Uncertainty in measurement represents the variability for any reported
value based on incomplete knowledge of the measurement. The
uncertainty value represents the sum of the unknown errors produced
between the transfer standards as it progresses down to the working
In reference to length the definition of a meter by the international
system of units is the distance that light travels in 1/299,792,458
seconds. Only a few artifacts are actually measured against the
speed of light since the equipment and expertise needed to perform
this measurement accurately would become very expensive and can likely
only be performed by a national measurement institute such as NRC or
NIST. The artifacts that are calibrated by the top level
national institutes are used as transfer standards to measure
additional artifacts which are then used to calibrate other artifacts
and eventually to the level of the working standard such as a gauge
Each measurement transfer can contribute an unknown amount of
measurement error. Each artifact has imperfections that may not
be fully understood at the time of measurement. The equipment
used to perform the comparison between the known and unknown also has
imperfections that can affect the results. This grey area is
considered to be a potential error that may or may not exist and
represents the uncertainty.
The more intermediate transfer layers between the primary reference
standard to the actual working standard then the greater the amount of
potential error or uncertainty. This system provides a way to
ensure that circular calibration does not occur where one company
calibrates the artifacts for a second company that in turn calibrates
the artifacts for the first company in an isolated manor. If
uncertainty is properly considered at each step eventually the
uncertainty in this situation will grow beyond reasonable
Some confuse measurement repeatability with measurement
uncertainty. Measurement repeatability provides an idea of
variability but cannot take into account the deviation from the true
value. Measurement repeatability is only a contributing factor
to measurement uncertainty.
Traceability as defined by NIST: "Traceability of measurement requires
the establishment of an unbroken chain of comparisons to stated
references each with a stated uncertainty.".
There are several requirements to produce a traceable
measurement. The biggest one is that a measurement must be
performed by an organization that is competent and recognized to do
so. Accreditation to ISO/IEC 17025 or an equivalent standard
with the measurement discipline listed on the accreditation scope is a
requirement for a traceable measurement.
In addition to recognition a reasonable estimation of the contribution
of measurement uncertainty must be produced. This uncertainty
value will contribute to the chain of uncertainties traced back to the
original definition of length by the international system of units.
If a calibration is performed by a company who is not recognized to
perform this work then these results, and any subsequent results, are
no longer considered traceable to the definition of length by the
international system of units.
In order to assign an uncertainty value to a measurement it is
necessary to list all the contributing errors associated with the
measurement and assign a value to each. The unknown part of the
measurement cannot be determined directly (it would no longer be
considered an unknown if it can be determined) but it can be estimated
indirectly by considering at all the sources that contribute to
potential error. This list of error sources, properly organized,
is referred to as an uncertainty budget.
The contribution from each value in the uncertainty budget is in the
form of a standard uncertainty. A standard uncertainty is
comparable to how standard deviation is calculated from a given set of
data. The conversion to a standard uncertainty from a potential
error source is done based on the distribution type associated with
the value. For example, a standard uncertainty from a non-normal
(rectangular) distribution of an error would be converted to a
standard uncertainty by dividing by the constant 1.73.
The type of contributing uncertainty can either be A or B. Type
A sources are from direct measurements such as repeatability.
Type B sources are values where only basic information is known about
the source such as the resolution of the measurement instrument or
calibration uncertainty of the reference artifacts used for the
All sources in the uncertainty budget should be listed even if the
influence is very small or negligible. An large uncertainty
source that has little effect on the total uncertainty can be handled
by setting a sensitivity value reflecting the amount of influence it
has on the measurement. Listing items that contribute only a
small amount to the total uncertainty at least acknowledges that the
contribution has been considered.
The total uncertainty is calculated as the square root of the sum of
the squares for each uncertainty component as described by GUM.
The reported uncertainty on the calibration report is the standard
uncertainty calculated from the uncertainty budget expanded to a
desired coverage level. The coverage describes the number of
standard uncertainties is represented by the uncertainty value.
A coverage value of 1 would represent 68% of the total range whereas a
coverage of 2 means that the uncertainty represents two standard
uncertainties or 95% of the total range and 3 represents 99% of the
total. The most common coverage value used is 2 and will appear
as (k=2) on the calibration
Uncertainty For CMM Calibration
In order to provide a good estimate of the measurement uncertainty
when calibrating CMM's a couple of problems needed to be solved.
One problem is that the calibration is always performed onsite so
there is no control over environmental conditions which frequently
vary from site to site. The machines often have differences that
can have an impact on the uncertainty such as scale resolution, type
of probing used, repeatability characteristics, selection of equipment
used for the calibration, and a variety of other contributing factors.
The solution used by Select Calibration Incorporated for this problem
is to use a dynamically calculated uncertainty budget. Instead
of assigning constants to the uncertainty budget each value is treated
as an expression with variables. The variables represent the
different aspects of the machine, environment, and selected equipment
along with anything else that can change from machine to machine.
For example, consider the scale resolution. The uncertainty from
a scale with a resolution of 0.001 mm is calculated by one of these
two methods (both results are identical but written differently
depending on interpretation):
Uc = 0.001 / (2 * sqrt(3))
Uc = (0.001 /2) / sqrt(3)
Instead of using the static constant of 0.001 in the uncertainty
budget expression a variable (Sr) is used instead to represent the
scale resolution so it would end up looking like this:
Uc = Sr / (2 * sqrt(3))
Uc = (Sr /2) / sqrt(3)
When moving from machine to machine only the variable information
describing the scale resolution needs to be entered which would
automatically update this component and all related components.
In addition to the scale resolution variable (Sr) other variables are
automatically recognized such as the temperature deviation from
reference (Td), range of temperature (Tr), expansion coefficient of
machine and equipment (CTEx), probe error (Pe1D, Pe2D, Pe3D),
uncertainty of support equipment such as the thermometer (UTt), and a
few other common items. Other user defined variables can be
added if required.
The measurement uncertainty is automatically calculated when a
calibration report is generated. The uncertainty budget used by
Select Calibration Incorporated is not setup as a spreadsheet which
allows seamless integration with the reporting software. The
reported uncertainty value using this setup is almost always unique to
Uncertainty expressions are a description of the measurement
uncertainty usually expressed with a formula such as Uc = A +
Bx. The measurement uncertainty for any given length would be
determined by substituting the length of the measurement into the
equation to produce a single value.
The calibration reports issued by Select Calibration Incorporated do
not use uncertainty expressions but instead show a single uncertainty
value attached to each measurand. It was decided to use this
method instead of an expression of the uncertainty in order to be more
precise by eliminating approximation errors.
The scope of accreditation uses expressions to represent the
uncertainty for length dependent values. The uncertainty
expressions used by Select Calibration Incorporated that include both
constant and length dependent components are in the format Uc = A + Bx
+ Cx^2. Using a curve fit of the actual uncertainty data
provides a more realistic representation of the uncertainty then using
only a line fit.
The graphs on the left show examples of the problem using a simple
line fit expression to describe the uncertainty data.
The first graph shows the contribution from all sources listed in the
uncertainty budget with the constant and length dependent components
identified in different colors. The combined value from the two
types of sources produces a curve.
The second graph shows the problem with trying to describe the curved
uncertainty shape using only a best fit line. The biggest concern with
this is that the line at length zero (Y Intercept) is almost always
lower than what could potentially be achievable by the
laboratory. In some cases this value may even show as a negative
if the curve is pronounced enough.
The third graph shows a set of actual uncertainty data compared to the
curve fit representation. Although the fit is not perfect it is
much better at describing the actual shape then what would be possible
using only a line. The expression used to generate the shape is
displayed below the graph with L representing the position in meters.