Other Dimensions: Uncertainty Budget

July 30, 2007
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Editor’s note: This is the second in a three-part series on measurement uncertainty.

The previous column in this series gave an indication of what measurement uncertainty was and referred to an uncertainty budget as the tool used to provide a value for it.

An uncertainty budget is similar to a financial budget in the type of information it can provide. But unlike a financial budget where every element is in dollar terms, the uncertainty budget has to deal with various elements and convert them into linear terms, such as millionths of an inch or microns, when doing gage calibration.

The uncertainty budget is an organized way to present and mathematically process the effects on a measurement from a range of elements. No, you don’t need software or special computers to do the job. A simple $5 calculator will take care of the computing-in fact, I have one that cost $1 and it works just fine. What you must have is knowledge of metrology in order to determine what elements to consider and the magnitude of their effects.

In a perfect world you could do an analysis for each element that will affect the measurement and arrive at a distribution factor you can use. Unfortunately, we do not live in such a world, but we can borrow some information from those who have done such studies and apply it to our situation.

What we have to do is bring each of the budget elements to a standard uncertainty that is then squared to provide the variance. Each of these could be considered a line item with variances shown at the end of each line. The variances are added and the square root of that total is the combined standard uncertainty. That value is doubled to provide the expanded uncertainty with a coverage factor of K=2. In this case, it means that the true value of the dimension will be within ± the value calculated for the expanded uncertainty 95% of the time. If the measurement was 0.2345 inch and the expanded uncertainty was 0.0002 inch, the true value is somewhere between 0.2343 to 0.2347 inch.

To some degree every measuring situation is unique so it is difficult to give a simple list of elements to consider for an uncertainty budget in dimensional work. There are some that are fairly common such as temperature, instrument error and resolution, master variations and repeatability. Measuring force may or may not be significant. Contact deformation, measuring face flatness and parallelism of the instrument may also need attention, along with the errors of auxiliary devices such as thread measuring wires or surface plate flatness. The geometry of masters and other items also may have to be taken into account.

Considerable attention has to be paid to the state of the item being measured. By that I mean you cannot declare the size of something if all you’ve done is take one or two diameter measurements. Errors in roundness undetected by diametrical measurements are part of the size of the item and should be accounted for in the budget.

The tolerances involved dictate whether or not all of these elements need to be included, but typically the finer the tolerances, the greater the number of elements that need to be considered. If you are interested in uncertainty in terms of the calibration of gages and instruments, the American Measuring Tool Manufacturers Association’s booklet “Searching for Zero” lists elements to consider for that level of work. You can get more information from their Web site, www.amtma.com.

One rarely mentioned benefit of an uncertainty budget is that it gives you an opportunity to evaluate the measurement process and equipment to make smart decisions if changes are needed. In the everyday world people tend to leap to finer resolution on equipment to improve their process. This may be the answer, but often it is not. If you have an uncertainty budget, the element showing the largest variance is what will need changing and that may be temperature, not resolution. Spending money for more digits means you’ll be no further ahead and the extra digits will be dancing around a lot more.

In my next column I’ll look at using measurement uncertainty to resolve measurement disputes and how it should be applied.

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