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					   MARLAP Chapter 19
Measurement Uncertainty
                      Keith McCroan

 Bioassay, Analytical & Environmental
    Radiochemistry Conference 2004
Outline

   What you should know
   The Guide to the Expression of
    Uncertainty in Measurement (the “GUM”)
   Uncertainty in the radiochemistry lab
   Summary of recommendations
What You Should Know

   Chapter 19 of MARLAP is the
    measurement uncertainty chapter

   It’s big and it has lots of equations

   What do you really need to know about it?
What You Should Know

   It has more than one target audience
       The first 3 sections present concepts and
        terms, with no math
       They are intended for readers who want to
        know what uncertainty means or who want to
        learn the terminology and notation
       The remaining sections contain the
        mathematical details for lab personnel who
        need to evaluate and report measurement
        uncertainty
What You Should Know

   At the end of the 3rd section, we
    summarize our major recommendations

   If you don’t like math, you can stop
    reading after the recommendations

   But the fun begins in Section 4
Top Recommendations
   Use the terminology, notation, and methodology
    of the GUM
   Report all results – even if zero or negative –
    unless it is believed for some reason that they
    are invalid
   Report the uncertainty of every result and
    explain what it is (e.g., 1σ, 2σ ?)
   Consider all sources of uncertainty and evaluate
    and propagate all that are believed to be
    potentially significant in the final result
All the Rest

   The chapter summarizes the GUM
       General information in Section 3
       Mathematical details in Section 4


   Section 5 discusses the evaluation of
    uncertainty for radiochemical
    measurements
Questions So Far?
Part 1
The GUM
What is the GUM?

   It is a guide, published by ISO and
    available from ANSI
   It presents terminology, notation, and
    methodology for evaluating and
    expressing measurement uncertainty
   It tries to get everyone speaking and
    writing the same language about
    uncertainty
The GUM

   Published in 1993 by ISO in the name of 7
    international organizations
   Revised and corrected in 1995
   Accepted by NIST and other national
    standards bodies
   Endorsed by MARLAP
   Gradually being adopted by ANSI & ASTM
Don’t Fight It

   The GUM approach is no harder than
    what you’ve done before
   More than anything else, you need to
    learn its terms and symbols
   You’re going to see it more and more
    (e.g., in ASTM documents)
   Resistance is futile
The GUM Approach

   What follows is an oversimplified summary
    of the terminology, notation, and
    methodology of the GUM
The Measurand

   Metrologists define the measurand for any
    measurement to be the “particular quantity
    subject to measurement”

   For example, if you’re measuring the
    specific activity of 137Cs in a sample of
    soil, the measurand is the specific activity
    of 137Cs in that sample
Uncertainty of Measurement
   The GUM defines uncertainty of
    measurement as a
       parameter, associated with the result of a
        measurement, that characterizes the
        dispersion of the values that could reasonably
        be attributed to the measurand
   An uncertainty could be (for example) a
    standard deviation, a multiple of a
    standard deviation, or the half-width of an
    interval having a stated level of confidence
Error of Measurement

   Statisticians and metrologists disagree about the
    meaning of the word “error”
   Statisticians use error to mean uncertainty, as in
    the “standard error” of an estimator
   To a metrologist, the error of a measurement is
    the difference between the result and the true
    value
   Metrological error is a theoretical concept – You
    can never know what its value is
Mathematical Model of Measurement
   Before one ever makes a measurement, one
    makes a mathematical model of the
    measurement
   Typically the value of the measurand is not
    measured directly but is calculated from other
    quantities (input quantities) that are measured
   The model is an equation or set of equations
    that determine how the value of the measurand,
    Y, is to be calculated from the values of the input
    quantities X1,X2,…,XN
Mathematical Model of Measurement
   When we talk about the model, we may
    also refer to the measurand Y as the
    output quantity
   Although the model may consist of one or
    more equations, we’ll denote it here
    abstractly as a single equation

               Y = f(X1,X2,…,XN)
Making a Measurement
   To make a measurement, one determines
    values for the input quantities and plugs them
    into the model to calculate a value for the output
    quantity, Y
   The values determined for the input quantities in
    a particular instance of the measurement are
    called input estimates and may be denoted by
    x1,x2,…,xN
   The value calculated for the output quantity is
    called the output estimate and may be denoted
    by y
Uncertainty Propagation

   Each input estimate has an uncertainty,
    and the uncertainties of the input
    estimates combine to produce an
    uncertainty in the output estimate
   The operation of mathematically
    combining the uncertainties of the input
    estimates to obtain the uncertainty of the
    output estimate is called propagation of
    uncertainty
    Steps in Uncertainty Propagation

   Determine values for the input quantities (the
    input estimates) and calculate the value of the
    output quantity (the output estimate)
                    y = f(x1,x2,…,xN)
   Evaluate the uncertainty of each input estimate
    and the covariance of each pair of correlated
    input estimates
   Propagate the uncertainties and covariances of
    the input estimates to calculate the uncertainty
    of the output estimate
Standard Uncertainty
   Before uncertainties can be propagated, they
    must be expressed in comparable forms
   The standard uncertainty of any 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)
   We express all the uncertainties as standard
    uncertainties when we propagate them
    Evaluating Uncertainties

   There are many ways to evaluate the standard
    uncertainty of an input estimate, xi
   For example, one might average the results of
    several observations and calculate the standard
    error of the mean, or take the square root of the
    number of counts observed in a single radiation
    counting measurement
   Or one might make a wild (educated) guess
    about the maximum possible error in a value
    and divide it by sqrt(3) or sqrt(6)
Type A and Type B Evaluations
   The GUM groups all evaluations of
    uncertainty into two categories: Type A
    and Type B
       A Type A evaluation of uncertainty is a
        statistical evaluation based on series of
        observations
       A Type B evaluation of uncertainty is anything
        else
Type A and Type B

   An uncertainty evaluated by a Type A
    method used to be called a “random
    uncertainty” (but not anymore!)

   An uncertainty evaluated by a Type B
    method used to be called a “systematic
    uncertainty” (but not anymore!)
Type A Evaluation

   Statistical evaluation of uncertainty
    involving series of observations

   Example:
       Make a series of observations of a quantity,
        then calculate the mean and the “standard
        error of the mean,” or what metrologists call
        the “experimental standard deviation of the
        mean”
Type B Evaluation

   Any evaluation that is not a Type A
    evaluation is a Type B evaluation

   Examples:
       Calculate Poisson counting uncertainty as the
        square root of the observed count
       Use professional judgment to estimate the
        maximum possible error in the value, then
        divide by sqrt(3) or some other constant
Covariance

   Correlations between input estimates affect the
    uncertainty of the output estimate
   The estimated covariance of two input
    estimates, xi and xj, is denoted by u(xi,xj)
   The estimated correlation coefficient is denoted
    by r(xi,xj)
   See the MARLAP text, the GUM, or Ken Inn for
    more information
Uncertainty Propagation

   Recall that the output estimate, y, is given by
                    y = f(x1,x2,…,xN)
   The following equation shows how the standard
    uncertainties and covariances of input estimates
    are propagated to produce the standard
    uncertainty of the output estimate
                                2
                  N
                       f      2           N 1 N
                                                          f f
    uc ( y )      x
                      
                 i 1 
                                u ( xi )  2 
                               
                                             i 1 j  i 1x i x j
                                                                    u( x i , x j )
                           i   
Combined Standard Uncertainty
   The standard uncertainty of y obtained by
    uncertainty propagation is called the
    combined standard uncertainty

   Notice that it is denoted here by uc(y), not
    u(y)

   The subscript “c” means “combined”
Sensitivity Coefficients

   Each partial derivative f / xi is called a
    sensitivity coefficient
   It equals the partial derivative of the function
    f(X1,X2,…,XN) with respect to Xi , evaluated at
    X1=x1, X2=x2, …, XN=xn
   It represents the sensitivity of y to changes in xi,
    or the ratio of the change in y to a small change
    in xi
Uncertainty Propagation

   All the standard uncertainties of the input
    estimates are treated alike for purposes of
    uncertainty propagation

   We do not distinguish between Type A
    uncertainties and Type B uncertainties
    when we propagate them
The “Law of Propagation of
Uncertainty”?
   The GUM calls the generalized equation
    for the combined standard uncertainty the
    “law of propagation of uncertainty”
   MARLAP prefers the less grandiose name
    “uncertainty propagation formula”
   It’s not a “law” – just a first-order
    approximation formula
    Uncertainty Propagation Formula

   The uncertainty propagation formula looks
    intimidating to most people
   If you learn examples of particular applications
    of it, you may be able to use them in many or
    most situations
   If you want to be able to handle any model
    thrown at you, either you need to know calculus
    or you need software for automatic uncertainty
    propagation
Examples

   If x1 and x2 are uncorrelated, then

       uc ( x1  x 2 )  u 2 ( x1 )  u 2 ( x 2 )
       uc x1x 2   u 2 ( x1 )x 2  x1 u 2 ( x 2 )
                                 2    2


                                          2
           x1        u ( x1 )  x1  u 2 ( x 2 )
                         2
       uc   
          x              2
                                
                                x          2
           2           x2      2      x2
Expanded Uncertainty

   One may choose to multiply the combined
    standard uncertainty, uc(y), by a number k,
    called the coverage factor to obtain the
    expanded uncertainty, U
   The expanded uncertainty is intended to
    produce an interval about the result that has a
    high probability of containing the (true) value of
    the measurand
   That probability, p, is called either the coverage
    probability or the level of confidence
Expanded Uncertainty

   Traditionally we have called expanded
    uncertainties “two-sigma” or “three-sigma”
    uncertainties

   For any number k > 1, what we have
    called a “k-sigma” uncertainty is an
    expanded uncertainty with coverage
    factor k
Expanded Uncertainty

   Reporting an expanded uncertainty, especially
    with k=2, usually suggests that you believe the
    result has a distribution that is approximately
    normal
       When k=2, you are implying that the coverage
        probability is about 95 %
       What are you implying if you use k=1.96?
   But reporting only the combined standard
    uncertainty (an estimated standard deviation)
    does not imply any particular distribution or
    coverage probability
Terms to Remember

   Measurand, Y
   Mathematical model of measurement
                Y = f(X1,X2,…,XN)
   Input quantities Xi, output quantity Y
   Input estimates xi, output estimate y
   Standard uncertainty, u(xi)
   Estimated covariance, u(xi,xj)
   Estimated correlation coefficient, r(xi,xj)
Terms - Continued

   Propagation of uncertainty
   Combined standard uncertainty, uc(y)
   Coverage probability, or level of
    confidence, p
   Coverage factor, k
   Expanded uncertainty, U = k × uc(y)
           Part 2
  Uncertainty in the
Radiochemistry Lab
Counting Error

   Well, first of all, MARLAP calls it “counting
    uncertainty,” not “counting error”
   We define it as the component of the
    combined standard uncertainty of the
    result due to the randomness of
    radioactive decay [and radiation emission]
    and radiation counting
   It’s only a portion of the total uncertainty of
    a measurement
Counting Uncertainty

   We admit that one can often evaluate the
    standard uncertainty of a total count, n, by taking
    the square root of n
   It is a convenient Type B method of evaluation,
    which doesn’t require repeated measurements
   It is based on the assumption that n has a
    Poisson distribution, which may not always be a
    good assumption
   Again, counting uncertainty is only a portion of
    the total uncertainty of the final result
Non-Poisson Example

   One of the best examples of non-Poisson
    counting statistics comes from alpha-counting
    222Rn and its progeny in a Lucas cell

   An atom of 222Rn may produce more than one
    count as it decays through a series of short-lived
    states from 222Rn to 210Pb
   Counts tend to occur in groups
   The counting uncertainty of n is usually larger
    than sqrt(n)
When n is Small

   If the Poisson model is valid, and if n, the
    number of counts, can assume values
    close to or equal to zero, we recommend
    evaluating the counting uncertainty as
    sqrt(n+1), not sqrt(n)

   Otherwise you may end up reporting
    results sometimes as 0 ± 0
Other Uncertainties

   MARLAP provides guidance about other
    uncertainty components
   The guidance is intended to be helpful, not
    prescriptive, and certainly not complete
   We deal with uncertainties for volume and
    mass measurements, which are relatively
    easy to handle but which also tend to be
    relatively insignificant
Laboratory Subsampling

   We also deal with an uncertainty that is neither
    insignificant nor easy to handle: the uncertainty
    associated with subsampling heterogeneous
    solid material for analysis
   Appendix F presents some highlights of Pierre
    Gy’s sampling theory as it applies to
    subsampling for radiochemical analysis
   We recommend that labs not ignore
    subsampling uncertainty, although it is hard to
    evaluate well
Subsampling - Continued

   We provide a reasonably simple equation
    for evaluating the standard uncertainty
    due to subsampling, which you can use by
    default if you don’t have a better approach
    of your own
   The equation (next slide) depends on the
    mass of the sample, the mass of the
    subsample, and the maximum particle
    diameter
The Equation

   mL = mass of entire sample
   mS = mass of subsample
   d = maximum particle diameter
   k = 0.4 g/cm3 by default
   u(FS) = relative standard uncertainty due to
    subsampling
                      1   1 
            u(FS )  
                     m       k d3
                      S   mL 
                              
Why This Equation?

   The form of the equation is derived from
    Gy’s theory
   The default value of k is somewhat
    arbitrary but should give OK results
   The equation rightly punishes one for
    taking too small an aliquant for analysis or
    failing to grind a lumpy sample before
    subsampling
Other Uncertainties

   Real time and live time
   Instrument background
   Radiochemical blank
   Calibration (detection efficiency)
   Half-life – easy but usually negligible
   Gamma-ray spectrometry (MARLAP
    chooses to punt this one)
                   Part 3
Summary of Recommendations
Recommendations

   Use the terminology, notation, and methodology
    of the GUM
   Report all results – even if zero or negative –
    unless you believe they are invalid
   Report either the combined standard uncertainty
    or the expanded uncertainty
   Explain the uncertainty – in particular state the
    coverage factor for an expanded uncertainty
Recommendations
- Continued
   Consider all sources of uncertainty and
    evaluate and propagate all that are
    believed to be potentially significant in the
    final result
   Do not ignore subsampling uncertainty just
    because it is hard to evaluate
   Round the reported uncertainty to either 1
    or 2 figures (we suggest 2) and round the
    result to match
Final Recommendation

   Consider all the preceding
    recommendations to be severable
   If you can’t do everything, do as much as
    you can
   But at least use the GUM’s terminology
    and notation so that we all speak and write
    the same language
Questions?

				
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