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Regardless of how careful or scientific, all measurements are subject to error and uncertainty. We can never be absolutely certain that a measurement result is true or finite. Therefore, we must establish a boundary for which we are confident that the measurement result will lie between. Error vs Uncertainty Error and uncertainty are two terms that have been used interchangeably in the past. However, the meaning of each term is distinctly different. Essentially, it expresses something about its quality. The Darkness Principle states that everything about a system can not be known.

Where we are able to quantify error through empirical experimentation, we can apply a correction. Since we are unable to know the true value, we will never know the error. Any error whose value is not known is a source of uncertainty. Therefore, we must account for what we do not know and what we can not quantify. Types of Errors Errors can be classified into two distinct types: random and systematic.

The classifications are determined by how the error impacts the measurement result. Sources of random error add a component to the measurement result that is unknown.

How to Calculate Standard Deviation (Uncertainty) for Measured Values

When random errors are small, it is possible to identify systematic errors. The classifications are determined by how the uncertainty is estimated. Sources of Error and Uncertainty A measurement result is never perfect. Internal and external factors impact the ability to achieve the ideal measurement results.

These factors are identified as sources of uncertainty. Propagation of Uncertainty Some physical measurements cannot be accomplished with a single direct measurement. Therefore, the measurement is calculated by the direct measurement of two or more independent variables. When a measurement result requires two or more steps, the estimation of uncertainty requires two or more steps. The following methods should be used to determine how the uncertainties of indirect measurements propagate through the calculations to produce an uncertainty in the final result.

Estimating Uncertainty Estimating uncertainty is accomplished through the combination and expansion of multiple contributing factors that introduce uncertainty in the measurement result. Uncertainty contributors are combined using the root sum of squares method. Under the Central Limit Theorem, the probability distribution of the combined uncertainty takes the form of a Gaussian or normal distribution.

Afterward, the combined uncertainty is expanded to a desired confidence level.


If the source provides the degrees of freedom, this should obviously be used. If not provided, scientific judgment is necessary to approximate the degrees of freedom. For example, for sources that warrant a high level of confidence, such as National Institute of Standards and Technology NIST , the degrees of freedom is conventionally set to , which may be a conservative estimate Taylor and Kuyatt However, it may be reasonable to use a lower value in the cases where it is common to have a small sample size.

Measurement uncertainty

An example of this is the level of confidence on DNA barcoding known for only a few taxonomically identified rare species. Periodic calibrations are necessary to maintain confidence in sensor measurements. Calibrations naturally change when sensors are subjected to environmental conditions and when materials degrade. For many environmental sensor networks, sensor calibrations are made on an annual basis e. Calibrations are made on a calibration fixture, each specific to the type of sensor, to assess the sensor functional performance f in Eqs.

Hence, each calibration will have its own set of sources of uncertainty e. The objective of the calibration is to make the sensor's measurement as unified with the reference standard as possible by exposing the sensor to a standard under a controlled and stable, repeatable environment and calibrating its response with an algorithmic fit between the reference and sensor. However, even standards have an uncertainty that contributes to the overall combined uncertainty.

Types of Errors

Additional examples of associated calibration uncertainties are the algorithm fit, repeatability, data acquisition system DAS , and reproducibility. These calibration uncertainty components are addressed below with a typical approach to quantifying them for a sensor. Calibrating a sensor sometimes referred to in metrology as a unit under test in theory brings the measurement as close to the true value as possible. However, the true value of a measurand can never be completely known. Therefore, a reference standard sensor is used to represent the true value, and the uncertainty associated with the reference standard measurand is a distribution Eqs.

The reference standard can be a primary standard, such as those based on first principles. Many times, the standard is a secondary or higher standard in which case the calibration of this standard can be traced to the primary standard Fig. For example, a secondary standard of a PRT calibrated to the first principles e.

ZERA GmbH: Measurement uncertainty

When assessing uncertainty for the standard, a Type B method is often used. However, sometimes they provide a calibration sheet reporting tolerance t or accuracy a. These have to be transformed into uncertainty. Tolerance is modeled by a uniform rectangular distribution, in which case 7a where t is the tolerance as quoted by the manufacturer or calibration facility. The transformation of accuracy is less straightforward because it has multiple definitions; it may be necessary to contact the manufacturer to determine the meaning.

If little information is available, 7c where a represents accuracy reported as the SD, in which case Eq. Calibration trueness is determined under stable conditions by the sensor measurement x i and the standard measurement S i in a Type A evaluation, such that: 8. By normalizing the sensor data to the standard, that is, detrending, Eq. The variation in the bath temperature does not impact the accuracy of the calibration, but rather, the difference between the standard and sensor is the impact on uncertainty. It is important to check for normally distributed results for the difference between the sensor and standard.

Variations that occur during typical operating conditions of a calibration should be evaluated using the Type A approach and reported by SD Eq. Here, the conditions to test depend on the sensor and the calibration protocol including the equipment hardware and software i. For example, the physical adjustment of a tipping bucket could vary with the operator, whereas for a PRT, the operator's only influence is the placement of the sensor in a controlled environment.

The analysis should include evaluating a sample of sensors that represents the population of sensors being evaluated.

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  4. ZERA GmbH: Measurement uncertainty;

A DAS, such as a data logger, receives a signal from a sensor and passes it along or logs it. Because the data logger is typically calibrated by an external facility, uncertainty related to the DAS is typically estimated by a Type B evaluation.

Uncertainty Analysis for Angle Calibrations Using Circle Closure

Radiofrequency interference, induced voltage spikes, and other electronic influences can affect a signal measured by a DAS and should be assessed for influence. For example, if analog signals from multiple sensors are multiplexed sensed sequentially but through one input channel , a stable reference signal i. Using this method to assess both calibration trueness and reproducibility, the variance terms for the signal influence should then be combined in quadrature Type A along with the DAS manufacturer's provided calibration Type B.

In some cases, the DAS uncertainty needs to be added twice in quadrature two times the variance as a component in Eq. Sources comprised in the combined uncertainty of a calibration include calibration trueness, calibration reproducibility, uncertainty of the reference standard, and the DAS.

All of these terms are added in quadrature to represent the combined calibration or method uncertainty, rf. Expanded uncertainty is found by multiplying the calibration combined uncertainty by the coverage factor Eqs. A gold standard or a sensor that remains in line with the calibration is used to monitor the calibration reproducibility through a redundancy test Taylor and Loescher Calibration and sensor repeatability parameters can be determined and controlled for all calibration set points and for every sensor for complete QC of the population of sensors.

If a DAS is used to log analog signals from the sensor, the uncertainty from the system is an additional component similar to that described in the section on DAS under Sensor calibration.