## 2019

MEASURE FOR MEASURE

# Balanced Budget

## A well-rounded approach to documenting measurement uncertainty

by Dilip Shah

In the last edition of this column (March 2009), I discussed Type-A and Type-B contributors of measurement uncertainty and what goes into a measurement budget. In this installment, I will outline a process for building that budget.

The process follows seven steps:

- Identify
the uncertainty contributors in the measurement process and classify the
uncertainty as Type A or Type B.
^{1} - Assign one of the four distribution types to uncertainty contributors (normal, rectangular, triangular or U-shaped).
- Convert the magnitude of the uncertainty contributor to standard uncertainty (see Table 1).

Typical uncertainty contributors that should be considered for any measurement process (calibration or test) are:

**Resolution
of the unit under test (UUT).**^{2} Resolution is classified
as Type B with a rectangular distribution. To convert resolution to standard
uncertainty, the smallest division or digit of the instrument is divided by the
square root of 12 if the information about the way the instrument resolves
itself (how the digit increments or decrements) is known. If no other
information is available, the resolution is divided by the square root of
three.

**The
uncertainty stated for the calibration of the standard used to calibrate the
UUT. **This standard is normally calibrated by an ISO 17025-accredited
laboratory (for traceable calibration) and is stated as an expanded uncertainty
at *k* = 2 with a
confidence interval of approximately 95%.

Because you normally would not know the
combined uncertainty of the standard, it is considered to be Type B. To convert
the expanded uncertainty to standard uncertainty, it is divided by the *k*-value on the
uncertainty report. If the *k*-value
is not provided, the uncertainty reported is treated as standard uncertainty.

**The
manufacturer’s specification for the accuracy of the instrument.**
Equipment manufacturers state specifications in different formats, so the user
has to interpret them correctly to apply the specifications to their measurement
uncertainty analysis.

The specification or tolerance is treated as Type-B uncertainty and is normally assigned a rectangular distribution. To convert the specification value to standard uncertainty, divide it by the square root of three.

**Personnel**.
Many test and calibration processes involve human interaction that can
influence the test and calibration results. If the repeatability analysis is
performed using statistical analysis, it is considered Type-A uncertainty and
is normally treated as Gaussian (normal) distribution.

If the repeatability or reproducibility
data is reported from published literature with no supporting data, it should
be considered as Type-B uncertainty with a rectangular distribution.^{3}
The variation in method is normally captured in the repeatability data. In some
cases, however, a separate analysis is required.

**The
environmental factors that have an impact on the measurement.**
Normally, one thinks of temperature (dimensional measurement) and relative
humidity. But there can be other factors, including vibration (mass
measurement), cleanliness (particle count), altitude and acceleration due to
gravity (mass measurement).

The temperature measurements are normally cyclical in nature, and the distribution attribute of the cyclical pattern is U-shaped. The U-shaped distribution is converted to standard uncertainty by dividing the magnitude of the contributor by the square root of 2.

If there isn’t sufficient information to
assign a distribution to an uncertainty contributor, ISO Guide 98:2008
recommends that you assign the most conservative distribution (rectangular
distribution).^{4}

Continuing the seven steps of the process:

- Document your findings in an uncertainty budget.
- Combine uncertainty using the root sum square method.
- Assign the appropriate k-factor multiplier to combined uncertainty to report expanded uncertainty.
- Document in an uncertainty report with the appropriate information.

It is good practice to document the uncertainty budgets using a spreadsheet template, which should be validated for calculations. One such example is shown in Figure 1. All spreadsheet cells with formulas should be protected from accidental alteration.

Measurement uncertainty budgets are estimates of the measurement error in a process and vary for the same piece of equipment or a process, depending on the laboratory’s operating environment. Thus, it is not a good idea to adopt someone else’s budget, as it may not be a true estimate of the user’s measurement error.

### References

- International Organization for
Standardization,
*ISO/IEC Guide 99: 2007—International vocabulary of metrology—Basic and general concepts and associated terms*. - Philip Stein, “All You Ever Wanted to Know About Resolution,”
*Quality Progress*, July 2001. - Philip Stein, “How to Write an Uncertainty Budget,”
*Quality Progress*, July 2003. - International Organization for
Standardization,
*ISO/IEC Guide 98-3: 2008—Guide to the expression of uncertainty in measurement*.

**Dilip Shah** is president of E = mc3 Solutions in Wadsworth, OH. He has more than 30 years of experience in metrology and applications of quality and statistics in metrology. He is a past chair of ASQ’s Measurement Quality Division and Akron-Canton Section 0810, and is co-author of The Metrology Handbook (ASQ Quality Press, 2004). Shah is an ASQ-certified quality engineer and calibration technician, and a senior member of ASQ.

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