uncertainty calculations

1. Measurement
Uncertainty

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
 We use 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.
 We try 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”
 Avoid 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
 We state 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
 We cite 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
 Our 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