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Uncertainty


Uncertainty Description


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 standards.

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 block.
uncertainty_ts
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 values. 

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


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.

Uncertainty Budget


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.

uncertainty_dt 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 measurement. 

uncertainty budget 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.

Expanded Uncertainty


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 report.

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))
or
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))
or
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 the customer.

Uncertainty Expressions


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.


uncertainty_g1 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.



uncertainty_g2 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.



uncertainty_g3 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.