Guide to the Expression of Uncertainty in Measurement

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Guide to the Expression of Uncertainty in
Measurement

• Keith D. McCroan
• MARLAP Uncertainty Workshop
• October 24, 2005
• Stateline, Nevada

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Introduction

• Guide to the Expression of Uncertainty in
  Measurement was published by the International
  Organization for Standardization in 1993 in the
  name of 7 international organizations
• Corrected and reprinted in 1995
• Usually referred to simply as the GUM

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The Seven Sponsors

• International Bureau of Weights and Measures
  (BIPM)
• International Electrotechnical Commission (IEC)
• International Federation of Clinical Chemistry
  (IFCC)
• International Organization for Standardization
  (ISO)
• International Union of Pure and Applied Chemistry
  (IUPAC)
• International Union of Pure and Applied Physics
  (IUPAP)
• International Organization of Legal Metrology
  (OIML)

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Stated Purposes

• Promote full information on how uncertainty
  statements are arrived at
• Provide a basis for the international comparison
  of measurement results

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Benefits

• Much flexibility in the guidance
• Provides a conceptual framework for evaluating
  and expressing uncertainty
• Promotes the use of standard terminology and
  notation
• All of us can speak and write the same language
  when we discuss uncertainty

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What Is Measurement Uncertainty?

• parameter, associated with the result of a
  measurement, that characterizes the dispersion of
  the values that could reasonably be attributed to
  the measurand GUM, VIM
• Examples
• A standard deviation (1 sigma) or a multiple of
  it (e.g., 2 or 3 sigma)
• The half-width of an interval having a stated
  level of confidence

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The Measurand

• In any measurement, the measurand is defined as
  the particular quantity subject to measurement
• For example, if youre trying to determine the
  massic activity of 239Pu in a specified sample of
  soil as of a specified date and time, that is the
  measurand

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Error of Measurement

• In metrology the error of a measurement is the
  difference between the result and the actual
  value of the measurand
• The error is treated as a random variable
• With mean and standard deviation
• True even for systematic error (discussed later)

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Error vs. Uncertainty

• In metrology, error is primarily a theoretical
  concept, because its value is unknowable
• Uncertainty is a more practical concept
• Evaluating uncertainty allows you to place a
  bound on the likely size of the error
• It is a critical aspect of metrology
• A measured value without some indication of its
  uncertainty is useless

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Random Systematic Errors

• Error can be decomposed into random and
  systematic parts
• The random error varies when a measurement is
  repeated under the same conditions (e.g.,
  radiation counting)
• The systematic error remains fixed when the
  measurement is repeated under the same conditions
  (e.g, error in a ?-ray emission probability)

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Correcting for Systematic Error

• If you know that a substantial systematic error
  exists and you can estimate its value, include a
  correction (additive) or correction factor
  (multiplicative) in the model to account for it
• Remember the correction term or factor itself
  has uncertainty
• A small residual systematic error generally
  remains after all known corrections have been
  applied

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Uncertainty

• Uncertainty of measurement accounts for random
  error and systematic error
• Does not account for blunders or other spurious
  errors, such as those caused by equipment failure
• Spurious errors represent loss of statistical
  control of the measurement process

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The Measurement Model

• Usually the final result of a measurement is not
  measured directly, but is calculated from other
  measured quantities through a functional
  relationship
• Well call this function a measurement model
• The model might involve several equations, but
  well follow the GUM and represent it abstractly
  as a single equation

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Example Radiochemistry

• In radiochemistry, a simple model might look like

where a denotes massic activity (the measurand),
Cs the sample count, ts the sample count time, e
the detection efficiency, etc.
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Input and Output Quantities

• In the generic model Y f(X1,,XN), the
  measurand is denoted by Y
• Also called the output quantity
• The quantities X1,,XN are called input
  quantities
• The value of the output quantity (measurand) is
  calculated from the values of the input
  quantities using the measurement model

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Input and Output Estimates

• When one performs a measurement, one obtains
  estimated values x1,x2,,xN for the input
  quantities X1,X2,,XN
• These estimated values may be called input
  estimates
• One plugs input estimates into the model and
  calculates an estimated value for the output
  quantity
• The calculated estimate may be called an output
  estimate

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Propagation of Uncertainty

• When a measurement model is used to estimate the
  value of the measurand, the uncertainty of the
  output estimate is usually obtained by
  mathematically combining the uncertainties of the
  input estimates
• The mathematical operation of combining the
  uncertainties is called propagation of uncertainty

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Standard Uncertainty

• Before propagating uncertainties of input
  estimates, you must express them in comparable
  forms
• The commonly used approach is to express each
  uncertainty in the form of an estimated standard
  deviation, called a standard uncertainty
• The standard uncertainty of an input estimate xi
  is denoted by u(xi)
• Radiochemists traditionally called this one
  sigma uncertainty

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Combined Standard Uncertainty

• The standard uncertainty of an output estimate
  obtained by uncertainty propagation is called the
  combined standard uncertainty
• The combined standard uncertainty of the output
  estimate y is denoted by uc(y)

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Methods for Uncertainty Evaluation

• The GUM classifies methods of uncertainty
  evaluation (for input estimates) as either Type A
  or Type B
• Type A method of evaluation by statistical
  analysis of series of observations
• Type B method of evaluation by any means other
  than statistical analysis of series of
  observations
• If it isnt Type A, its Type B

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Combining Uncertainties

• All uncertainty components are treated alike for
  the purpose of uncertainty propagation
• One does not distinguish between Type A
  uncertainties and Type B uncertainties when
  propagating them to obtain the combined standard
  uncertainty

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Random Systematic

• Twenty years ago, it was common to call a Type A
  uncertainty a random uncertainty and a Type B
  uncertainty a systematic uncertainty
• The GUM explicitly disparages those terms now
• So avoid them
• But recall that the terms random error and
  systematic error are still accepted (when
  referring to error, not uncertainty)

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Examples Type A

• Make a series of observations of an input
  quantity Xi
• Let xi be the arithmetic mean and let u(xi) be
  the experimental standard deviation of the mean
  (the standard error of the mean)
• Least-squares regression can also be a Type A
  method
• If there is a well-defined number of degrees of
  freedom (number of observations minus number of
  parameters estimated), its probably a Type A
  method of evaluation

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Examples Type B

• Often a Type B evaluation involves estimating a
  bound, a, for the largest possible error in the
  estimate, xi, and dividing by an appropriate
  constant based on an assumed distribution for the
  error
• For example, if you believe the true value lies
  within a of the estimated value, xi, but you
  know nothing more than that, assume a rectangular
  distribution, and divide a by to obtain
  u(xi)
• Example Uncertainty associated with rounding on
  a digital display

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Rectangular Distribution
xi a
xi - a
xi
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Triangular Distribution

• Sometimes you can estimate a bound, a, for the
  error, but you believe that values near xi are
  more likely than those farther away
• In this case, you might assume a triangular
  distribution for the error
• If so, you divide a by to obtain u(xi)
• Example Capacity of a pipette, with a specified
  nominal volume and tolerance

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Triangular Distribution
xi a
xi - a
xi
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Imported Values

• There are many other possible Type B methods
• E.g., using the value and standard uncertainty of
  the half-life of a radionuclide published by NNDC
• A calibration certificate for a standard might
  provide a confidence interval for the value with
  some specified level of confidence, such as 95
• Assume a normal distribution and derive standard
  uncertainty from percentiles of that distribution
  (e.g., if the confidence level is 95 , divide
  the half-width of the confidence interval by 1.96)

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What about Counting Uncertainty?

• Make a radiation counting measurement, where C
  counts are observed
• Let xi C and u(xi)
• What type of uncertainty evaluation is this Type
  A or Type B?
• This method of evaluation presumes Poisson
  counting statistics
• Beware Sometimes the distribution isnt Poisson
• Note Counting uncertainty isnt the total
  uncertainty

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Correlations

• An issue sometimes neglected in uncertainty
  evaluation is the fact that some input estimates
  may be correlated with each other
• May either increase or decrease the uncertainty
  of the final result
• One common example is the correlation that often
  exists between the parameters for a calibration
  curve fit by least squares

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Notation for Correlations

• If you know there is a correlation between two
  input estimates xi and xj, you should evaluate it
  and propagate it
• Estimated correlation coefficient (a number
  between -1 and 1) is denoted by r(xi,xj)
• The estimated covariance of xi and xj is denoted
  by u(xi,xj)
• u(xi,xj) r(xi,xj) u(xi) u(xj)

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Uncertainty Propagation Formula

• Most commonly used equations for uncertainty
  propagation are based on the general equation
  shown below, which the GUM calls the law of
  propagation of uncertainty
• MARLAP prefers uncertainty propagation formula

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Sensitivity Coefficients

• The partial derivatives ?f/?xi that appear in the
  uncertainty propagation formula are called
  sensitivity coefficients
• These derivatives are evaluated at the measured
  values of the input estimates
• OK to approximate them You dont necessarily
  have to calculate them using formulas from
  calculus

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Components of Uncertainty

• The term component of uncertainty means several
  things, but one definition is explicit in the GUM
• The component of the combined standard
  uncertainty, uc(y), generated by the standard
  uncertainty u(xi) is the product of the absolute
  value of the sensitivity coefficient ?f/?xi and
  u(xi), which may be denoted by ui(y)

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Uncertainty Propagation

• Uncertainty propagation formula is derived from a
  first-order Taylor-polynomial approximation of f
• It is commonly used, but the approximation is not
  great in some situations (e.g., dividing one
  value by another value with a very large relative
  uncertainty)

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Automatic Uncertainty Propagation

• Many find the uncertainty propagation formula
  intimidating, but it is actually straightforward
• Simple enough to be done automatically in most
  cases by software libraries
• In the presenters opinion, uncertainty
  propagation is one of the easiest aspects of
  uncertainty evaluation
• The hard part is understanding the measurement
  process well enough to recognize and evaluate
  uncertainties that ought to be propagated

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Expanded Uncertainty

• It is common to multiply the combined standard
  uncertainty, uc(y), by a factor, k, chosen so
  that the interval y kuc(y) has a specified high
  probability of containing the true value of the
  measurand
• GUM calls product U kuc(y) an expanded
  uncertainty
• Factor k is called a coverage factor (often k2
  or 3)
• The probability that y U contains the true
  value is called the coverage probability, p

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Summary of Steps

• Define the measurand and construct the
  mathematical model of the measurement
• Obtain estimates, xi, of the input quantities
• Evaluate the standard uncertainties u(xi), by
  Type A or Type B methods, and evaluate the
  covariance u(xi,xj) for each pair of correlated
  input estimates xi and xj
• Apply the model to evaluate the output estimate, y

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Summary of StepsContinued

• Propagate the standard uncertainties u(xi) and
  covariances u(xi,xj) to obtain the combined
  standard uncertainty uc(y)
• Optionally, multiply uc(y) by a coverage factor,
  k, to obtain an expanded uncertainty, U
• Report the result, y, with either the combined
  standard uncertainty, uc(y), or the expanded
  uncertainty, U
• Explain the uncertainty clearly

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Questions?