Upload
gloria-hopkins
View
216
Download
2
Embed Size (px)
Citation preview
MARLAPMeasurement Uncertainty
Keith McCroan
U.S. Environmental Protection Agency
National Air and Radiation Environmental Laboratory
Topics CoveredBrief overview of concepts and terms of
probability and statisticsMeasurement uncertaintyDetection and quantification limitsMiscellaneous
Target Audience(s)Project planners and managersRadiochemists and techniciansComputer programmersData validators and assessorsMetrologists?
Measurement UncertaintyMARLAP 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
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.
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).
Type A EvaluationStatistical evaluation of uncertainty
involving a series of observationsAlways has an associated number of
degrees of freedomExamples include simple averages and
least-squares estimatesNot “random uncertainty”
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
CovarianceCorrelations 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).
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.
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.
RecommendationsFollow 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.
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.
Detection and QuantificationThere 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.
DetectionA 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.
“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.
Misuse of the MDCMARLAP 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.
Quantification LimitsMARLAP 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).
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.
The MQC - ContinuedMARLAP’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.
Use of the MQCThe 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.
Other TopicsEffects of nonlinearity on uncertainty
propagationLaboratory subsampling – based on
Pierre Gy’s sampling theoryTests for normalityExample calculations
Other Topics - ContinuedDetection 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)