=x1, X2=x2, …, XN=xn
It represents the sensitivity of y to changes in xi, or the ratio of the change in y to a small change in xi

32. Uncertainty Propagation
All the standard uncertainties of the input estimates are treated alike for purposes of uncertainty propagation
We do not distinguish between Type A uncertainties and Type B uncertainties when we propagate them

33. The “Law of Propagation of Uncertainty”?
The GUM calls the generalized equation for the combined standard uncertainty the “law of propagation of uncertainty”
MARLAP prefers the less grandiose name “uncertainty propagation formula”
It’s not a “law” – just a first-order approximation formula

34. Uncertainty Propagation Formula
The uncertainty propagation formula looks intimidating to most people
If you learn examples of particular applications of it, you may be able to use them in many or most situations
If you want to be able to handle any model thrown at you, either you need to know calculus or you need software for automatic uncertainty propagation

35. Examples
If x1 and x2 are uncorrelated, then

36. Expanded Uncertainty
One may choose to multiply the combined standard uncertainty, uc(y), by a number k, called the coverage factor to obtain the expanded uncertainty, U
The expanded uncertainty is intended to produce an interval about the result that has a high probability of containing the (true) value of the measurand
That probability, p, is called either the coverage probability or the level of confidence

37. Expanded Uncertainty
Traditionally we have called expanded uncertainties “two-sigma” or “three-sigma” uncertainties
For any number k > 1, what we have called a “k-sigma” uncertainty is an expanded uncertainty with coverage factor k

38. Expanded Uncertainty
Reporting an expanded uncertainty, especially with k=2, usually suggests that you believe the result has a distribution that is approximately normal
When k=2, you are implying that the coverage probability is about 95 %
What are you implying if you use k=1.96?
But reporting only the combined standard uncertainty (an estimated standard deviation) does not imply any particular distribution or coverage probability

39. Terms to Remember Measurand, Y Mathematical model of measurement
Y = f(X1,X2,…,XN)
Input quantities Xi, output quantity Y
Input estimates xi, output estimate y
Standard uncertainty, u(xi)
Estimated covariance, u(xi,xj)
Estimated correlation coefficient, r(xi,xj)

40. Terms - Continued Propagation of uncertainty
Combined standard uncertainty, uc(y)
Coverage probability, or level of confidence, p
Coverage factor, k
Expanded uncertainty, U = k × uc(y)

41. Uncertainty in the Radiochemistry Lab
Part 2
Uncertainty in the Radiochemistry Lab

42. Counting Error
Well, first of all, MARLAP calls it “counting uncertainty,” not “counting error”
We define it as the component of the combined standard uncertainty of the result due to the randomness of radioactive decay [and radiation emission] and radiation counting
It’s only a portion of the total uncertainty of a measurement

43. Counting Uncertainty
We admit that one can often evaluate the standard uncertainty of a total count, n, by taking the square root of n
It is a convenient Type B method of evaluation, which doesn’t require repeated measurements
It is based on the assumption that n has a Poisson distribution, which may not always be a good assumption
Again, counting uncertainty is only a portion of the total uncertainty of the final result

44. Non-Poisson Example
One of the best examples of non-Poisson counting statistics comes from alpha-counting 222Rn and its progeny in a Lucas cell
An atom of 222Rn may produce more than one count as it decays through a series of short-lived states from 222Rn to 210Pb
Counts tend to occur in groups
The counting uncertainty of n is usually larger than sqrt(n)

45. When n is Small
If the Poisson model is valid, and if n, the number of counts, can assume values close to or equal to zero, we recommend evaluating the counting uncertainty as sqrt(n+1), not sqrt(n)
Otherwise you may end up reporting results sometimes as 0 ± 0

46. Other Uncertainties
MARLAP provides guidance about other uncertainty components
The guidance is intended to be helpful, not prescriptive, and certainly not complete
We deal with uncertainties for volume and mass measurements, which are relatively easy to handle but which also tend to be relatively insignificant

47. Laboratory Subsampling
We also deal with an uncertainty that is neither insignificant nor easy to handle: the uncertainty associated with subsampling heterogeneous solid material for analysis
Appendix F presents some highlights of Pierre Gy’s sampling theory as it applies to subsampling for radiochemical analysis
We recommend that labs not ignore subsampling uncertainty, although it is hard to evaluate well

48. Subsampling - Continued
We provide a reasonably simple equation for evaluating the standard uncertainty due to subsampling, which you can use by default if you don’t have a better approach of your own
The equation (next slide) depends on the mass of the sample, the mass of the subsample, and the maximum particle diameter

49. The Equation mL = mass of entire sample mS = mass of subsample
d = maximum particle diameter
k = 0.4 g/cm3 by default
u(FS) = relative standard uncertainty due to subsampling

50. Why This Equation?
The form of the equation is derived from Gy’s theory
The default value of k is somewhat arbitrary but should give OK results
The equation rightly punishes one for taking too small an aliquant for analysis or failing to grind a lumpy sample before subsampling

51. Other Uncertainties Real time and live time Instrument background
Radiochemical blank
Calibration (detection efficiency)
Half-life – easy but usually negligible
Gamma-ray spectrometry (MARLAP chooses to punt this one)

52. Summary of Recommendations
Part 3
Summary of Recommendations

53. Recommendations
Use the terminology, notation, and methodology of the GUM
Report all results – even if zero or negative – unless you believe they are invalid
Report either the combined standard uncertainty or the expanded uncertainty
Explain the uncertainty – in particular state the coverage factor for an expanded uncertainty

54. Recommendations - Continued
Consider all sources of uncertainty and evaluate and propagate all that are believed to be potentially significant in the final result
Do not ignore subsampling uncertainty just because it is hard to evaluate
Round the reported uncertainty to either 1 or 2 figures (we suggest 2) and round the result to match

55. Final Recommendation
Consider all the preceding recommendations to be severable
If you can’t do everything, do as much as you can
But at least use the GUM’s terminology and notation so that we all speak and write the same language

56. Questions?