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)

Concerns & QuestionsOverkill? Is anything important missing?

E.g., a table of “typical” uncertainties More real-world examples of good

uncertainty evaluationHow can the examples be improved?Contradictory standards on detection

limits