# How does measurement uncertainty combine with tolerances?

Given a tolerance within which your workpiece should be manufactured, say some length should be $10\pm1$mm. If you determine that your uncertainty in measuring this length is $0.2$mm (at 95%). How should a measurement of $9.1$mm be treated?

Clearly there is a significant probability that this value will actually be outside of tolerance. Do you need to decrease you tolerance range based on the uncertainty in your measurement?

• Should the tolerance be +/- 0.1 (currently says +/- 1)? In which case 9.9 with an uncertainty of 0.2 would be well within tolerance (min 9.7 max 10.1) easily between 9 and 11. Commented Jan 22, 2015 at 23:59
• @jhabbott Yeah obviously what I wrote doesn't make much sense. I think I meant to say how should a length of 9.1mm (or 10.9mm) be treated. If it was $\pm 0.1$mm your uncertainty would be bigger than the tolerance where you are obviously doing something wrong in you measurement setup. Commented Jan 23, 2015 at 9:47

You need to ensure that even in the worst case scenario, you still meet your measurement spec of $10 \pm 1\text{mm}$. If your tolerance is $0.2\text{mm}$ of your measurement, then a measurement of $11\text{mm}$, while may look like it meets spec, it doesn't because it could be $11.1\text{mm}$.

So the worst case that still meets your spec is a measurement of $10.9\text{mm}$, because then with a max tolerance of $0.2\text{mm}$, you still meet $11\text{mm}$.

With a $0.2\text{mm}$ tolerance, your $10 \pm 1\text{mm}$ spec becomes $10 \pm 0.9\text{mm}$.

How should a measurement of $9.9\text{mm}$ be treated?

So revised spec is between $9.1\text{mm}$ and $10.9\text{mm}$, so $9.9\text{mm}$ is within spec.

• I'd add to this that, while this is the right way to approach the inspection of the part, I think it's a bad idea to alter the drawing to fit this new tolerance, unless you never plan on supplying this print to anyone else. If you've got a +/-0.1mm uncertainty on your measurement, you adapt the drawing to this, and then give it to your supplier, who has a +/-0.2mm measurement uncertainty, they can measure the dimension and still be out of spec for the same reason. Leave this adjustment to the inspector, put the range you want for the part (not the measurement) on the print. Commented Jan 22, 2015 at 20:00
• This answer would be more complete if it addressed the stochastic nature of random uncertainties, and product reject rates. If a 5% reject rate is acceptable, then using the 95% uncertainty values to produce acceptable limits is great, but if a 5% reject rate puts you out of business, not as much. Commented Dec 5, 2015 at 19:27

There are two different aspects in your measurement. On the one hand, you are dealing with tolerances. On the other hand, you cover probabilities in measurement systems.

Just for a rough calculation: The probability of the real length to be within 9.8mm and 10mm is 95%. The certainty of this measurement depends on the distribution of your probability. For example, assuming a Gaussian Distribution (or any other symmetric distribution), your certainty is higher than the 95%. If you're lucky, you can get the certainty range for 99% or 99.5% percent as well from the supplier. Another option is to do a lot of measurements and find the certainty ranges on your own.

Measurement variations are very common and should be taken in to consideration when engineering systems. In most cases high precision equipment is available but might be cost prohibitive to justify purchasing for the project. Therefore, the goal of the engineer is to design the system to account for measurement variation. In this case the min and max limits are 9mm and 11mm with 10mm being nominal. There are few strategies that can be used. They are

• Define an upper and lower control measurement limit to account for maximum 0.2 mm variation. Therefore the LCL and UCL would be 9.2mm and 10.8mm. This will guarantee the work pieces are always $10\pm1$mm

• Another would be to perform a gauge R&R study to understand true measurement variation and include this data in the design. Make sure calibration is included in the preventive maintenance schedule.

• Using a Design for Six Sigma (DFSS) might be a better approach. Hopefully, the design is 6 sigma capable, after accounting for 0.2mm worst-case measurement variation. If so measurement variation might be insignificant.

In most case a combination of the above plus other strategies will be required to achieve a good design

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