Measurement Uncertainty -- Quantification of Systematic Errors

Once all practical steps have been taken to eliminate or reduce the magnitude of systematic errors, the final action required is to estimate the maximum remaining error that may exist in a measurement due to systematic errors. This quantification of the maximum likely systematic error in a measurement requires careful analysis.

Quantification of Individual Systematic Error Components

The first complication in the quantification of systematic errors is that it’s not usually possible to specify an exact value for a component of systematic error, and the quantification has to be in terms of a "best estimate." Once systematic errors have been reduced as far as reasonably possible using the techniques explained in Section 3.3, a sensible approach to estimate the various kinds of remaining systematic error would be as follows.

Environmental condition errors:

If a measurement is subject to unpredictable environmental conditions, the usual course of action is to assume midpoint environmental conditions and specify the maximum measurement error as _x% of the output reading to allow for the maximum expected deviation in environmental conditions away from this midpoint. Of course, this only refers to the case where the environmental conditions remain essentially constant during a period of measurement but vary unpredictably on perhaps a daily basis. If random fluctuations occur over a short period of time from causes such as random draughts of hot or cold air, this is a random error rather than a systematic error that has to be quantified according to the techniques explained in Section 3.5.

Calibration errors:

All measuring instruments suffer from drift in their characteristics over a period of time. The schedule for recalibration is set so that the frequency at which an instrument is calibrated means that the drift in characteristics by the time just before the instrument is due for recalibration is kept within an acceptable limit. The maximum error just before the instrument is due for recalibration becomes the basis for estimating the maximum likely error. This error due to the instrument being out of calibration is usually in the form of a bias. The best way to express this is to assume some midpoint value of calibration error and compensate all measurements by this midpoint error. The maximum measurement error over the full period of time between when the instrument has just been calibrated and time just before the next calibration is due can then be expressed as _x% of the output reading.

Example 2:

The recalibration frequency of a pressure transducer with a range of 0 to 10 bar is set so that it’s recalibrated once the measurement error has grown to +1% of the full-scale reading. How can its inaccuracy be expressed in the form of a _x% error in the output reading?

Solution:

Just before the instrument is due for recalibration, the measurement error will have grown to +0.1 bar (1% of 10 bar). An amount of half this maximum error, that is, 0.05 bar, should be subtracted from all measurements. Having done this, the error just after the instrument has been calibrated will be _0.05 bar (_0.5% of full-scale reading) and the error just before the next recalibration will be +0.05 bar ( +0.5% of full-scale reading). Inaccuracy due to calibration error can then be expressed as _0.05% of full-scale reading.

System disturbance errors:

Disturbance of the measured system by the act of measurement itself introduces a systematic error that can be quantified for any given set of measurement conditions. However, if the quantity being measured and/or the conditions of measurement can vary, the best approach is to calculate the maximum likely error under worst-case system loading and then to express the likely error as a plus or minus value of half this calculated maximum error, as suggested for calibration errors.

Measurement system loading errors:

These have a similar effect to system disturbance errors and are expressed in the form of _x% of the output reading, where x is half the magnitude of the maximum predicted error under the most adverse loading conditions expected.

Calculation of Overall Systematic Error

The second complication in the analysis to quantify systematic errors in a measurement system is the fact that the total systemic error in a measurement is often composed of several separate components, For example, measurement system loading, environmental factors, and calibration errors. A worst-case prediction of maximum error would be to simply add up each separate systematic error. For example, if there are three components of systematic error with a magnitude of _1% each, a worst-case prediction error would be the sum of the separate errors, that is, _3%. However, it’s very unlikely that all components of error would be at their maximum or minimum values simultaneously. The usual course of action is therefore to combine separate sources of systematic error using a root-sum-squares method. Applying this method for n systematic component errors of magnitude _x1%, _x2%, _x3%, _____ _xn%, the best prediction of likely maximum systematic error by the root-sum-squares method is ....

Before closing this discussion on quantifying systematic errors, a word of warning must be given about the use of manufacturers' datasheets. When instrument manufacturers provide data sheets with an instrument that they have made, the measurement uncertainty or inaccuracy value quoted in the data sheets is the best estimate that the manufacturer can give about the way that the instrument will perform when it’s new, used under specified conditions, and recalibrated at the recommended frequency. Therefore, this can only be a starting point in estimating the measurement accuracy that will be achieved when the instrument is actually used. Many sources of systematic error may apply in a particular measurement situation that are not included in the accuracy calculation in the manufacturer's data sheet, and careful quantification and analysis of all systematic errors are necessary, as described earlier.

Example 3:

Three separate sources of systematic error are identified in a measurement system and, after reducing the magnitude of these errors as much as possible, the magnitudes of the three errors are estimated to be System loading: _1.2%

Environmental changes: 0.8% -- Calibration error: 0.5%

Calculate the maximum possible total systematic error and the likely system error by the root-mean-square method.

Solution:

The maximum possible system error is _(1.2 + 0.8 + 0.5)% =_2.5% -- Applying the root-mean-square method, likely error =_

NEXT: Sources and Treatment of Random Errors

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