# How to Calculate Test Uncertainty Ratio Test Uncertainty Ratio or TUR is a common term used in calibration. It is the ratio of the tolerance or specification of the test measurement in relation to the uncertainty in measurement results. It is used to evaluate measurement risk and validate the suitability of calibration methods. The most common requirement for many calibrations is a 4:1 TUR. However, not all calibrations meet a 4:1 TUR. Let’s learn how to calculate TUR, so you can evaluate calibration results like a pro.

To calculate the Test Uncertainty Ratio, we must know the value of:
•    The tolerance or specification; and
•    The uncertainty in measurement.

Once this information is known, we can use the following equation to calculate TUR. Now that we have seen the TUR equation, let’s put it use with the following example.

Example:
Imagine you are calibrating a digital multimeter at 10-volts. To conform to specifications, the digital multimeter must measure the sourced voltage between 9.995 and 10.005 volts. The estimated uncertainty of the measurement result is 0.0012 volts. Calculate the Test Uncertainty Ratio. The Test Uncertainty Ratio for the measurement result in our example is 4.2:1. According to most Metrologists, this would yield comparative measurement results with a high degree of confidence for determining conformance or non-conformance to specification.

Now that you know the equation and how to use it, plug it into your excel spreadsheets during your next calibration to calculate TUR. ## =ROUND((F2-E2)/(2*D2),1)&”:1″

If your TUR is 4:1 or greater, Awesome! If not, then you may want to evaluate your results further to determine your measurement risk.

FREE TUR CALCULATOR: Calculate TUR quickly using MS Excel.

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### Richard Hogan

Richard Hogan is the CEO of ISO Budgets, L.L.C., a U.S.-based consulting and data analysis firm. Services include measurement consulting, data analysis, uncertainty budgets, and control charts. Richard is a systems engineer who has laboratory management and quality control experience in the Metrology industry. He specializes in uncertainty analysis, industrial statistics, and process optimization. Richard holds a Masters degree in Engineering from Old Dominion University in Norfolk, VA. Connect with Richard on Google+ and LinkedIn.

1. Richard,

I did calibration for 22 years in the Navy and now I am a Lab Manager for a civilian company. For the last few months I have been giving myself a crash course in CMC’s, Uncertainty Budgets, TUR’s, and TAR’s. The question I have will involve all of the above for a 0-1″ outside micrometer.

My Uncertainty Budget looks a little like this (after distribution):

Component Std Uncert
Uncert of Standard 3µ”
Resolution 28.9µ”
Repeatability 0 <~Due to course measurement
Uncompensated Error 1.15µ"
Temp diff 2.37µ"
Temp variance from 68 0.853µ"

Combined Std Uncert: 29.2µ"
Exp Uncert (k=2) 58.4µ"

OK, now that I have that out of the way… Let's say the micrometer has a resolution of .0001" (100µ") and it's tolerance is also ±.0001" (which is the case 95% of the time), how do I achieve a 4:1 TUR? When I calculate it I get about 1.7:1, or do I not use the entire expanded uncertainty in my budget to calculate TUR? I feel that I am doing something wrong since no one would be able to achieve a 4:1 or better since the resolution of the unit under test is always going to impede it. TAR is easy, 6µ" vs 100µ" = almost 17:1. I just want to make sure that I am calculating the TUR correctly. Any help would be greatly appreciated. Please let me know if you need any additional info.

-Mat

• Hi Matt,

Unfortunately, some calibrations will not achieve a 4:1 TUR when you use the calibration uncertainty to calculate TUR instead of the process or CMC uncertainty. The good thing is that you are not doing anything wrong. Manufacturer’s of mechanical and dimensional devices have been known (for years) to claim accuracy specifications that are close to the resolution of the instrument. In your case, you may be able to evaluate resolution uncertainty as 0.5R, or one-half the resolution. This practice is typically acceptable for mechanical and dimensional devices and may increase your TUR slightly.

If you have any additional questions, please feel free to email me at rhogan@isobudgets.com. I will be glad to help you.

2. Uncertainty
1. Type A
2. Type B
C A R= Calibration, Accuracy, Resolution
Comb Uncertainty
Expanded Uncertainty

3. but if i calculate TUR in this way how mutch it is wrong ..??

Tur=Tol/Ue

Tol= tollerance given by the method or laboratory
Ue = that is the uncertanty of measurement that i have calculated related to the thing i am calibrating.

• Hi Stefano,

Most the time, your calculation will not be wrong. The only time that it will be wrong is when you analyze a tolerance that is asymmetrical, like gauge blocks where the maximum positive tolerance may +2µin and the maximum negative tolerance is -4µin. In this case, you will want to consider the full tolerance range (i.e. upper limit – lower limit) divided by the twice the expanded uncertainty. I hope this helps.

4. i forgot to say that my lab is takeing in cout only the tools uncertanty.
the method uncertanty and as well the variability of the measurment are not taken in count

5. HI,
I have question..
The measurement uncertainty includes effect of resolution of UUT.
So my UC for Torque is 0.6 %+0.6R
I have dial torque wrench of Proto 6121A, 0-175 Ft-Lbf. It has resolution of 5 Ft-Lb
This makes my cal Uncertainty at 50 ft-Lbf =0.3+3 =3.3 Ft-Lb. I can never match TUR 0f 4:1 even though my cal instrument is 40 times better due to 0.6 R value of UUT.
Same things normally happens with Analog pressure gauges where accuracy is significantly lower then resolution.

Any comment or opinion or solution for this pl

Yes, this is becoming more common today since TUR is supposed be calculated using the calibration uncertainty and not the CMC uncertainty. See number 2 in section T(5) of the A2LA P102: https://www.a2la.org/policies/A2LA_P102.pdf

I find that TUR is commonly less than 4 to 1 with most analog scale measurement instruments and digital display measurement equipment where the accuracy is almost equal to the resolution. It does not mean that your calibration process is substandard. It is almost unavoidable because it is based on the design of the unit under test.

I hope this helps.

6. Hi, I have a question.
Why do we have 2 formulas to calculate TUR? (uls-lsl/2u | tol/u)
What’s the difference between them? And, why do you divide by the twice the expanded uncertainty?

• Hi Samara,

There are two equations that you can use depending on the application of analysis.

“Tolerance / Uncertainty” is an easy calculation of TUR when the tolerance in symmetrical; equal in both directions. Example: ±1µm

“(USL – LSL) / (2U)” is the calculation of TUR that you would use when your tolerance is asymmetrical; not equal in both directions. Example: +4µm, -2µm

I hope this helps.

Rick

• Hi
If we take the formula ” Tolerance / Uncertainty” should we multiply the tolerance by 2 ?
I mean if the tolerance is ±1µm and expanded uncertainty is
0.5 µm;
Then TUR=1/0.5 or (1*2)/0.5.

Please tell me which one is correct.

• Hi Charan,

For your example, TUR = 1/0.5 = 2.0:1

You only need to multiply the uncertainty by two if your tolerance is listed as the full range (i.e. Upper Limit – Lower Limit).

I hope this helps.

Rick

7. Hi, I have to calibrate a digital multimeter and I need some help about TUR. Hope you can help me, here’s what I’ve done:

Digital multimeter (pattern): Accuracy 0,5%
Digital multimeter (UUT): Accuracy 0,75%

I have made 10 measurements for 10VDC as parameter. From this I have calculated u1 as uncertainity Type B (using pattern’s accuracy 0,5*10VDC/sqrt(3)) and u2 as standard deviation of my measurements (Type A), then:

combined uncertainity = sqrt(u1^2+u2^2) = uc
expanded uncertainty = 2*uc

So here’s my question, How do I calculate TUR?

8. Hi Richard,

In your formula you define u as expanded uncertainty but the denominator is 2u.

• Hi Chris,

Yes. Maybe I should have used 2U. My apologies for the confusion.

In the equation in this article, I recommend that you use the full span of the tolerance (i.e. USL minus LSL). In most cases, this will be twice the tolerance. So, you would need to divide by twice the expanded uncertainty.

Most people just divide the tolerance by the uncertainty. This method works as long as your tolerance band is symmetrical (which it will be in most cases). However, every once in a while you will encounter an asymmetrical tolerance and you will need to use the equation that I listed in this article to avoid errors.

I hope this helps.

Best regards,

Rick