INTRODUCTION
TO
MEASUREMENT UNCERTAINTY
Steve Clyens, CET
Gas Specialist, Ontario Region
CGA Gas Measurement School
Edmonton, June 2006
REFERENCES
1. Guide to the Expression of Uncertainty in Measurement. BIPM, IEC,IFCC, ISO,
IUPAC, IUPAP, OIML, 1995.
2. International Vocabulary of Basic and General Terms in Metrology (VIM).
International Organization for Standardization, 1993.
3. Measurement Good Practice Guide No. 11 (Issue 2), A Beginner's Guide to
Uncertainty of Measurement, Stephanie Bell
4. The Uncertainty of Measurements: Physical and Chemical Metrology: Impact and
Analysis, Kimothi, S. K., 2002
DISCLAIMER
This presentation does not represent or communicate Measurement Canada policy
relating to the determination, evaluation, expression and/or use of measurement
uncertainty in trade measurement and related applications.
This presentation is intended for those people who know little or nothing about
Measurement Uncertainty but are involved in measurement in some form or other as
part of their daily activities.
INTRODUCTION
Many technical papers and guides on this subject are loaded with advanced
statistical terminology and concepts, complex-looking mathematical equations, not
to mention the language of differential calculus used by scientists and engineers.
The subject of Measurement Uncertainty can be somewhat intimidating to many
people.
This presentation WILL NOT teach you how to perform measurement uncertainty
evaluations and calculations.
The main objective is to get you thinking more about the concept of measurement
uncertainty and why it is important and necessary to include it in your own particular
measurement activities.
HOW LONG IS THE PAPER CLIP?
ruler may have inherent error, was it calibrated?
ruler has limited resolution, limited precision (# significant digits)
length of ruler and/or the paper clip may change with a change in
temperature
paper clip may not be lined up parallel with the ruler
bottom of paper clip may not be lined up with zero of the ruler
different people may report different results (parallax error, etc.)
and other things we haven't considered or don't know about
These are all potential sources of error that, if not fully accounted for and/or
corrected, can cause doubt or uncertainty about the quality of the measurement
result.
If we want a more accurate and precise result, we need to consider certain factors
and quantities that could influence the outcome of the measurement.
If all we want is a ball park figure we can look at the scale and take our best guess.
THE PERFECT MEASUREMENT
It is these imperfections that give rise to error in the result of a measurement.
Making a perfect measurement would require:
perfect measurement equipment,
perfect measurement processes,
perfect conditions for measurement,
perfect people making the measurement
Of course, none of these exist. All have imperfections to some degree or other.
We can report that a measurement result is 100% accurate but we can never be
100% certain about such a result. In fact, we can never be 100% certain about any
measurement result.
This would require a perfect measurement. Unfortunately, there is no such thing.
MEASUREMENT ERROR
Error (of measurement) = Systematic Error + Random Error
Error (of measurement) = Measurement Result* - True Value
Measurand
Particular quantity subject to measurement.
The length of the paperclip at 70 oF in centimeters and expressed to 2 decimal places.
* May be based on a single measurement or the mean of a series of repeat measurements.
Traditionally, error of measurement is viewed as having a systematic component
and a random component.
Error (of measurement)
Result of a measurement minus the true value of the measurand.
SYSTEMATIC ERROR
remains constant over a series of repeat measurements, or
varies in a predictable way over a series of repeat measurements, or
may occur randomly over repeat measurements, samples, time etc.
The recognized systematic effect could be a positive or negative offset of the
measurement result from the true value that:
readily identifiable (and so correctable), or
very difficult if not impossible to identify
Systematic errors may be:
Systematic errors arise from recognized (systematic) effects of influence factors or
quantities on a measurement result.
SYSTEMATIC ERROR
bias in measuring equipment, ie. zero offset error, span error, errors in
calibration, hysteresis
incorrect measurement methods, ie. selection and use, temperature effects on
dimensional measurements
human factors, ie. parallax error, incorrect assumptions of linear response
use of measurement equipment under conditions differing from calibration
conditions
time, ie. drift of metrological characteristics such as equipment bias
Some of the influence factors or quantities that can lead to systematic error are:
The greater the magnitude of the overall systematic error, the poorer the accuracy of
the measurement result unless the systematic error is corrected.
RANDOM ERROR
Random errors arise from the (random) effects of unpredictable variations of
influence factors or quantities on a measurement result.
When a series of repeat measurements of the same measurand are made, random
errors:
If all systematic error could be accounted for and eliminated, the expected value of
the mean of a very large (approaching infinite) number of normally distributed
random errors would be zero.
are characterized by random variation in the individual measurement results
will have varying magnitude and sign (+ or -)
will tend to be normally distributed about the mean (average) of the individual
measurements
RANDOM ERROR
unpredictable variations in environmental and/or measurement conditions,
ie. temperature, pressure, humidity, vibration, etc.
inherent instability of measuring equipment, ie. repeatability & reproducibility
differences between persons making measurements, ie. degree of expertise
and performance
instability of the "thing" being measured
Some of the influence factors or quantities that can lead to random error are:
The greater the magnitude of the random errors:
the greater the degree of variation amongst repeat measurements
the wider the dispersion of individual measurement results about their mean
QUANTIFYING VARIATION - Standard Deviation
where,
x i is the result of the i'th measurement
Sample (or experimental) standard deviation is used to quantify the extent of
variation in a sample of repeat measurements of the same measurand.
is the sample mean (average)
n is the sample size
Standard deviation has the same units of measure as the measurement result.
In metrology, standard deviation is used as an indication of:
repeatability - closeness of agreement between results of measurements of the
same measurand carried out under same conditions of measurement
reproducibility - closeness of agreement between results of measurements of
the same measurand carried out under changed conditions of measurement
standard uncertainty - established using statistical or non-statistical methods
REDUCING MEASUREMENT ERROR
removing known systematic error from the measurement process,
making calibration adjustments to the measuring device,
applying corrections from calibration certificates to measurement results
increasing the number of repeat measurements (sample size) and taking the
mean as the measurement result (not always practical or cost effective)
identifying the measurement's random influence factors or quantities and
attempting to minimize their variation during the measurement process
Random error cannot be completely eliminated but can be reduced by:
Systematic error cannot be completely eliminated but can be reduced by:
BEST ESTIMATE OF THE MEASURAND
imperfect determination of corrections for systematic effects, and
incomplete knowledge of all existing systematic effects, and
random variations of repeat measurements
There will always be some doubt or uncertainty about the goodness or quality
of an estimate of the value of the measurand; that is, a measurement result.
Even after reducing recognized systematic and random errors in a measurement
process, any subsequent measurement result is still only an estimate of the value of
the measurand because of:
The best estimate of the measurand is considered to be the mean of a sample of
repeat measurements.
The larger the sample size the better the estimate.
UNCERTAINTY OF MEASUREMENT
Parameter, associated with the result of a measurement, that characterizes the
dispersion of the values that could reasonably be attributed to the measurand.
Measurement Result (MR) ± Uncertainty (U)
This provides for an interval estimate that allows us to state, with a certain level of
confidence, that the measurand lies somewhere between:
The ISO definition of Measurement Uncertainty is:
When a measurement result is reported with no indication of its uncertainty, ie. as a
point estimate of the measurand, we have zero confidence in its goodness or
quality.
Uncertainty of measurement is due to the unrecognized and/or uncorrected errors in
the measurement result.
(MR - U) and (MR + U)
WHY IS IT IMPORTANT?
Any decision made on the basis of measurement results requires some indication of
the quality of those results.
Decision Making
Without such an indication, it is impossible to judge the fitness of the measurement
results as a basis for making decisions relating to health, safety, commerce or
scientific research.
Most, if not all, of our companies or agencies have quality programs, quality controls,
quality manuals and quality procedures all with the aim of producing quality products
and/or quality test results.
Uncertainty is used as a quantitative indication of the goodness or quality of a
measurement result.
The measurements we make should be quality measurements.
Quality
WHY IS IT IMPORTANT?
Fit For Purpose
An evaluation of the overall uncertainty of a measurement process (methods,
equipment, procedures and operators) provides a means of establishing that the
process will allow valid measurements and results to be obtained.
Tolerances and Specifications
The uncertainty of a measurement result should be taken into account when it is
compared against performance specifications or tolerance limits.
The smaller the uncertainty, the closer the measurement result can approach the
tolerance limit without being rejected.
An estimation of measurement uncertainty is required for testing and calibration
laboratories with ISO/IEC 17025 accreditation.
Laboratory Accreditation
In calibration, uncertainties have to be stated in the calibration certificate as they
are required by the user of the calibrated equipment.
WHY IS IT IMPORTANT?
Uncertainty allows meaningful comparison of results from different facilities or within
a facility or with reference values given in specifications or standards.
Comparisons
66.80 oF
67.00 oF
± 0.15 oF
± 0.19 oF
An uncertainty budget (a list of the uncertainty components and their respective
uncertainty contributions) can be used to determine which sources of uncertainty
contribute most so that financial and other resources can be applied wisely.
Process Planning and Improvement
A consideration of individual uncertainty components also indicates aspects of the
measurement process to which attention should be directed to improve
procedures.
How do these values compare? How about now?
UNIFORM METHOD OF DETERMINATION
Ideally, the method for evaluating and expressing uncertainty should be readily
implemented, easily understood and generally accepted internationally.
In 1993 a guide, prepared by a joint working group of experts nominated by the
BIPM, IEC, ISO and OIML organizations, was published.
The GUM recommends that all recognized systematic error be corrected first.
Guide to the Expression of Uncertainty in Measurement (GUM), 1993
The GUM establishes general rules for evaluating and expressing uncertainty in
measurement and is analytical in nature.
The need for such a method was recognized by the Comite International des Poids
et Measures (CIPM), the world's highest authority in metrology, in 1977.
1. Specify or define the measurand
THE GUM METHOD
The GUM method of uncertainty evaluation consists of eight generalized steps.
What follows is a very basic description of each of these steps.
2. Express mathematically the relation between the measurand and any input
quantities
State the quantity subject to measurement and, where applicable, the conditions of
measurement.
Quantity of interest is either:
a. measured directly, ie. length of paperclip, or
b. the output, y, of an equation composed of a number of input quantities, x n.
y = f (x 1, x 2, x 3, ..... x n)
Each input value, x i, is an estimate of a measurand with its own uncertainty.
THE GUM METHOD - Sources of Uncertainty
a. incomplete definition of the measurand; (significant unaccounted for influences)
b. imperfect realization of definition of measurand; (limitations of test conditions)
c. nonrepresentative sampling; (conditions of use differ from those of calibration)
d. inadequate knowledge of the effects of environmental conditions on the
measurement or imperfect measurement of environmental conditions;
e. personal bias; (influence introduced by person(s) making measurements)
f. finite resolution; (reading a scale, quantity per A/D count)
g. inexact values of reference standards and materials (inherited uncertainty);
h. inexact values of constants and other parameters obtained from external
sources (value or constants determined empirically);
i. estimates and assumptions used in the measurement process (linear response,
no hysteresis);
j. variations in repeat measurements under apparently identical conditions
(random error).
3. Identify all sources of uncertainty
Some potential sources of uncertainty in a measurement result, include:
4. Evaluate the input quantities and quantify the standard uncertainty of each
THE GUM METHOD - Type A and B Evaluations
The GUM does not use a systematic and random uncertainty approach to
evaluation. Instead, individual uncertainty components are evaluated using one of
two methods:
Once evaluated, an uncertainty component's quantified uncertainy, when expressed
as one standard deviation, is called its standard uncertainty.
Type A - Uncertainty is evaluated based on the statistical distribution of the results
of a series of measurements and can be characterized by experimental
standard deviations.
Type B - Uncertainty is evaluated from assumed probability distributions based on:
past experience of the measurements and/or behaviour of equipment
data provided in calibration and other certificates,
manufacturer’s specifications,
uncertainties assigned to reference data taken from handbooks
5. Identify the covariances of correlated input quantities
THE GUM METHOD - Combined Uncertainty
6. Calculate Combined Uncertainty (for each input quantity and the final output)
Does any input quantity influence the behaviour of any other input quantity?
Individual uncertainty components, expressed as standard uncertainties, are
combined (by Root Sum Square) to give the combined standard uncertainty.
The sensitivity of the final output to each input quantity (change in output caused by
a given change in an input) is accounted for in this step thru the application of
sensitivity coefficients.
These coefficients can be established using partial derivatives (calculus ) or
numerical methods (algebra).
Any correlation between input quantities and the resulting covariances (+ or -) are
also accounted for in this step.
If yes, they are said to be correlated and their covariance must be quantified.
7. Calculate the Expanded Uncertainty
THE GUM METHOD - Expanded Uncertainty
For degrees of freedom >29,
Level of Confidence Coverage Factor, k
68.26% 1
95.46% 2
99.73% 3
Expanded Uncertainty = Coverage Factor (k) x Combined Standard Uncertainty
A coverage factor, usually k=2, is applied to the final combined standard uncertainty
to give the expanded uncertainty.
The expanded uncertainty, when applied to the measurement result, provides for
an interval estimate of the measurand at the desired level of confidence (usually
approx. 95%).
THE GUM METHOD - Reporting Result and Its Uncertainty
This device's corrected indication has an expanded uncertainty of ±0.15 oC based on a
standard uncertainty multiplied by a coverage factor k = 2, providing a level of confidence
of approximately 95%.
This uncertainty was estimated according to the ISO Guide to the Expression of
Uncertainty in Measurement (GUM).
Corrected Indication
15.00 oC
with associated Uncertainty
± 0.15 oC or ± 0.3% Full Scale
This means approximately 95 times out of 100, the measurand will be within:
A thermometer's calibration certificate might include the following statement of
uncertainty:
8. Report the measurement result with the estimated expanded uncertainty
14.85 and 15.15 oC
Let's see how this might be applied.
CONFIDENCE IN OUR MEASUREMENT RESULTS
How do we establish some degree of confidence in our estimate of the value of the
measurand (measurement result)?
1. Take all reasonable steps to reduce error (systematic and random):
(a) in the measurement equipment
(b) in the measurement process
2. Quantify the total uncertainty associated with the estimate (measurement result)
produced by the measurement equipment / process.
3. Establish an interval estimate of the value of the measurand at a pre-determined
level of confidence using the measurement result and the quantified expanded
uncertainty.
QUESTIONS ??
Steve Clyens, CET
Gas Specialist, Ontario Region
Measurement Canada
232 Yorktech Drive
Markham, ON
L6G 1A6
905-943-8737 (office)
416-938-5712 (cell)
clyens.steve@ic.gc.ca
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