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

**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

**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

**Setting up the Study**

**Viewing the Preliminary Uncertainty Budget**

Click the **Calculate** button and the software generates a bias

**What does an uncertainty budget do?**

An uncertainty budget makes a list of all the relevant sources of variation (uncertainty), and for each

**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

**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:**

**Uncertainty Contribution:** Amount of variation, after

**Df:**

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

**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.

**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 (

**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

The following example doesn’t relate to Figure 4. It illustrates how to find expanded uncertainty for your reference

**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.

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:

**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.

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:

Then the table entries would look like this:

**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

**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 AIAG.org. 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 BIPM.org. It can be read online or downloaded free. It is also available as an ISO document.

**Credits**

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

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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