Evaluating Uncertainty of Measurement. Muelaner

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Dr Jody Muelaner
Evaluating Uncertainty of Measurement
This page is the second part of a series of pages explaining the science of good mea-
surement. In Part 1: Key Principles in Metrology and Measurement Systems Analysis
(MSA)  concepts such as uncertainty of measurement, confidence and traceability
were introduced. This page goes into greater depth about uncertainty of measure-
ment introducing types of uncertainty was well as some basic statistics used in un-
certainty evaluation. It is followed by Part 3: Uncertainty Budgets, Part 4: MSA and
Gage R&R, and Part 5: Uncertainty Evaluation using MSA Tools.
All measurements have uncertainty which arises from many sources
such as repeatability, calibration and environment. Calculating the
total uncertainty resulting from all sources involves first estimating
the contribution from each source and then determining how these
will combine to give a combined standard uncertainty.
Type A and Type B Sources of Uncertainty
Sources of uncertainty are classified as Type A if they are estimated by statistical
analysis of repeated measurements or Type B if they are estimated using any other
available information. For example to find the repeatability of an instrument we
could simply measure the same quantity 20 times and analyse the different results
we got, this would be a Type A uncertainty. To find the uncertainty in our calibration
reference standard by this method would be impractical, for this we simply look up
the uncertainty on it’s calibration certificate, this is a Type B uncertainty.
Basic Statistics required for Uncertainty Evaluation
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In order to give values for the uncertainties an understanding of basic statistics is re-
quired; standard deviations, probability distributions such as the normal distribu-
tion and the central limit theorem. These are explained below.
Standard Deviation
In order to find the repeatability of an instrument we might measure the same quan-
tity 20 times. We will then have 20 slightly different measurement results which give
an idea of how much the instrument varies when measuring the same thing, we
need to find a number which represents   this variation. The simplest number to
find would be the range in the data; the biggest value minus the smallest value, but
this wouldn’t really represent the spread of all the measurements very well and the
more measurements we made the bigger it would get which suggests it is not a reli-
able method. The standard deviation is a number which can be calculated which is
essentially the mean average of how far each individual measurement is from the
mean average of all measurements. This is best understood with an example….
The standard measure of uncertainty is the standard deviation so one ‘standard un-
certainty‘ means one standard deviation in the measurement.
Probability Distributions and the Central Limit Theorem
I’m going to explain what a  probability distribution is by asking you to imagine
rolling some dice. First just roll one dice. You can roll a 1, 2, 3, 4, 5 or 6, and you have
an equal chance of rolling any one of them. If you roll the dice 100 times and plot a
bar graph of the score against the frequency (the number of times you got that score)
then you will end up with 6 bars all of equal height. These bars form a rectangular
shape and this is known as a rectangular distribution; the result can be any value be-
tween some limits. Uncertainty due to rounding to the nearest increment on an in-
strument’s scale has a rectangular distribution.
Now consider rolling 2 dice; A and B. you can score anywhere between 2 and 12. But
now there is not an equal chance of getting any score. There is only one way to score
2 (A=1 and B=1), there are two ways to score 3 (A=1 and B=2, A=2 and B=1) and so on.
The chances keep increasing until you get to a score of 7, which can arise from six
different permutations of dice, and then the odds decrease until we get to a score of
12 for which there is again only one way to roll it; both dice must be sixes. If we plot
these scores on a bar chart we get a triangular distribution. This is interesting be-
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cause it doesn’t only apply to dice,   if we combine the effects of any two random
events with rectangular distributions or similar magnitude then we get a triangular
distribution.
If we add more dice things get even more interesting. The peak of the triangle starts
to flatten and the ends start to trail off to form a bell shape known as the normal dis-
tribution. In fact the central limit theorem states that a number of independent dis-
tributions of any shape will combine to give a normal distribution. Combined uncer-
tainty can therefore generally assumed to be normally distributed even when many
of the components of uncertainty are not. This is useful as if we know our uncer-
tainty is normally distributed then we know what the probability is of a measure-
ment being in error by more than one standard uncertainty (68%), by more than two
standard uncertainties (95%) etc
The Central Limit Theorem – One Die Gives and
Equal Chance of a Number Between 1 and 6 but the
Sum of 2 Dice is a Triangular Distribution between 2
and 12. More Dice tend towards a Normal Distribu-
tion
Some typical sources of uncertainty
Uncertainty of measurement can arise from many sources. I tend to focus on dimen-
sional measurements but the principles of uncertainty evaluation and many of the
sources of uncertainty can be applied to any measurement for example time, temper-
ature, mass etc. Some sources of uncertainty which found in all virtually all traceable
measurements are the uncertainty of the reference standard used for calibration, the
repeatability of the calibration process and the repeatability of the actual measure-
ment. Environmental uncertainties such as the temperature will be significant
sources of uncertainty for many measurements. Resolution or rounding may not be a
significant source of uncertainty for digital instruments reading out to many decimal
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places beyond the repeatability of the system but is often significant for instruments
with a manually read scale such as rulers, dial gauges etc. Alignment is often a major
source of uncertainty for dimensional measurements.
Alignment Errors
Alignment is a common source of uncertainty in dimensional measurement. For ex-
ample when a measurement of the distance between two parallel surfaces is made it
should be perpendicular to the surfaces. Any angular deviation from the perpendicu-
lar measurement path will result in a cosine error in which the actual distance is the
measured distance multiplied by the cosine of the angular deviation.
Cosine Error is a Common Source of Uncertainty of Measurement
Another common alignment error is parallax error. This results from viewing a
marker, which is separated by some distance from the scale or object being mea-
sured, at an incorrect angle. Parallax error commonly observed when a passenger in
a car reads the speedometer. Another common example is when the markings on the
upper surface of a ruler are used to measure between edges on a surface. If the view-
ing angle is not perpendicular to the ruler this will result in parallax error.
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Parallax error is commonly observed when a passen-
ger in a car reads the speedometer orwhen using a
ruler with the markings on the upper surface
Abbe Error is similar to parallax error but rather than resulting from alignment of
viewing angles it results from alignment of machine axes. The distance between the
axis along which an object is being measured and the axis of the instruments mea-
surement scale is known as the Abbe Offset. If the distance along the object is not
transferred to the distance along the scale in a direction perpendicular to the scale
then this will result in an error. The size of this error will be the tangent of the angu-
lar error multiplied by the Abbe Offset. Instruments such as Vernier callipers are sus-
ceptible to Abbe Error as the measurement scale is not co-axial with the object being
measured. Micrometers are not susceptible to Abbe Error.
An Instrument is Susceptible to Abbe Error if the
Measurement Scale is not Co-Axial with the Axis of
Measurement
Repeatability and Reproducibility
Repeatability is estimated by making a series of measurements, generally by the
same person and under the same conditions, and then finding the standard devia-
tion of these measurements. Reproducibility is estimated by making a series of mea-
surements, each by a different person.
One challenge in evaluating uncertainty of measurement is determining which
sources of uncertainty contribute to the observed repeatability. For example it may
be that alignment errors vary randomly, contributing to repeatability, and therefore
do not need to be evaluated as a separate component of uncertainty. Experience and
judgement often play a role in such evaluations.
One method of establishing both repeatability and reproducibility in a single test is a
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‘Gage Repeatability and Reproducibility (Gage R&R) Analysis of Variance (ANOVA)’.
Using this method several different parts are measured by a number of different
people but each part is usually only measured 2 or 3 times by each person. The order
of the parts is randomized. The ANOVA statistical analysis is then used to separate
out the variation in the results which is due to three sources; the actual component
variation; the repeatability of the measurement system; and the reproducibility of
results between different people. Gage R&R is a very good way to establish the re-
peatability and reproducibility components of measurement uncertainty but effort
spent on this should not be seen as a substitute for evaluating other sources of uncer-
tainty such as calibration and environment.
Resolution
Uncertainty of measurement due to resolution is a result of rounding errors. For
many digital instruments the readout resolution is many times smaller than the ac-
tual instrument uncertainty. In such cases rounding errors due the instrument reso-
lution are insignificant.
For more traditional instruments resolution is often a significant source of uncer-
tainty. The maximum possible error due to rounding is half of the resolution. For ex-
ample when measuring with a ruler which has a resolution of 1 mm the rounding er-
ror will be +/- 0.5 mm which has a rectangular distribution. Converting a tolerance
with a rectangular distribution into a standard uncertainty is covered later.
Rounding Errors Can be up to Half of the Resolution
of an Instrument
Temperature
Temperature variations effect measurements in a number of ways:-
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Thermal expansion of the object being measured and of the instrument used to
measure it
For interferometric measurements changes in the refractive index
For optical measurements which depend on light following a straight line path
temperature gradients will cause refraction leading to bending of the light and
therefore distortions
Calibration
Any errors in the reference standard used to calibrate a measurement instrument
are transferred during calibration. Instruments therefore inherit uncertainty from
their calibration standard. The actual process of calibration is also not perfectly re-
peatable; therefore additional uncertainty is introduced through the calibration
process.
If calibration has been carried out by an accredited calibration lab then an uncer-
tainty will be given on the calibration certificate. This is not the uncertainty for mea-
surements made using the instrument; it is simply the component of uncertainty due
to calibration. This point is often overlooked.
When carrying out a calibration a complete uncertainty evaluation must be carried
out for the calibration process. The combined uncertainty for the calibration then be-
comes a component of uncertainty for measurements taken using the instrument.
Combining individual sources of Uncertainty of Mea-
surement
Once the individual sources for the uncertainty of a measurement have been identi-
fied and quantified a combined uncertainty should be calculated. This is the actual
uncertainty of measurement for the process being considered.
As an introduction to the process of combining uncertainty we will first assume that
all sources are normally distributed. Considering the measurement of a bolt, let’s as-
sume that there are just two sources of uncertainty, the thermal expansion of the bolt
and the uncertainty of the calliper measurement itself. The measurement result (y)
will therefore be given by
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y=X+Δx +Δx
where x is the true length, Δx is the error due to thermal expansion and Δx is the
error due to the calliper measurement itself.
A Simple Example of Combining Uncertainty for the
Measurement of a Bolt using Callipers
The error due to each source is not known; each error source has an uncertainty
which defines the range of values which we might expect it to take. The probability
that both errors will be maximal or minimal at the same time is very small. There-
fore to simply add up the uncertainties would be overly pessimistic. Instead the com-
ponent uncertainties are combined statistically to give a combined uncertainty.
The measurement result is given by y=f(x) where x1, x2 etc are inputs such as the
true length and the various errors. Each input has an associated uncertainty u(xi).
The combined uncertainty is then given by:-
For simple cases, such as our example of the callipers measuring the bolt, where
y = x + x 2 … x
The partial derivatives will all be equal to one so that
Applying this to the example
y=X+Δx +Δx
The true length (X) has no uncertainty, leaving two sources of uncertainty; the uncer-
tainty in the error due to thermal expansion u(Δx ); and the uncertainty in the error
due to the calliper measurement itself u(Δx ). The combined uncertainty is therefore
simply
T C
T C
1 2 n
T C
T
C
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This information can be used to create a simple uncertainty budget. First the stan-
dard uncertainty for each source of uncertainty is estimated.
In this simple uncertainty budget for a calliper mea-
surement each source of uncertainty is first esti-
mated
There is a simple functional relationship where the errors are simply added to the
true value to give the measurement result. The combined uncertainty is therefore
simply the square root of the sum of each component uncertainty squared (RSS). The
combined uncertainty is multiplied by a coverage factor to give the uncertainty at a
required confidence level (the expanded uncertainty).
The simple uncertainty budget is completed by cal-
culatingthe combined standard uncertainty and ex-
panded uncertainty
The example on this page was a simplification to introduce the concepts. An example
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of a proper uncertainty budget is given in the next section.
Part 3: Calculating an Uncertainty Budget…
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