1. MARLAP Measurement Uncertainty
Keith McCroan
U.S. Environmental Protection Agency
National Air and Radiation Environmental Laboratory

2. Topics Covered
Brief overview of concepts and terms of probability and statistics
Measurement uncertainty
Detection and quantification limits
Miscellaneous

3. Target Audience(s) Project planners and managers
Radiochemists and technicians
Computer programmers
Data validators and assessors
Metrologists?

4. Measurement Uncertainty
MARLAP endorses the Guide to the Expression of Uncertainty in Measurement (ISO-GUM).
International guidance – years of development and review by seven international organizations
Strongly recommended by NIST
Best way to ensure consistency among labs in the U.S. and the rest of the world

5. Measurement Model
Define the measurand – the quantity subject to measurement
Determine a mathematical model, with input quantities, X1,X2,…,XN, and (at least) one output quantity,Y.
The values determined for the input quantities are called input estimates and are denoted by x1,x2,…,xN.
The value calculated for the output quantity is called the output estimate and denoted by y.

6. Standard Uncertainty
The standard uncertainty of a 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).
The standard uncertainty of the output estimate, y, determined by uncertainty propagation, is called the combined standard uncertainty, and is denoted by uc(y).

7. Type A Evaluation
Statistical evaluation of uncertainty involving a series of observations
Always has an associated number of degrees of freedom
Examples include simple averages and least-squares estimates
Not “random uncertainty”

8. Type B Evaluation
Any evaluation that is not a Type A evaluation is a Type B evaluation.
Not “systematic uncertainty”
Examples:
Calculating Poisson counting uncertainty (error) as the square root of the observed count
Using professional judgment combined with assumed rectangular or triangular distributions
Obtaining standard uncertainties in any manner from standard certificates or reference books

9. Covariance
Correlations among input estimates affect the combined standard uncertainty of the output estimate.
The estimated covariance of two input estimates, xi and xj, is denoted by u(xi,xj).

10. Uncertainty Propagation
“Law of Propagation of Uncertainty,” or, more simply, the “uncertainty propagation formula”
Standard uncertainties and covariances of input estimates are combined mathematically to produce the combined standard uncertainty of the output quantity.

11. Expanded Uncertainty
Multiply the combined standard uncertainty, uc(y), by a number k, called the coverage factor to obtain the expanded uncertainty, U.
The probability (or one’s degree of belief) that the interval y +- U will contain the value of the measurand is called either the coverage probability or the level of confidence.

12. Recommendations Follow ISO-GUM in terminology and methods.
Consider all sources of uncertainty and evaluate and propagate all that are considered to be potentially significant in the final result.
Do not ignore subsampling uncertainty just because it may be hard to evaluate.

13. Recommendations - Continued
Report all results – even if zero or negative
Report either the combined standard uncertainty or the expanded uncertainty.
Explain the uncertainty – in particular state the coverage factor for an expanded uncertainty.
Round the reported uncertainty to either 1 or 2 figures (suggest 2) and round the result to match.

14. Detection and Quantification
There are several standards on the subject of detection limits.
MARLAP tries to follow the principles that are common to all.
We follow IUPAC (more or less) for quantification limits.

15. Detection
A detection decision is based on the critical value (critical level, decision level) of the response variable (e.g., instrument signal, either gross or net).
The minimum detectable concentration (MDC) is the smallest (true) analyte concentration that ensures a specified high probability of detection.

16. “A Priori” vs. “A Posteriori”
MARLAP avoids the “a priori” vs. “a posteriori” distinction.
We recognize:
Many labs report a sample-specific estimate of the MDC
Many experts insist it should not be done
We take no firm position except to state that the sample-specific MDC has few valid uses and is often misused.

17. Misuse of the MDC
MARLAP states that no version of the MDC should be used in deciding whether an analyte is present in a laboratory sample.
The MDC cannot be determined unless the detection criterion has already been specified.

18. Quantification Limits
MARLAP cites IUPAC’s guidance for defining quantification limits.
The minimum quantifiable concentration (MQC) is the analyte concentration that gives a relative standard deviation of 1/k, for some specified number k (usually 10).

19. The MQC
We hoped to unify the approaches to uncertainty and to detection and quantification limits.
ISO-GUM in effect treats all error components as random variables.
Is this approach consistent with IUPAC’s approach to quantification limits? We proceeded as if the answer were yes.

20. The MQC - Continued
MARLAP’s MQC is based on an overall standard deviation that represents all sources of measurement error – not just “random errors.”
This standard deviation differs from the combined standard uncertainty, a random variable whose value changes with each measurement.

21. Use of the MQC
The MQC is almost unknown among radiochemists but should be a useful performance characteristic.
The MDC is well-known and is sometimes used for purposes that would be better served by the MQC.
E.g., choosing a procedure to measure Ra-226 in soil.

22. Other Topics Effects of nonlinearity on uncertainty propagation
Laboratory subsampling – based on Pierre Gy’s sampling theory
Tests for normality
Example calculations

23. Other Topics - Continued
Detection decisions based on low-background Poisson counting or few degrees of freedom
Expressions for the critical net count in the pure Poisson case
Well-known (so-called “Currie’s equation”)
Not so well-known (Nicholson, Stapleton)

24. Concerns & Questions Overkill? Is anything important missing?
E.g., a table of “typical” uncertainties
More real-world examples of good uncertainty evaluation
How can the examples be improved?
Contradictory standards on detection limits