Estimating Measurement Uncertainty
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.
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.
Parallax is the apparent displacement or shift in an object caused by a change in position of the observer. It is the source of error in a measurement when a scale is read at a slight angle.
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.
the Combined Standard Uncertainty in Terms of Uncertainty Interval
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%.
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, operator skill, sample variation, and environmental factors. When a measurement has more than one identifiable source of measurement uncertainty, then the combined standard uncertainty (uc) must be calculated.
Calculating the combined standard uncertainty is a two step process. The first step is to determine the uncertainties measured directly and the second step is combine the uncertainties using summation in quadrature, which is also known as root sum of the squares. For example, is a measurement of a measurand x, has three sources of uncertainty for which the three standard uncertainties u1(x) , u2(x) and u3(x) have been determined, then the combined standard uncertainty uc(x) for the measurement is given by:
Uncertainty contributions from both Type A and Type B evaluations may be combined as long as they are expressed in similar terms before they are combined. Thus, all the uncertainties must be expressed as one standard uncertainty and in the same units.
Propagation of Uncertainty
Many physical quantities are not determined from a single direct measurement but instead are calculated by combining two or more separate measurements. Therefore, it is important to understand how measurement uncertainty propagates when mathematical operations are performed on measured quantities, so that a final combined uncertainty can be calculated. Consider the determination of the velocity of a sound wave as it travels through a medium. The velocity (V) is calculated by dividing the measured distance (d) traveled by the measured time (t) that it took the sound to travel the distance. This calculation of velocity is easy enough but the measured quantities (d and t) each have a measurement uncertainty that must be combined to arrive at an uncertainty for the velocity calculation.
The propagation of uncertainty is treated differently depending on the mathematical operation(s) performed. The simplest case is where the result is the sum of a series of measured values (either added together or subtracted). The combined standard uncertainty is found by squaring the uncertainties, adding them all together, and then taking the square root of the total. For more complicated cases, such as multiplication and division where mixed units are often involved, it is necessary to work in terms of relative uncertainties. The formula for calculating the combined standard uncertainty for basic mathematical operations are shown in the table below.
Note 1: The result of this calculation is the relative combined uncertainty. The absolute combined uncertainty can be calculated by multiplying uc by the best approximation of the measurand.
The equations in the table above or only valid if the contributing uncertainties are not correlated. Factors leading to measurement error are often independent, but sometimes they are correlated of inter-related. For example, a temperature shift could have a similar effect on several uncertainty contributors. If two or more sources of uncertainty are believed to be correlated, consult the references for additional information on dealing with the correlation.
The Uncertainty Budget
An uncertainty budget is simply a way of organizing and summarizing the uncertainty analysis in tabular form. An uncertainty budget lists all the contributing components of uncertainty and these components are used to calculate the combined standard uncertainty for the measurement. The table can consist of as few as two columns, one for listing the source of uncertainty and the second for recording the standard uncertainty. However, more involved tables such as the one shown below can be helpful.
Consider the need to determination of the remaining wall thickness between the bottom of a drilled hole and the surface. This determination would require the depth of the hole to be measured and subtracted from the measured value of the total thickness of the block. Five readings for each measurement were taken and summarized in the table below.
For this example, two possible source of uncertainty in the measurement will be considered: the resolution of the dial gage and the repeatability of the measurement. Since there are multiple sources of uncertainty in this calculation, the evaluation process will be broken down into the following steps:
Uncertainty of Individual Measurements Due to Resolution of Dial Gage
Uncertainty of Individual Measurements Due to Measurement Repeatability
Combined Uncertainty of Individual Measurements
Combined Uncertainty of Calculated Remaining Wall Thickness
State the Uncertainty in Terms of an Uncertainty Interval and Level of