## 2019

MEASURE FOR MEASURE

# Standard Definition

## Getting to the bottom of measurement uncertainty

by Dilip Shah

In the last edition of this column (January 2009), I emphasized
the importance of documenting measurement uncertainty to establish metrological
traceability as it is defined in *ISO/IEC Guide 99:2007*. In this column, other *ISO/IEC Guide 99:2007* definitions pertaining to measurement uncertainty are discussed.

Measurement uncertainty is documented in an uncertainty budget, which is defined as a statement of measurement uncertainty, of the components of that measurement uncertainty, and of their calculation and combination.

Note: An uncertainty budget should include the measurement model, estimates and measurement uncertainties associated with the quantities in the measurement model, co-variances, type of applied probability density functions, degrees of freedom, type of evaluation of measurement uncertainty and any coverage factor.

On Sept. 30, 2008, the International Organization for Standardization
(ISO) published *ISO/IEC Guide 98-3:2008—Guide
to expression of uncertainty in measurement*,
which replaced *ISO/IEC Guide 98:1995* (the U.S. equivalent guide is *ANSI/NCSL
Z540-2-1997*).

To develop a measurement uncertainty budget, all known error sources contributing to the measurement process should be evaluated and estimated as either Type A or Type B evaluations of measurement uncertainty.

Type A evaluation is defined as the evaluation of a component of measurement uncertainty by a statistical analysis of measured quantity values obtained under defined measurement conditions.

Note 1: For various types of measurement conditions, see repeatability condition of measurement, intermediate precision condition of measurement and reproducibility condition of measurement.

Note 2: For information about statistical analysis, see *ISO/IEC Guide 98-3*.

Note 3: See also *ISO/IEC Guide 98-3:2008*, 2.3.2, ISO 5725, ISO 13528, ISO/TS 21748, ISO 21749.

Type B evaluation is defined as the evaluation of a component of measurement uncertainty determined by means other than a Type A evaluation of measurement uncertainty.

This includes evaluation based on information associated with authoritative published quantity values, associated with the quantity value of a certified reference material, obtained from a calibration certificate, about drift, obtained from the accuracy class of a verified measuring instrument or obtained from limits deduced through personal experience.

Note: See also *ISO/IEC Guide 98-3:2008*, 2.3.3.

The Type A uncertainty that is obtained by statistical analysis is expressed as a standard deviation and is called standard measurement uncertainty, standard uncertainty of measurement, standard uncertainty or measurement uncertainty expressed as a standard deviation.

Type B uncertainty estimates are also converted to standard uncertainty estimates by correction factors based on their estimated probability distributions.

Combined standard measurement uncertainty (or combined standard uncertainty) is defined as standard measurement uncertainty that is obtained using the individual standard measurement uncertainties associated with the input quantities in a measurement model.

Note: In case of correlations of input quantities in a measurement
model, co-variances also must be taken into account when calculating the
combined standard measurement uncertainty; see also *ISO/IEC Guide 98-3:2008*, 2.3.4.

When Type A and B uncertainty estimates are
quantified and expressed as standard uncertainty, they are normally combined
via the root sum square (RSS) method to derive the combined uncertainty. The combined
uncertainty is denoted by *u _{c}*.

Expanded measurement uncertainty (or expanded uncertainty) is defined as the product of a combined standard measurement uncertainty and a factor larger than one.

Note 1: The factor depends on the type of probability distribution of the output quantity in a measurement model and on the selected coverage probability.

Note 2: The term factor in this definition refers to a coverage factor.

Note 3: Expanded measurement uncertainty is referred to as "overall uncertainty" in paragraph 5 of Recommendation INC-1 (1980) and as "uncertainty" in IEC documents.

The combined uncertainty is normally multiplied by a coverage factor (*k*) to report
the expanded measurement uncertainty at approximately 95% confidence interval
level (*k *= 2).
The expanded uncertainty is denoted by:

*U* = *k* • *u*_{c}

Table 1 shows the values for the coverage factors for reporting expanded measurement uncertainty at different confidence interval levels.

Therefore, when a traceable measurement *x* is made, it is reported with its associated expanded measurement
uncertainty as:

*x* ± *U _{k}*

_{=2}

This means that the measurement *x* can be anywhere within the interval of:

*x* + *U _{k}*

_{=2}and

*x*–

*U*

_{k}_{=2}

at 95.45% confidence interval.

The error associated with this measurement is the uncertainty, *U _{k}*

_{=2}, stated at 95.45% confidence interval. This is shown graphically in Figure 1.

The documented uncertainty budget is maintained for future reference when making measurements using the same measurement process (equipment, environment, operator and any other associated components). The uncertainty budget is a live document and needs to be evaluated whenever a change in the process occurs. Typical triggers for evaluation are:

- When equipment is calibrated.
- When equipment is replaced by another instrument.
- When the operating environment is changed.
- When operator interaction is changed.
- When any significant change in the process is made.

The general process of documenting measurement uncertainty with its associated definitions was outlined in this column. In a future column, the process of quantifying Type A and B uncertainties and developing an uncertainty budget will be discussed in detail using a measurement example.

### Bibliography

- American
National Standards Institute,
*ANSI/NCSL Z540.2-1997 U.S. Guide to Expression of Uncertainty in Measurement*. - International
Bureau of Weights and Measures,
*Recommendation INC-1 (1980)—Expression of experimental uncertainties*. - International
Organization for Standardization,
*ISO/IEC Guide 99: 2007—International vocabulary of metrology—Basic and general concepts and associated terms*. - 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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