All manufacturing companies that get audited require some or all of their calibration certificates to specify the calibration uncertainty. At a minimum, some manufacturers only need certified uncertainty for gages that are reference standards, which are used to calibrate other gages. Those companies would usually send their reference standards to an outside calibration source for certification. Often it would be convenient, however, to know how to certify your own gages. For example, you might need to calibrate some pin gages, but you don’t have a certified micrometer, because you don’t send micrometers out. For those companies, as well as companies that have to certify all of their calibrations, this article uses popular software to show the steps required to do your own uncertainty certifications.


The Basics (Preliminary Study)

The first step would be to measure repeatability on a reference standard and calculate the difference between the average measurement and the reference value. This difference is called “bias.” The study is usually called a “bias study.” Sometimes, you may see it referred to as a “repeatability study” or a “type 1 study.”

A more elaborate form of this study, called a “linearity study,” is seldom used and is not included in this article. The linearity study should be used when you have reason to suspect that bias might be significantly different for a large reference standard, compared to a small reference standard. A linearity study would use three or more reference standards.


Who does the bias study?

The bias study is often done by a calibration technician or a lab technician; however, it is important to note that the measurement technique used for the preliminary bias study is completely different from the technique used for calibration.


What do they have to know?

Because we will always know what number to expect when measuring a reference standard, it is often easy to make some or all of the variation go away. Calibration technicians become especially good at this. Since a main purpose of our preliminary study is to measure repeatability variation on a reference standard, technicians will have to modify their technique. They do this by making sure they can’t see the readout until they have finished aligning the part and gage. This is surprisingly difficult to do. Some technicians may find they like to cover the readout with a sticky note until they get the hang of it. Besides learning the measurement technique, they need to be introduced to the tool that lists and combines all the sources of uncertainty. This tool is a “form” called an “uncertainty budget.” This article will show how to use it.


Setting up the Study

Typical sample size is 25 repeated measurements on a reference standard, and official recommendations vary from 15 to 30. The study uses one operator and one reference standard or calibrated part. Once we have the measurements, we would open a GAGEtrak Calibration Management Software module called Linearity, Bias and Uncertainty, select study type Bias Only and enter a reference value and the measurements. See Figure 1.


Measurements for Bias Study


Viewing the Preliminary Uncertainty Budget

Click the Calculate button and the software generates a bias analysis, and offers to set up a preliminary uncertainty budget. We will introduce the uncertainty budget first, and look at the bias analysis later. Figure 2 shows the uncertainty budget. It is called preliminary, because it is not yet complete.


Preliminary Uncertainty Budget


What does an uncertainty budget do?

An uncertainty budget makes a list of all the relevant sources of variation (uncertainty), and for each source it provides information the software will need to calculate the “uncertainty contribution.” It then adds up all the contributions by a special method called Root Sum Square or RSS, to get “combined uncertainty.” Finally, it multiplies combined uncertainty by a “coverage factor” (usually 2) to get “expanded uncertainty.” Expanded uncertainty is the number we need to put on a calibration certificate, when a certificate is required.


Description of Uncertainty Budget Columns

Uncertainty Contributors: Sources of variation.

Type: “A” means calculated by a current study. “B” means determined by means other than calculation, or not current.

Plus or minus: The amount of variation, expressed as a standard deviation, or as an extreme value, whichever is available.

Probability Distribution: The software must assume a distribution “shape” in order to convert all plus-or-minus values to a common basis.

Divisor: The software will select a divisor based on distribution shape.

Sensitivity Coefficient: Optional column that can be used to insert a multiplier.

Uncertainty Contribution: Amount of variation, after all plus-or-minus values are converted to a common basis called “standard uncertainty.”

Df: Optional column for degrees of freedom. Df is a modified sample size, which is different than actual sample size, for statistical purposes.


Where do the plus-or-minus values come from?


Basics of Gage Uncertainty - Bias Analysis

Linearity: Blank, because we assumed it was negligible.

Bias: Labeled Standard Error in Figure 3.

Resolution: Looked up in a database, or observed.

Repeatability: Labeled Standard Dev in Figure 3.

Ref. Standard: Will be looked up in a calibration certificate.


Finished Version of Uncertainty Budget

The finished version includes the following changes, compared to Figure 2. The changes were done just by simply editing Figure 2.


Basics of Gage Uncertainty - Finished Uncertainty Budget

Required Change: Reference Standard (Done)
Add an item for the uncertainty of the reference standard. Refer to the certificate to get the plus-or-minus value. Assume it had been multiplied by 2, unless otherwise specified, and choose probability distribution Normal (2). You can also include degrees of freedom (df), if available.

Recommended Change: Resolution (Done)
Current practice is to use half of this item; do that by entering a 0.5 multiplier in the sensitivity coefficient column. Change type to B.

Potential Improvements: (Optional)

Linearity: (Done–type B only)
Consider adding an allowance for linearity, but only if recommended by the gage manufacturer or by your procedures. Change type to B.

Bias: (Done)
Consider substituting Bias instead of Standard Error for the bias plus-or-minus value. See Figure 3. This change would require changing probability distribution to rectangular. This change would allow treating bias as a random variable instead of a systematic error.

Part Characteristic: (Not done)
Consider making a copy of the general uncertainty budget and customizing it for a specific part characteristic to address tolerance and part geometry issues, etc.


Preparing Calibration Certificates

Calibration certificates that relate to this uncertainty budget should list the expanded uncertainty, and preferably, the coverage factor k and degrees of freedom (df). If you will be doing calibrations with the software we used here, there is a button to automate linking to the relevant uncertainty budget.


How to Read an Outsourced Uncertainty Certificate

We will assume all of your outside calibration sources follow the Guide to the Expression of Uncertainty in Measurement (GUM) – see next section. However, this document leaves plenty of room for individuality. If you have multiple sources for outside calibration, you will see multiple ways of presenting the information. In general, they will not give you a copy of their uncertainty budget, and they may not give you a complete uncertainty calculation, but they will give you information you can use to figure out an entry for you own uncertainty budget.

The following example doesn’t relate to Figure 4. It illustrates how to find expanded uncertainty for your reference standard, when your calibrator doesn’t give you exactly what you want.

Example 1, Calibration Certificate for 85-Piece Gage Block Set, in Inches
This certificate provides three groups of information (author’s comments are in parentheses):

a. Tolerance: Plus or minus 50 microinches. (This could be converted to a complete uncertainty estimate, but they haven’t done that for us.)

b.  “Calibration uncertainty, including coverage factor k = 2: 3 microinches (0 – 1 inch), 2.0 + 1.5L microinches (2 – 4 inches) where L is gage block length in inches.” (This is a partial uncertainty estimate that doesn’t include any allowance for gage block deviations from nominal [bias].)

c. A list of measured deviations (bias) for each of the gage blocks. The maximum absolute value of the deviations is 33 microinches. (This could be converted to the missing part of the partial uncertainty estimate, but they haven’t done that for us.)

By coincidence, there are also three different ways to get the uncertainty for the reference standard.

1. Maximum permissible error method (MPE). Enter information from group “a” as follows.

Basics of Gage Uncertainty - Table 1

Note that we did not convert the tolerance to uncertainty before entering the information; the uncertainty budget will take care of that for us. If you would prefer to convert it before entry and perhaps write it on the certificate, use a calculator with the formula: 50 ÷ 1.732 × 2 = 57.7. If you would prefer not to use a calculator, there are other options. GAGEtrak has a place to calculate formulas and store them for future use. You could also get the answer by making a temporary entry in a blank uncertainty budget. If you convert before entry the table entries would look like this:

Basics of Gage Uncertainty - Table 2

2. Combining information from groups b and c. The largest uncertainty from group b occurs when “L” = 4 inches. Then expanded uncertainty = 2.0 + 1.5 × 4 = 8 microinches. From group c, the largest bias is 33 microinches. As is, these entries would require 2 separate rows.

Basics of Gage Uncertainty - Table 3

For those who would prefer to convert before entry, use 33 ÷ 1.732 × 2 = 38.1. If you would also want to combine the two expanded uncertainties to a single row, use:

Basics of Gage Uncertainty - Formula 1

Then the table entries would look like this:

Basics of Gage Uncertainty - Table 4

3. For special situations, when you need the smallest uncertainty you can get. Use the uncertainty for an individual gage block, rather than the uncertainty for the whole set. For example, suppose we want to use the 1 inch gage block, and we are willing to use the corrected value. From the group c list (not shown) the corrected value would be 0.999967, and from group b, the expanded uncertainty for the corrected value would be 3 microinches. Then the table entries would look like this:

Basics of Gage Uncertainty - Table 5


Where to Get More Information

The following manuals are available in multiple languages.

For information about the studies used to generate the numbers, see Measurement Systems Analysis 4th Edition, published by This manual does not cover uncertainty budgets.

The automotive industry manual that does cover uncertainty budgets is published by the German Association of the Automotive Industry as VDA Volume 5, Capability of Measurement Processes 2011. Examples are included in the manual. The example in this article is based on VDA 5, Annex F.6, on pages 134-135.

More detail on basic uncertainty budgets and gage capability formulas is available in ISO 22514-7 Capability of Measurement Processes 2012.

Guide to the Expression of Uncertainty in Measurement (GUM) (2008) is published by It can be read online or downloaded free. It is also available as an ISO document.


Software used for this article is GAGEtrak Calibration Management Software furnished by CyberMetrics, Phoenix, Arizona.


About The Author
Gary Phillips has been in the quality field for nearly 50 years. Previously with GM’s Cadillac division, Gary has now been a consultant for over 30 years and has trained well over 20,000 people worldwide, primarily in technical subjects related to quality and reliability engineering, such as designed experiments, engineering testing, statistical process control and measurement systems analysis.


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