Measuring Confidence

Determining a measuring device’s accuracy

by Dilip Shah

During the construction of the pyramids in Egypt, the cubit was used as the measurement of length. It was based on the length from the ruling pharaoh’s middle fingertip to his elbow, as shown in Figure 1. Despite the cubit’s length changing from one pharaoh to the next, the pyramids were built with a 0.05% accuracy on their geometry.

Figure 1

This accuracy was made possible through the practice of comparing the craftsman’s wooden cubit to the master craftsman’s granite cubit every full moon. According to legend, a craftsman who failed to submit to this practice was punishable by death.

Today, this practice is referred to as a recalibration interval. Luckily, the punishment is not as harsh for failing to regularly submit a measurement device for recalibration. When a product is manufactured to a specification, it’s measured for compliance at various stages of the process. There should be confidence in the reported results of physical measurements, and this confidence comes from the accuracy of the measurement device.

Measuring equipment should be verified at a specified interval based on the application. This is because modern-day instruments are more complex and delicate and are subject to change from various influencing factors.

Tracing precision

For all measuring equipment, having metrological traceability is equal in importance to having accuracy. Metrological traceability is defined as a "property of a measurement result whereby the result can be related to a reference through a documented unbroken chain of calibrations, each contributing to the measurement uncertainty."1

Metrological traceability is the modern equivalent of comparing the wooden cubit to the granite cubit. The granite cubit’s traceability extended to the fingertip-to-elbow ruling of a pharaoh’s arm. Today, a production instrument’s traceability may extend through several stages, including the international system of units (SI) of measurement, comparison and other means. An example of a traceability method is illustrated in Figure 2.

Figure 2

In this example, the documented unbroken chain of calibrations is verified through review of documentation—usually the calibration certificates—from one measurement laboratory to another. For verifying each calibration contributing to the measurement uncertainty, examinations are performed on the measurement uncertainty provided on the calibration certificates, including the uncertainty of the end-user laboratory to provide its overall measurement uncertainty. This satisfies the metrological traceability requirement of the instrument.

Table 1 shows an example of the propagation of measurement uncertainty at various stages of the calibration chain. Using the root sum square method to combine the uncertainties, the primary laboratory’s uncertainty would be calculated by:

Table 1

If a multifunction, multirange instrument is used, all the ranges’ and the functions’ uncertainties require measurement uncertainty estimation. A simple multimeter uncertainty analysis may involve 30 individual measurement uncertainty budgets. The use of spreadsheets or software simplifies this task.

The verification of metrological traceability may be carried out by accrediting bodies as a third-party assessment to ISO/IEC 17025 (or equivalent) standard. This is shown as a parallel activity in Figure 2, starting with International Laboratory Accreditation Cooperation as the international body overseeing the accreditation process.

For the production of goods and services requiring the use of measurement instruments to comply with specifications set by the manufacturer, it is important to ensure that uncertainty of the instrument is much less than the specification for the part being measured.

Specifying confidence

The craftsmen building the pyramids had to measure each block of stone for its position. If it was too large or too small, it would compromise the integrity of the structure. They probably had some specification—written or otherwise—determined for the dimensions of the block of stone. Worker craftsmen had wooden cubits with less accuracy and resolution than the granite cubit used by the master craftsmen. Yet, they must have determined the wooden cubit’s fitness of use to be satisfactory for the measurement of the stone blocks.

Specifications today are much smaller due to tightened requirements for accuracy and precision. No measuring instrument will provide perfect results, but it will measure with a degree of confidence. A measurement reported with its measurement uncertainty provides the measurement confidence range—usually at 95% confidence interval.

If someone reports a measurement at 10 millimeters, the measurement is meaningless. It does not provide an uncertainty associated with it. If the measurement is reported at 10 millimeters with an uncertainty of +/– 0.01 millimeter, it means that the reported measurement will fall within the 9.99 to 10.01 millimeters with a 95% confidence interval.

If the specification for this measurement is 10 +/– 0.001 millimeters, the instrument is not capable of making the measurement. Figure 3 (p. 47) illustrates how the uncertainty will consume all of the specification.

Figure 3

We would need an instrument with a better resolution and uncertainty. The rule is: We would need an instrument with at least a 0.0001 millimeter resolution and an uncertainty that is four times less than the specification (0.00025 millimeters) or a test uncertainty ratio of 4-to-1 or better. This would provide for more leeway in making the measurement with more confidence (see Figure 4).

Figure 4

If an instrument is used with the same resolution (0.001 millimeters) as the specification, the uncertainty contribution of the resolution will consume the specification (see Figure 5).

Figure 5

This is why it is important that we discontinue the process of examining the calibration label on the instrument and the calibration certificate to ensure matching information during audits, such as the date due for calibration and serial numbers. These activities alone aren’t sufficient for checking that the product quality requirements will be verified by the instrument used.

More scrutiny is needed to verify that the instrument used for monitoring the quality of the process in production is fit for the purpose—by examining the measurement uncertainty, resolution and metrological traceability—and the specification of the process.

Measurement confidence only can be gained by ensuring there is metrological traceability of an instrument and its measurement uncertainty is quantified.


  1. Joint Committee for Guides in Metrology, International Vocabulary of Metrology—Basic and General Concepts and Associated Terms (VIM), third edition, JCGM, 2012.

Dilip Shah is president of E = mc3 Solutions in Medina, OH. He is the past chair of ASQ’s Measurement Quality Division and past chair of Akron-Canton Section. Shah, an ASQ fellow, is also co-author of The Metrology Handbook (ASQ Quality Press, 2012), and an ASQ-certified quality engineer, auditor and calibration technician.

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