# Presentation: 433639

## MARLAPMeasurement Uncertainty

- Keith McCroan
- U.S. Environmental Protection Agency
- National Air and Radiation 	Environmental Laboratory

## Topics Covered

- Brief overview of concepts and terms of probability and statistics
- Measurement uncertainty
- Detection and quantification limits
- Miscellaneous

## Target Audience(s)

- Project planners and managers
- Radiochemists and technicians
- Computer programmers
- Data validators and assessors
- Metrologists?

## 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

## Measurement Model

- Define the measurand – the quantity subject to measurement
- Determine a mathematical model, with input quantities, *X*1,*X*2,...,*X**N*, and (at least) one output quantity,*Y.*
- The values determined for the input quantities are called input estimates and are denoted by *x*1,*x*2,...,*x**N*.
- 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, *x**i*, is denoted by *u*(*x**i*).
- The standard uncertainty of the output estimate, *y*, determined by uncertainty propagation, is called the combined standard uncertainty, and is denoted by* u**c**(y).*

## 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”

## 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

## Covariance

- Correlations among input estimates affect the combined standard uncertainty of the output estimate.
- The estimated covariance of two input estimates, *x**i* and *x**j*, is denoted by *u*(*x**i*,*x**j*).

## 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, *u**c*(*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.

## 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.

## 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 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.

## 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.

## “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 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.

## 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).

## 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 - 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.

## 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.

## Other Topics

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

## 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)

## 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