Understanding and Expressing Measurement Uncertainty
associated with DC and Low Frequency Metrology
Edward Brown
Quality Systems Lab
Boca Raton, FL 33487

Introduction
Periodic calibration of electronic systems isrequired due to the large number and wide
variety of components in basic measurement
and sourcing systems. In metrology today acalibration laboratory needs to understand how
to quantify all the influences and componentsof measurement uncertainty as they pertain to
the laboratory’s environment.
These
uncertainty influences and components are as
varied as the cables used in a measurement
system to the training level of the technicians
tasked with operating this system. Consistent
results and confidence in the reported values ofa measurement can be achieved with due
consideration to all the contributors ofuncertainty. This paper is intended to help thelaboratory professional begin evaluation of thepossible sources of uncertainty and how toformulate a measurement uncertainty budget.
Common terms:
Accuracy is a qualitative expression of thecloseness of a measurement’s results to the truevalue.
Precision is a measure of repeatability. A highprecision indicates the ability to repeat
measurements within narrow limits.
Resolution is the smallest change that can bedetected.
Generally today with modern
instruments this is the smallest increment thatcan be displayed or LSD.
Uncertainty is a quantitative term that
represents a range of values wherein the true
value may lie. Uncertainty and confidence isdetermined using statistical techniques.

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Traceability is the ability to relate individual
measurement results to national standards ornationally accepted measurement systems
through an unbroken chain of comparisons.
Requirements for sound analysis:
Stable Environment; before a laboratory canbegin to evaluate the components ofuncertainty in a measurement or calibration
system, data must be collected to determine if
the system is stable. This data may be as
simple as monitoring the ambient environment
using a temperature and RH logger or ascomplex as repetitive measurement schemes ofthe equipment under evaluation.
Proper Training of Personnel; all personnel
tasked with performing measurements toassimilation of collected data should beproperly trained and evaluated on theirunderstanding of the tasks assigned to them.
Traceable Standards; all standards used in an
uncertainty-testing scheme must be traceable
for the results to be meaningful.
Uncertainty considerations; all possible
sources of uncertainty should be consideredfrom AC line voltage fluctuations to theresolution of the measurement system. A
source of uncertainty such as cable EMF may
be discarded after determining that theuncertainty is insignificant.
It may be
appropriate in some test schemes to combine
all of the insignificant uncertainties and createa label for this combined uncertainty.
Instrument specifications are the most common
source of uncertainty data, however properconsideration must be given to themanufacturer’s stated confidence level. If themanufacturer did not specify a confidence
level, then a rectangular distribution should beassumed, more on distributions later.
Sources of Uncertainty
Uncertainty in the results of a measurement
can be affected by many factors, someconsiderations:
Reference standards and measurement
equipment: Uncertainty in their calibration;
long term drift; resolution; vibration; electro
magnetic interference; sensitivity to change
during transportation and handling.
Measurement Setup: cables; shielding; warmup time; thermal voltage influences;measurement probes.
Measurement Process: duration of themeasurement; number of measurements;
conditioning of standards.
Environmental Conditions: temperature;temperatureoscillations;humidity;electromagnetic influences; transients in powersource.
Measurement errors
A measurement is subject to many sources oferror, some of which can cause an over orunder statement of the measurement quantity.
While the goal of any metrology lab is to keepthese errors small, they cannot be reduced tozero. The challenge for any metrology lab is tofind out the quantity of these errors and how
large they may be. Measurements are affectedby three types of errors; Random, Systematic,and Gross.
Random errors are due to unknown causes andare
only
detectablewhen
repeatedmeasurements are made with a stablemeasurement
setupandconsistentmeasurement technique. This type of error will
result in readings that, when repeated, are notalways the same. If the reason for the variation
is not obvious, then it falls into the category ofa random error. Note: Random errors cannot
be quantified without a stable environment andconsistent measurement technique.

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Systematic errors relate to the equipment used in
the measurement process or external influenceson the equipment. Examples include: loading
effects, thermals, drift-rate, leakage currents, andexternal noise.
Gross errors are caused by the technician andcan be strictly controlled with proper training.
Examples include: misreading of instrument
results, incorrect adjustments, using the wrong
instrument, errors in recording calibration data,and computational errors. All of these errors canbe avoided with proper training and attention todetail.
Classifications of uncertainty
Type “A” evaluation method; the method ofevaluation of uncertainty of measurement by thestatistical analysis of a series of measurements.
An example would the standard deviation of aseries of measurements taken by a laboratory
technician.
Type “B” evaluation method; the method ofevaluation of uncertainty of measurement bymeans other than the statistical analysis of aseries of observations. An example would be themanufacturer’s published specifications for aninstrument.
Methods of determining uncertaintyPublished specifications; as mentioned earlier
published specifications are the most common
source of uncertainty data used by commercial
calibration laboratories. This method is the mostappropriate for laboratories that take only
“simple measurements”. Simple measurements
can be defined as any measurement that is within
the common functional capabilities of aninstrument. This method of determination isconsidered a Type “B” uncertainty.
Statistical methods; this method requires thetaking of a series of measurements over a
specified length of time. This method is themost robust and is appropriate for any laboratorythat requires high confidence in theirmeasurement uncertainty statements. This is a
Type “A” uncertainty.

Distributions Associated with Measurement
Uncertainty
The last piece of information that is needed todetermine standard uncertainty is thedistribution of the Type B uncertainty. There
are four types of distribution:• Normal distribution• Rectangular distribution• Triangular distribution• U-shaped distribution
Normal distribution is usually associated withType A uncertainty and has a divisor of one.
Rectangular distribution is used where there isan equal probability of a measurementoccurring within the bound limits. This type ofdistribution is normally associated withmanufacturer specifications.
The GUMsuggests assuming the rectangular distribution
when the frequency distribution is not known.
Rectangular distribution assumption will allow
the laboratory to err on the conservative side.
To convert a Type B specification with arectangular distribution, divide the stateduncertainty by the square root of 3 to arrive at
the standard uncertainty.
Triangular and U-shaped distributions will
not be discussed in this paper, the assumptionof these distribution classifications requires asound understanding of statistical techniquesthat is beyond this paper’s intent.
Uncertainty Budget
In the process of defining an uncertainty
budget all of the most important contributors touncertainty in the measurement must beconsidered. Once all of the contributors aredefined they need to be normalized to standard
uncertainty. The GUM provides the following
correction factors for non-normal distributions.
Distribution
Rectangular
Triangular
U-Shaped

Divide by
Square root of 3
Square root of 6
Square root of 2

Divisor1.73212.44951.4142

An example of this conversion; amanufacturer’s accuracy specification for amultimeter at 100 volts is ±.5 volts. To
convert this rectangular distribution to aPage 3 of 4

standard uncertainty, divide .5 volts by thesquare root of 3 (1.7321) for a normalized
uncertainty of .2886.
Combining Uncertainties
Once all contributors in an uncertainty budget
have been converted to a standard uncertainty thestandard uncertainties must then be converted toa unified unit of measure. The final step in any
uncertainty budget is the combining ofuncertainties.
The process of combining
uncertainties is called Root Sum Square or RSS.
This means that each of the standard
uncertainties is squared before adding all of thesquared components together. The square rootof the result is taken as the total combined
standard uncertainty.
Uncertainty Formula:U=

S12+ S22+ S32...

The expanded uncertainty is obtained bymultiplying the resulting value U by a factor oftwo (k=2), approximating a 95% confidence
level. This expanded uncertainty is shown in theuncertainty statement of a scope of accreditation.
Reporting
It is important that all of the steps used to arrive
at a final uncertainty value be documented in a
final report. This document can be as simple as atable outlining all of the considerations and thereasoning behind their consideration or removal.
This report should be reviewed periodically by
the laboratory management and updated asmeasurement equipment, personnel and orprocedures change.
Creating a budget
1. List all possible contributors.
2. Decide on the Uncertainty Unit of measure.
3. Define the magnitude of the uncertainty
contributorsand
theirprobability
distributions.
4. Convert to standard uncertainty using theappropriate divisor and combine using RSS.
5. Document the basis for your estimates.
6. Review your uncertainty tables regularly.

Uncertainty Table for DC voltage standard
Uncertainty
Uncertainty Probability
TypeDivisor
Contributors @ 1 VDC(mV)Distribution
1
Repeatability.0002
NormalA
1
2
Long Term Drift.00001
NormalA
1

Standard
Uncertainty.0002.00001345.0002309.0000057.0000057

Specification
Thermal Stability
EMF, Cables.0004.00001.00001

Rectangular
Rectangular
Rectangular

B
B
B

1.7321
1.7321
1.7321

Combined Standard Uncertainty (RSS method)
Expanded Uncertainty (k=2).000305745.000611489

Notes:
1 – Repeatability was determined using an HP 3458A multimeter and taking readings twice a day for one week.
2 – Long term drift was determined using data from two years of intermediate testing.
3 – Specification was taken from the manufacturer’s specifications
4 – Thermal stability was taken from the manufacturer’s specifications
5 – EMF, Cable specification was taken from the manufacturer’s specifications

Table 1.0

Rounding
Rounding should only be performed once all
calculations have been completed. As seen inTable 1.0 the expanded uncertainty is reflected
to 9 places after the decimal. This number can
now be rounded to a more practical value
before being expressed on a scope ofaccreditation. The final expanded uncertainty
should reflect the resolution of the laboratory’s
indicating device, plus one decimal place if
desired. In the example above the rounded
expression would be, ± .000612 mV DC.
Summary
There are many factors to take into
consideration when expressing total andexpanded uncertainties in DC and Low
frequency metrology. This paper is intended togive the new technician some insight into thecomplexity of this discipline and start theprocess of critical thinking. This paper does
not, by any means, cover the diversity andcomplexity of calibrations in today’s metrology
laboratory.

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For those desiring more extensive information
websites such as NCSLI.org or NIST.gov aregood places to start.

References
ANSI/NCSL Z540-2-1997, U.S Guide to theexpression of uncertainty in measurement.
Boulder, CO: NCSL International (GUM)
Fluke Corporation. 1994.
Calibration
Philosophy in Practice, second edition.
Everett, WA: Fluke Corporation.