Uncertainty

Uncertainty Description

Uncertainty in measurement represents the variability for any reported value based on incomplete knowledge of the measurement.  The final uncertainty value represents the sum of the unknown errors accumulated between the transfer standards and any contribution from the measurement the uncertainty value is associated with.

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.

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 number of intermediate transfer standards between the primary and the working standard affects 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 back to the definition of the measurement unit by the international system of units.   If a calibration is performed by a company who is not recognized to perform this work then the results, and any subsequent results, are no longer considered traceable to the SI units.

Uncertainty Budget

In order to assign an uncertainty value to a measurement it is necessary to understand and 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 by considering at all the sources that contribute to potential error.  This list of error sources and contribution values, when 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 a standard deviation for a given set of data.  The conversion to a standard uncertainty from a potential error source is done based on the distribution type and range associated with the value.  For example, a standard uncertainty from a non-normal (rectangular) distribution of some range could be converted to a standard uncertainty by dividing the range with 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.

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 for measurement uncertainty when calibrating CMM's a couple of problems need to be solved.  One important problem is that the calibration is always performed onsite and many of the contributing uncertainty sources change value from site to site.  For example, environmental conditions are frequently different between different customers which potentially has the largest impact on the final measurement uncertainty.  Individual machines can have differences such as scale resolution, type of probing, and even repeatability characteristics that can have an impact on measurement uncertainty.  The selection of equipment used for testing by the calibration service provider will also contribute to the final measurement uncertainty.   Uncertainty budgets, especially when dealing with a variety of CMM's for different customers, takes on the form of an abstract list of error sources that cannot be estimated in advance.

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 an expression with the variable Sr is used to represent the contribution from the scale resolution.  The expression 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 provided which would automatically update this component and all related components.

In addition to the scale resolution, variables can be used for temperature deviation from the nominal, temperature gradient, uncertainty of the expansion coefficient of machine and equipment, uncertainty of the selected probe error, uncertainty of the calibration equipment such as the thermometer, step gauges, gauge blocks, laser, and any other items that vary from machine to machine.

The measurement uncertainty is automatically calculated when a calibration report is generated.  The uncertainty budget used by Select Calibration Incorporated has seamless integration with the reporting software.  The reported uncertainty value using this setup is almost always unique to the customer.

Uncertainty Expressions

Uncertainty expressions is a description of the measurement uncertainty expressed as 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 measured result.  It was decided to use this method instead of an uncertainty expression in order eliminate approximation errors.

The scope of accreditation for Select Calibration Incorporated uses expressions to represent the uncertainty over an unspecified length.  The uncertainty expressions used by Select Calibration Incorporated include both constant and length dependent components in the format Uc = A + Bx + Cx^2.  Using a curve fit of the uncertainty data provides a more realistic representation as compared to a line fit of the same data.  The following shows an example of the why a curve fit of the uncertainty expression is used and not a simple line fit:
 Contribution from all sources listed in the uncertainty budget with the constant and length dependent components shown in different solid colors.  The combined value from the constant and length dependent components produce a curve. Result of using a best fit line to describe the curved shaped of the uncertainty data. 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 be negative if the curve is pronounced enough. Comparison of actual uncertainty data calculated at different lengths as compared to the best fit curve uncertainty expression.  Although the fit is not perfect it is much better at describing the real 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.