MARLAP Chapter 19 Measurement Uncertainty

1. MARLAP Chapter 19Measurement 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. 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