Measurement Uncertainty (PowerPoint)

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```					   MARLAP Chapter 19
Measurement Uncertainty
Keith McCroan

Bioassay, Analytical & Environmental
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
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
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
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
   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
   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
   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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