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 A Short Guide to No measurement device produces perfect results. Uncertainty analysis is one way to define how confident you are of your measurements. Stephen Humpage, When you take a measurement and then state the uncertainty associated with it, you aren’t saying that the quantity you’re attempting to measure has no single, definite value (see the sidebar, “Glossary,”); you’re simply accepting the fact that no measuring device can produce perfect results. When you state measurement uncertainty, you’re reporting, in mathematical terms, how confident you are of the measurement. Consider the following example: The current flowing through a resistor is 16.73 mA. It’s not supposed to be a measurement. Rather it states what is actually happening in the real world. It’s a statement of fact (called the measurand). When you try to measure a quantity, you have no way of knowing the real value (the fact). The best that you can do is to use the most accurate equipment you have to interpret the value and then state how accurate you think the interpretation is. So, in the case of the example, you would report an assessment of the quantity as follows: The current flowing through a resistance was measured as being between 16.72 mA and 16.74 mA.
When you talk about a measurement error, you don’t include such things as: • Misreading instruments, such as analog gauges • Failing to apply known correction factors to raw data • Recording incorrect or misinterpreted experimental data These are simply mistakes that must be avoided. For uncertainty analysis to be of use, you must be totally honest. There is nothing to be gained in reporting an uncertainty that is an overoptimistic view of reality. If an uncertainty appears high, but is derived accurately, this is no imputation of either the equipment or the operator. Truthful analysis reveals the major contributors to the uncertainty. And this information can often lead to clues on how to refine methods and procedures to ensure that each contribution to the total uncertainty is minimized. The Relationship Between Repeatability and Uncertainty If a measurement is subject to random (and hence unpredictable) influences, there will always be a degree of uncertainty in the measurement. This will reveal itself if repeated observations of the same measurand are all different. The stronger the influence of random factors, the less repeatable your observations will become. By their very nature, you cannot correct for them; you can only attempt to quantify how strong their influence is. On the other hand, if your observations are subject to nonrandom (and hence predictable) influences, your readings will at least be repeatable. Furthermore, you can correct for their effects. For example:
• If for some unknown reason, a thermometer always underreads by 0.2°C, you can add this to the indicated value as an offset. Take great care, though, to ensure that the offset does not possess any drift or randomness. • A compass always points to magnetic north, whereas maps refer to true north. The difference between the two is called magnetic variation—it’s a nonrandom, known quantity. A navigator can therefore correct for variation and accurately plot a course between two points. From this information, you can conclude the following important points concerning randomicity, repeatability, and uncertainty: • If an observation is subject to random influences, there will be a lack of repeatability in the observation. • If a measurement is unrepeatable, there will be an uncertainty associated with that measurement. • Nonrandom influences do not by themselves produce measurement uncertainties. Put another way, repeat observations of the same measurand are not subject to measurement uncertainties if the measurement error is constant. Mean Value and Standard Uncertainty If you make repeated observations of the same measurand and each observation yields a different result, you need to find a method of expressing your conclusions as to the value of the measurand and how your conclusion relates to the spread of the observed values. Screen 1 shows a set of repeat observations of several calibration points, much like those that would be produced in many calibration procedures. The results derived by the calculator (indicated by the blue cells) are the mean value and the standard uncertainty. It also shows the nominal value. The mean value is the simple arithmetic average of the repeat observations for each calibration point. The standard uncertainty is the internationally accepted method of expressing the dispersion of the observed data about the mean value. For the mathematically minded, it is the standard deviation of the observations from the mean value. The important thing to grasp, though, is that the greater the value of the standard uncertainty, the less repeatable the observations are, and vice versa. Coverage Factor and Expanded Uncertainty
Measurement Error. The difference between the observed value and the measurand. Nominal Value. Each observation must have a nominal, or reference, value. This can, for example, be the calibrated dimension of a gauge-block or a temperature indicated by a UKAS calibrated thermometer. In each case, the value must be known to at least an order of magnitude higher that the accuracy required of the observations. Observed Data. The data must be genuine repeat observations. For example, in the case of a thermometer calibration, it is not acceptable to leave the thermometer in the temperature bath and record the value every minute or so. You must remove the device, wait, replace it, wait again for stability, and then make the observation. For a micrometer, the device must be closed onto the gauge block, a measurement recorded, and the jaws re-opened before repeating the procedure. Observed Value. The value that you measure as the interpretation of the measurand. Spread of Data. As the unrepeatability of the observations increases, so does the number of repetitions required to produce a meaningful sample. Statistical analysis can show us that, in a properly carried out repeat measurement: • ~68% of the observations will lie in the range: mean ±1 3 the standard uncertainty. • ~95% of the observations will lie in the range: mean ±2 3 the standard uncertainty. The values by which the standard uncertainties are multiplied are called coverage factors. In calibration work, they normally have the values of 1 and 2. Screen 2 shows the graphical output from the Caliso Uncertainty Cal culator, using the first calibration point in the previous example. Clearly seen are the data points, the mean line, and the uncertainty bands produced by coverage factors of 1 and 2. The calibration observations and the results of the calculations are stored in a Caliso Uncertainty File (*.CUF). The full filename of the .CUF file is stored in Caliso for each device. The Uncertainty Calculator can then be launched, with the appropriate data, by simply double clicking the filename. Summary When you make a measurement, you have no way of knowing how accurate the value is. All you can do is interpret the value and gauge the level of accuracy. To make uncertainty analysis as useful as possible, avoid taking an overoptimistic view of reality. By performing truthful analysis, you can uncover the major sources of the uncertainty. In doing so, you acquire information that can help you refine your procedures and minimize sources of error. Acknowledgment NIST Technical Document 1297—Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results. Stephen Humpage is cofounder, Caliso Software, 46 Rockspray Grove, Walnut Tree, Milton Keynes MK7 7EA, England, UK; 44-1908-675240, fax 44-1525-854056, support@calisouk.com. |
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