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# Estimating Measurement Uncertainty

##### - Terms - Uncertainty - PDFs   - Gaussian   - Uniform   - Triangular - Procedure- Combined - Example - References

For the uncertainty to be truly meaningful, it must address the entire measuring process, which may have uncertainties associated with factors such as equipment calibration, equipment resolution, operator skill, sample variation, and environmental factors. In many cases, sample variability and operator skill are the largest sources of uncertainty and they are often the only source considered when only the repeatability of a measurement is evaluated. However, a more thorough analysis will consider other sources of uncertainty. As a minimum, in addition to the repeatability evaluation, an uncertainty analysis should consider the instrumentation/standard calibration and the resolution of the instrumentation.

## The following a summary of the main steps in an uncertainty evaluation.

1. Define the measurand. (Determine exactly what needs to be measured.)
2. Carry out the needed measurements.
3. Evaluate the uncertainty due to the calibration standard and/or instrumentation used for the measurement.
4. Evaluate resolution/readability of all instrumentation.
5. Evaluate the repeatability.
6. Identify and evaluate other sources of uncertainty.
7. Prepare uncertainty budget documentation.
8. Evaluate reasonability of budget.
9. Determine combined standard uncertainty.
10. Express the uncertainty in terms of uncertainty interval and the confidence level.

Each of these steps will be looked at in more detail in the following sections

## Define the Measurand and Carry Out the Needed Measurements

The first step controlling and characterizing uncertainty in a measurement is to define the measurand as explicitly as possible. It is very important to know exactly what requires measuring. If possible, determine the required level of precision and accuracy and whether they can be obtained with the available equipment. Verify that equipment is properly calibrated and/or perform user calibration check using NIST or ISO traceable reference standard. If systematic error (bias) is found to exist, record the value so that it can be eliminated with a correction to the measurement data. It is also important to ensure that test samples are identified to prevent mix-ups and to mark measurement locations to allow for repeat measurements of the same locations. Finally, make the measurements with care and correct for any bias that has been identified.

## Evaluate the Uncertainty Due to the Calibration Standard and/or Instrumentation

The next step is to review the calibration data from the calibration certificate for the calibration standard or test instrumentation that will be used for the measurement. Typically, when uncertainty is stated directly on calibration certificates it will be the expanded uncertainty with a coverage factor of two and a confidence level of 95%. The uncertainty value will need to be divided by the coverage factor to get back to the standard uncertainty before the combined uncertainty can be calculated.

If a statement of tolerance or accuracy is presented on the certification certificate, this is describing the interval of values that the true value is asserted to exist. Since nothing more is known about this interval, a uniform pdf should be used to calculate the standard uncertainty associated with it. To calculate the standard uncertainty, the half interval will be divided by √3. For example, an instrument with a reported tolerance or accuracy of ±0.004mm will have a full interval of 0.008mm and a half interval of 0.004. The standard uncertainty will be 0.008mm/2√3 or 0.004mm/√3, which is 0.0023mm.

## Evaluate resolution/readability of all instrumentation

Digital instrumentation provides a discrete value but due to rounding, the true value could lie within ± 0.5 times the resolution of the display. To calculate the standard uncertainty for digital device, simply divide the display resolution by √3. See the information of uniform probability density functions for more information.>

The resolution or readability of an analog device depends on the ability to estimate to the nearest scale division mark or fraction of a division. Since there is a higher probability that the true value of the measurement will occur near the best estimate of the value than near the limits of the interval of possible values, a triangular probability density function is used. To calculate the standard uncertainty associated with a triangular pdf, the interval of possible values is divided by 2√6. As an example, consider a measurement made with a dial caliper that has division marks in 0.01mm increments. Say that it is allowable to estimate to one-half of a increment or 0.05mm. The range possible values associated with this resolution is 0.05mm or ± 0.025mm. The standard uncertainty is then 0.05mm divided 2√6 or 0.011mm. See the information of triangular probability density functions for more information

## Evaluate the Repeatability

To evaluate sources of uncertainty due to factors such as sample variability, placement of the measurement instrument, and operator skill and consistency in making the measurement, a repeatability study should be conducted. A repeatability study is only useful when the measurement device is sensitive enough to produce scatter in the readings. Repeat readings that all produce the same value may improve overall confidence in the measurement, but they don't provide any additional information about the probability density function. Therefore, the measurement must be sensitive enough to produce scatter in the reading so that the shape of the distribution can be determined and the standard deviation can be properly calculated. When repeat readings produce scatter that is distributed approximately equally about the mean value of the sample, then a Gaussian pdf can be used to evaluate the repeatability of the measurement. See the Gaussian probability density functions for more information.

## Identify and Evaluate Other Sources of Uncertainty

Other source of uncertainty will likely be small compared to the instrumentation and repeatability evaluations discussed above, but all identifiable sources of uncertainty should be addressed. Other possible sources of uncertainty may include: parallax, thermal expansion and other temperature effects, voltage drift and etc.

## Combined Standard Uncertainty

Once the standard uncertainties for all the sources of uncertainty in a measurement or set of measurements have been calculated, then the combined standard uncertainty can be calculated. The combined standard uncertainty is the total uncertainty in the measurement and can come from a combination of type A and B evaluations.
Learn more on calculating the combined standard uncertainty on the next page.

## Express the Combined Standard Uncertainty in Terms of Uncertainty Interval and the Confidence Level

The results of the measurement and uncertainty analysis should be reported in terms of the uncertainty interval and the confidence level. Since more than one type of pdf likely contributed to the combined uncertainty, the type of pdf used cannot be stated. Therefore, a measurement might be reported as 18.2 ± 0.15mm with a confidence level of 68%. The measurement could also be reported as 18.2 with a relative uncertainty of ± 0.0083. The uncertainty could also be expressed in terms of percent uncertainty as ± 0.83%.

Last updated on November 24, 2010