Fluke Digital Multimeter Calibration Manual

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1

Applying Measurement Uncertainty To

Digital Multimeter Calibration

An introductory study of measurement

uncertainty and its application to digital

multimeter calibration

Teleconference:

US & Canada Toll Free Dial-In Number: 1-[phone]

International Dial-In Number:[phone]

Conference Code: 1010759559

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 2

Welcome

Greetings from –
Fluke Corporation

Everett, Washington, USA
We are very pleased to bring you this

presentation on measurement

uncertainty for DMM Calibration.

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 3

Welcome

This presentation is based on Fluke’s

extensive experience with:

− Use and design of calibration
Instruments

− Our experience and understanding of the
problems faced when applying
measurement uncertainty for both
regular and accredited metrology

Thanks for your time, we hope you find it
both valuable and useful.

Welcome and Thanks!

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 4

Presented by

Fluke’s Calibration Business Unit

and Jack Somppi

Electrical Calibration Instruments
Product Line Manager

[e-mail]

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 5

Web seminar etiquette

• Choice of Audio – VOIP or Teleconference

− VOIP receives audio only while teleconference is two way
sound

• Don’t mute your phone if you have background

music enabled

• Use Q&A or chat to send me questions or request
clarification

• There will be an opportunity throughout the

discussion to pause and ask questions.

• You can view the material using either full screen
or multi window methods

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 6

Applying Measurement Uncertainty To

Digital Multimeter Calibration

An introductory study of measurement

uncertainty and its application to digital

multimeter calibration

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 7

Objectives

In this session you will -

• Be introduced to the concept of measurement

uncertainty and why it is important

• Observe the basic elements that influence

measurement uncertainty for DMM calibration
applications

• Study a simple but detailed example of calculating

measurement uncertainty

• Consider some benefits of automating measurement

uncertainty calculations

• Receive a variety of references for further research on

this topic

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8

Benefits

• Introduce measurement uncertainty to

those in calibration/metrology who are
not familiar with it

• Understand why measurement

uncertainty is important for quality
metrology

• Understand measurement uncertainty

with respect to DMM calibration

• Appreciate to the benefits of automation

• Have technical references for more

detailed information

• Obtain copies of this presentation via

email

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9

Measurement Uncertainty

& Why It Is Important

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10

Facts regarding measurement -

• Can you ever measure the true value of
something?

− No, there will always be errors

• How important is this fact?

− Very important, as measurement is never complete
unless you know how good it is!

• How is this taken into account in today’s

calibration & metrology?

− By applying & documenting the measurement uncertainty
process to the tests being done

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11

Measurement uncertainty in

metrology today…

Measurement errors were not rigorously evaluated in all
cases. Often in industrial labs, accuracy ratio analysis

(referred to as TUR’s or TAR’s or TSR’s) had been frequently

used to evaluate the significance of the calibrator’s errors on

the measurements. Other errors were sometimes ignored.

Individually analyzed, calculated, & documented measurement

uncertainties are more thorough and are required to be

considered - as stated in

− ANSI/ISO/IEC 17025:2005 General Requirements for the
Competence of Testing and Calibration Laboratories

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12

ISO 17025 –

about measurement uncertainty…

5.4.6 Estimation of uncertainty of
measurement

− 5.4.6.1 A calibration laboratory, or a testing

laboratory performing its own calibrations, shall
have and shall apply a procedure to estimate the

uncertainty of measurement for all calibrations

and types of calibrations.

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13

… about the sources of

uncertainty…

ISO 17025, Section 5.4.6.3:

− NOTE 1: Sources contributing to the uncertainty

include, but are not necessarily limited to,

• The reference standards and reference

materials used

• Methods and equipment used

• Environmental conditions

• Properties and condition of the item being

tested or calibrated

• Operator

There are many contributors to uncertainty

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14

ISO 17025, Section 5.10.4

Calibration Certificates shall include …
for the interpretation of calibration results

a. The conditions of the test
b. The uncertainty of measurement &

compliance statements to metrological standards
c. Evidence of traceability
When statements of compliance are made, the

uncertainty of measurement shall be taken into account

…about calibration certificates…

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15

An example of an accredited

calibration certificate –

“Measurement uncertainties at the

time of test are given in the following

pages, where applicable. They are
calculated in accordance with the
method described in NIST TN1297,

for a confidence level of 95% using a

coverage factor of approximately 2
(K=2).”

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of

measurement uncertainty….

From the NPL UK - “A Beginner's Guide to

Uncertainty of Measurement”

• Uncertainty of a measurement tells us something about

its quality

• Uncertainty of measurement is the doubt that exists

about the results of any measurement

• For every measurement – even the most careful – there

is always a margin of doubt

• You need to know the uncertainty before you can

decide whether the tolerance is met

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

“How is this Measurement Uncertainty

obtained?”

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement

Uncertainty – a topic often discussed &

debated among metrologists

Initially, there were no standardized

process to quantify measurement

uncertainty….

But a standard technique was agreed

upon & published in October 1993:

ISO Guide 98 - Guide to the

Expression of Uncertainty in
Measurement (a.k.a. GUM)

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA, refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSI/NCSL Z540.2-1997 (R2002) U.S.

Guide to Expression of Uncertainty in
Measurement

http://www.ncsli.org and find it in the store

under NCSLI publications

NIST Technical Note 1297

http://www.physics.nist.gov/Pubs/guidelines/
contents.html

Recommendation:
Refer to the GUMs -

Internationally, many metrology

organizations publish similar GUMs

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions?

- about measurement uncertainty

or why it is important

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty &

Calibrating DMMs

A study of applying the GUM to DMM

calibration

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First – lets look at the concept

Our initial look –

• Consider verifying a

precision digital multimeter

• With a hypothetical study

of verifying the DMM’s

measurement performance

at 100 millivolts DC

• Let’s briefly look at what

measurement uncertainty

could be in this case

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement

“doubt” when verifying a DMM

• The most obvious & significant sources of doubt:

− Inaccuracy of the calibrator’s output value

• 100.0000 mV might actually be 100.0000 mV .0030 mV

− Repeatability or randomness in measurement values from the DMM

• 100.0003 mV, 99.9995 mV, 100.0010 mV, etc.

− Resolution or sensitivity limits on the DMM

• It’s value is ½ the least significant digit,

• in this example it represents 0.05 V

• Many other factors that could also contribute to uncertainty:

− ambient temperature effects, thermal emfs, noise, loading, power line

conditions, etc.

• Consider all factors and include if they significantly contribute to

measurement uncertainty

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of

measurement uncertainty

• Type A uncertainty – errors that can be statistically

evaluated from the set of measurement data (Often
considered as random uncertainty)

− For example: Repeatability of the measurement (influenced by dmm
characteristics, signal stability, jitter, noise, etc.)

• Type B uncertainties – estimates of errors influencing the

measurement that are not directly observed from the
measurement data (Often considered as systematic
uncertainty)

− Errors of the calibrating standards (performance specifications for
accuracy changes over time and other conditions)

− Inherent limitations of the unit being tested (DMM resolution
limitations)

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

• To quantify uncertainty, the various sources of uncertainty need
to be quantified, evaluated, & combined

• Calculate a combined estimate of all the individual A and B types
of uncertainties

• This combined uncertainty is:

− a basic estimate (representing one statistical standard deviation)

− usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1

...

n

c uuuuu 

Combining all the uncertainties

cu

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

• As mentioned, calculations for u

c

pertain to ± one standard

deviation of measurement uncertainties (covering 68% of the

population of measurements)

• Usually it is desired to express uncertainty for a larger population or
condition, say 95% or 99%.

• Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - U

m

.

• A coverage factor, k, (often equal to 2), would indicate a 95%
confidence.

cku

U

m



68%

95%

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now, returning to the …
statement of uncertainty

• ... A measurement is complete only when
accompanied by a statement of the uncertainty of the
estimate. For example:

VDMM = 100.0051mV 0.0004 mV

• In this case,  0.0004 mV would be the resulting value
of U

m,

calculated as shown below:

ckumV

U

m

0004.0

22

3

2

2

2

1

...

n

uuuuk 

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general
process – are we okay so far?

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next, a different and more
detailed example…

Examine the use of a Fluke 5500A to verify a 3.5 digit

DMM at 10 Amps of Alternating Current at 50 Hz

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

• Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements).

• Type A uncertainties includes effects from:

− Variations of multiple repeated readings from the UUT

− Effects of the system noise

− Noise and short term variation of the standard

• Now let’s examine the basic statistics …

The “A” portion…

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement

Value

1

10.07

2

10.02

3

10.01

4

10.06

5

10.04

Average

10.04

Measured value: the average
of a series of measurements

AIavg 04.10

• An average of multiple measurements is
a better estimate of the true value than
any individual value

• As a rule of thumb, taking between 4 &
10 measurements are sufficient.

• Uncertainty improvements for more than

10 have diminishing results

• In our example, 5 readings are
sufficient. Any improved uncertainties
for more readings are not significant

versus required measurement

tolerances (a typical DMM specification
for this example test is ~ ±2.5%).

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement

Value

Deviation from

Average

x1

10.07

+0.03

x2

10.02

-0.02

x3

10.01

-0.03

x4

10.06

+0.02

x5

10.04

0.00

Calculating the uncertainty due
to measurement repeatability

• The uncertainty is statistically

analyzed from the measurement
data series

u

1

– for a normally distributed

population, the best estimate of
uncertainty is the experimental
standard deviation of the mean

NOTE: In the unusual case where

1. the calibrating standard is extremely accurate &
stable, and

2. the repeated test measurement values are

unchanged (or even with only a ± one digit
change)

Then this uncertainty can be considered as non

significant

• One measurement value would be sufficient

• The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean





















1

)(

1

2

n

xx

n

i

i

s

n

s

u 1

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement

Value

Deviation from

Average

x1

10.07

+0.03

x2

10.02

-0.02

x3

10.01

-0.03

x4

10.06

+0.02

x5

10.04

0.00

x

(Average)

10.04

s (Estimated Std. Dev.)

0.02549

The estimated standard deviation

 









1

1

2

)(

n

i

n

i

xx

s

25.5 mA

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u

1

– estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider:

− Probability Distribution = Normal

− Sensitivity Coefficient = 1

− Degrees of Freedom = 4

mA

mAs

n

u 4.11

5.25

5

1 

What are these?

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms & concepts

• Probability Distribution: “the scatter of the values”

− Normal or Gaussian

− Rectangular or Uniform

− Triangular, U or bi-modal, …

• Degrees of Freedom: “how many”

− A value related to the amount of information that was employed in
making the estimate.

− Usually equals the sample size minus one (n-1) for type A uncertainties,
and is often considered infinite ( ) for parameters such as

manufacturer specifications

• Sensitivity Coefficient: “how influential”

− Change in measurement response divided by the corresponding change
in stimulus (usually a value of 1 in the case we are considering)

For more information, see technical references on statistics



©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u

1

– estimated standard uncertainty

Calculate the Standard Deviation of the Mean

− Probability Distribution = Normal

− Sensitivity Coefficient = 1

− Degrees of Freedom = 4

mA

mAs

n

u 4.11

5.25

5

1 

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The ”B” type of uncertainties …

All the other uncertainties that cannot be determined statistically during

the measurement process, such as -

− Calibrator inaccuracy or error

− Measurement errors due to limitations of the DMM’s resolution

− lead effects, thermal emfs, loading, etc.

• Estimates here are based on scientific judgment using all relevant

information

• Numerically, these are expressed as one standard deviation
estimates for each different uncertainty

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u

2

- uncertainty due to the

calibrator inaccuracy

u

2

is the ±1 sigma estimate of the calibrator error,

• (estimates a ±1 standard deviation coverage of

the errors - for 68% of all possible values),

• based on the specifications for performance at the
specific test setting

− Start with the manufacturer’s recommended specifications

at the test point

− Adjust as required for any appropriate factors such as

legal traceability limitations, improvements for output

characterizations, etc.

− Convert to a ± one sigma confidence interval basis

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator
specifications

• For this example, assume it is a certified calibrator that is routinely
calibrated every year.

• The absolute uncertainty specifications for 10 Amps, 50 Hz:

0.06% of output plus 2000 Amps

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u

2

• Step 1: Calculate the maximum instrument error per

manufacturer’s specifications at the point of test

5500A – 1 year specs @10 A, 50 Hz

±(0.06% of 10 A + 2000 μA)

is calculated to be:

±(6 mA + 2 mA) = ±8 mA

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u

2

• Step 2: Convert the specified
error to an error value that
covers ±one standard deviation
(or a ±1 sigma confidence

interval)

− If no other information is provided
by the manufacturer, assume a
rectangular distribution

±1σ = ±spec / (√3)

− If manufacturer specifies a different
distribution, such as a normal
distribution, then calculate as

appropriate.

For example with a normal
distribution at 99%

±1σ = ±spec / (2.58)

Normal Probability Distribution

1

2 3-123

Uniform or Rectangular

Probability Distribution

Probability of Occurrence

Value of Reading

Full width

Mean or

Average reading

-a

+a

±spec

limits

±spec limits

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Fluke’s 5500A specifications

The manufacturer’s specs document that specifications are

based on a normally distributed, 99% confidence interval

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u

2

•The value of u

2

is the ±1 sigma calibrator spec:

5500A – 1 year specs @10 A, 50 Hz

This u

2

value should be smaller than the published spec!

With a spec of ±8 mA at 99% confidence

divide by 2.58 to convert to a ±1 sigma spec

u

2

= 8 mA / 2.58 mA = 3.1 mA at ±1 std. dev.

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u

2

–

u

2

is the ±1 sigma estimate of calibrator

specification uncertainty

− Probability Distribution = Normal – as stated in the

manufacturer’s information

− Sensitivity Coefficient = 1

− Degrees of Freedom =

mA

u

2

1.3



©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u

3

- uncertainty due to UUT

measurement limitations

• Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

• The LSD of resolution for this UUT measuring 10 Amps is 10 mA

10.00

10.00000

LSD (least significant digit)

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u

3

The formula for u

3

is:

Calculates the standard

uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 2.9 mA at a ±1 std. dev.

3LSD

2

1

3 u

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u

3

–

u

3

is the ±1 sigma estimate of dmm LSD resolution

uncertainty

− Probability Distribution = Rectangular

− Sensitivity Coefficient = 1

− Degrees of Freedom =

mA

u

3

9.2



©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the “B” portion…

u

2

= 3.1 mA at ±1 standard deviation

u

3

= 2.9 mA at ±1 standard deviation

• There are no other “B” uncertainties which are

significant for this particular test

(Note: It is often good to identify and document the

other possible uncertainties deemed insignificant.)

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties …

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1

...

n

c

uuuu

u



12.16 mA

222

9.21.34.11 

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty
budget

Source of

Uncertainty

Type

Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability

A

u

1

11.410-3

1

Normal

1

11.410

-3

4

Calibrator

B

u2

810

-3

1

Normal

2.58

3.110

-3

Resolution

B

u3

510-3

1

Rectangular

2.910

-3

Current

Measurement

Combined

u

C

- -

Assumed

Normal

-

12.1610-3

5.2

How do you calculate the overall

effective Degrees of Freedom?



3



©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

• is the overall effective
degrees of freedom for the

combined uncertainty (u

c

).

• The formula considers each
uncertainty, each sensitivity
coefficient and each

uncertainty’s specific value

for degrees of freedom to

calculate





















N

i

i

ii

c

eff

v

xuc

yu

v

1

44

4

)(

)(

v

effveff

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula
in our example case

2.5

)10(2.91)10(3.11

4

)10(11.41

)10(12.16

434434434

43

























v

eff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yu

c

eff

v





Our effective degrees of freedom considering all our uncertainties

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

c

m

ku

U



Calculating the expanded
uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

68.27% 1

90% 1.645
95% 1.960

95.45% 2.0
99% 2.576

99.73% 3

k

is the coverage factor

• How confident should you be with your measurement results?

(68%, 95%, 99%....)

• 95% confidence is commonly accepted as appropriate.

• Um expresses the uncertainty, expanded from a single standard

deviation of 68%, to uncertainty value with a higher confidence.

• For a large population with a normal distribution, 95% coverage
is calculated by k with a value of 1.96
(or sometimes 2 for convenience – giving 95.45%)

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of

measurements or samples

• Adjusting k is done using the:

students’ t distribution table

• A coverage factor adjustment
is needed because our data
set had a fewer number of
values, rather than a larger set
(such as 20, 50, or 100)

• The table lists the proper
coverage factor for populations
with smaller degrees of
freedom

Fraction p in percent

Degrees of

freedom 

68.27 90 95 95.45 99 99.73
1 1.84 6.31 12.71 13.97 63.66 235.8
2 1.32 2.92 4.3 4.53 9.92 19.21
3 1.2 2.35 3.18 3.31 5.84 9.22
4 1.14 2.13 2.78 2.87 4.6 6.62
5 1.11 2.02 2.57 2.65 4.03 5.51
6 1.09 1.94 2.45 2.52 3.71 4.9
7 1.08 1.89 2.36 2.43 3.5 4.53
8 1.07 1.86 2.31 2.37 3.36 4.28
9 1.06 1.83 2.26 2.32 3.25 4.09

10 1.05 1.81 2.23 2.28 3.17 3.96

20 1.03 1.72 2.09 2.13 2.85 3.42

50 1.01 1.68 2.01 2.05 2.68 3.16

100 1.005 1.66 1.984 2.025 2.626 3.077



1 1.645 1.96 2 2.576 3

For our example with the effective degrees of freedom (V

eff

) of 5.2,

a coverage factor of 2.57 expands u

c

to a value with 95% confidence

(compared to 1.96 for an infinite set of measurements/samples).

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

c

m

ku

U



Expanded measurement
uncertainty calculation

 57.2

U

m

12.16 mA



U

m

31.26 mA

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty

Type

Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability

A

u

1

11.410-3

1

Normal

1

11.410

-3

4

Calibrator

B

u2

710

-3

1

Normal

2.58

2.710

-3

Resolution

B

u3

510

-3

1

Rectangular 2.910

-3

Current

Measurement

Combined

u

C

- -

Assumed

Normal

-

12.110-3 5.2

Current

Measurement

Expanded

U

m

31.2610-3

-

Assumed

Normal

2.57 - 5.2



3



©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg

UII 

Final results -

• The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 04.10 

0.031

At a level of confidence of 95%

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken, does

that improve the uncertainty?

Increased degrees of freedom
V

eff

= 5 10, 20 or 100

Causes marginal improvements
in k and in

• 5 measurements, V

eff

= 5.2

− k = 2.57, = 31 mA

• 9 measurements, V

eff

= 10.3

− k = 2.23, = 27 mA (4 mA better)

• 17 measurements, V

eff

= 20.7

− k = 2.09, = 25 mA (2 mA better)

• 78 measurements, V

eff

= 100.9

− k = 1.984, = 24 mA (1 mA better)

Fraction p in percent

Degrees of

freedom 

68.27 90 95 95.45 99 99.73
1 1.84 6.31 12.71 13.97 63.66 235.8
2 1.32 2.92 4.3 4.53 9.92 19.21
3 1.2 2.35 3.18 3.31 5.84 9.22
4 1.14 2.13 2.78 2.87 4.6 6.62
5 1.11 2.02 2.57 2.65 4.03 5.51
6 1.09 1.94 2.45 2.52 3.71 4.9
7 1.08 1.89 2.36 2.43 3.5 4.53
8 1.07 1.86 2.31 2.37 3.36 4.28
9 1.06 1.83 2.26 2.32 3.25 4.09

10 1.05 1.81 2.23 2.28 3.17 3.96

20 1.03 1.72 2.09 2.13 2.85 3.42

50 1.01 1.68 2.01 2.05 2.68 3.16

100 1.005 1.66 1.984 2.025 2.626 3.077



1 1.645 1.96 2 2.576 3

U

m

UmUmU

m

U

m

So improves only 7 mA by taking

73 more measurements

U

m

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond ±31 mA by taking more

measurements have any practical value?

What’s the value of increasing
V

eff

from 5 to ?????

The test tolerance is ±250 mA

• 5 measurements, V

eff

= 5.2

− k = 2.57, = 31 mA

• With a = 31mA,

the test ratio is already 8:1

(TUR = Test Spec ÷ Total Uncertainty

0.25A ÷ 31mA = 8.06)

Fraction p in percent

Degrees of

freedom 

68.27 90 95 95.45 99 99.73
1 1.84 6.31 12.71 13.97 63.66 235.8
2 1.32 2.92 4.3 4.53 9.92 19.21
3 1.2 2.35 3.18 3.31 5.84 9.22
4 1.14 2.13 2.78 2.87 4.6 6.62
5 1.11 2.02 2.57 2.65 4.03 5.51
6 1.09 1.94 2.45 2.52 3.71 4.9
7 1.08 1.89 2.36 2.43 3.5 4.53
8 1.07 1.86 2.31 2.37 3.36 4.28
9 1.06 1.83 2.26 2.32 3.25 4.09

10 1.05 1.81 2.23 2.28 3.17 3.96

20 1.03 1.72 2.09 2.13 2.85 3.42

50 1.01 1.68 2.01 2.05 2.68 3.16

100 1.005 1.66 1.984 2.025 2.626 3.077



1 1.645 1.96 2 2.576 3

U

m

U

m

AmpsI 04.10 

0.031

So to satisfy a minimum test ratio of 4:1,

5 measurements are more than adequate!

U

m

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions?

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of

Measurement Uncertainty Simpler

What can you do to automate this?

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

• A custom program
designed for a specific
requirement

• A custom spreadsheet for

analysis

• A commercial metrology

based software package

such as
Fluke’s MET/CAL Plus

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

MET/CAL automates the

uncertainty calculations

Post test summary of

10.000A @50Hz
Including:

5 reading average
Calculated combined

standard uncertainty

How does this work?

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

MET/CAL manages &

analyses the uncertainties

Number of Measurements = 5
Value 1 = 10.07
Value 2 = 10.01
Value 3 = 10.02
Value 4 = 10.04
Value 5 = 10.06

UUT Indicated = 10.04

Standard Deviation = 0.02549509757
Standard uncertainty = 0.01140175425
Sensitivity Coefficient = 1
Degrees of Freedom = 4

System Actual = 10
System Accuracy = 0.008
Confidence interval of spec = 2.58
1 Sigma Spec = 0.003126379456
Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200

UUT Resolution = 0.01

Resol. Standard Uncertainty. = 0.002886751346

Sensitivity Coefficient = 1
Degrees of Freedom = 1e+200

Combined Std. Uncertainty = 0.01216490061
Effective Deg. of Freedom = 5.186506
Standard Uncertainty = 0.01207040471
Coverage Factor = 2.567104753
Expanded Uncertainty = 0.031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With MET/CAL the user configures:

• Specific statistics used

• Confidence / Coverage

• Number of measurements

• Accuracy of the standard

In the cal or test procedure you also
specify test parameters:

• Test point

• UUT resolution

In the test process, MET/CAL provides
the uncertainty details (our example is
shown to the right)

Details are permanently stored in the
data base. They accessible for reports
& future analysis.

MET/CAL Data for

our example

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

“Automation” –

some words of wisdom

• Remember, it is always the metrologist’s responsibility to

insure proper calculation of measurement uncertainty

− Every lab has unique characteristics which must be supported

− Configuring the measurement characteristics is also unique

− Defining the specific error budget for the test

− Configuring the specific measurement uncertainty parameters

• There should be definite information to support answering

any auditor’s questions

• Keep records of the procedure’s measurement design with

an uncertainty error budget

• Be able to demonstrate the reasonableness of the test’s

uncertainties

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of MET/CAL

automation

• Automation simplifies a structured
calculation process

• Usable for manual, semi automated,
or fully automated testing methods

• MET/CAL provides flexibility to
customize the calculation process &

factors

• MET/CAL’s database stores all the

information for future reference

• Report writing flexibility permits

properly configured certificates and

data summaries

• Lets the technical staff concentrate
on the test quality rather than the
rote mathematical & statistical

processes

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions?

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion & Review –
What have we done?

• Topics

− Measurement uncertainty & why it is important

− How measurement uncertainty obtained

− Examples on measurement uncertainty & calibrating DMMs

− Benefits of automating

• Measurement Uncertainty is becoming an essential

consideration in all metrology & calibration measurements

• Measurement results are considered incomplete without a

quoted uncertainty

• Calculations usually require a statistical process on

multiple measurements for each test

• Automation can be a valuable support for measurement

uncertainty calculations

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs &
other references for details:

ANSI/NCSL Z540.2-1997 (R2002) U.S.

Guide to Expression of Uncertainty in
Measurement

http://www.ncsli.org and find it in the store

under NCSLI publications

NIST Technical Note 1297

http://www.physics.nist.gov/Pubs/guidelines/
contents.html

Where to go from here?

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

• Chapters 20-22 on Statistics &

Uncertainty in the text book

Calibration: Philosophy in
Practice 2nd. Edition

• Fluke’s Training Course – Cal Lab

Management for the 21st Century

• Various reference material under

technical papers at the resource
library on Fluke’s web site:

http://www.fluke.com

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

• EA-4/02 “Expression of the Uncertainty of Measurement of
Calibration”

http://www.european-accreditation.org

• UKAS Publication LAB-12 “The Expression of Uncertainty

In Testing”

http://www.ukas.com/

• NPL UK - “A Beginner's Guide to Uncertainty of
Measurement”

http://www.npl.co.uk/npl/

• Fluke’s “Calibration – Philosophy in Practice, Second
Edition”

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

• NCSL International: RP-12 - Determining &

Reporting Measurement Uncertainties

https://www.ncsli.org/

• NIST Website: Essentials of expressing

measurement uncertainty

http://physics.nist.gov/cuu/

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions?

22

3

2

2

2

1

...

n

c uuuuu 

©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration

Web Seminar Series

For information & reservations to attend our

seminars, go to www.flukecal.com, click

on the menu selection “Events &

Training”, and click on the “Web
Seminars” selection, and again click on

the desired seminar selection,

Our Seminar Topics Include:

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©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

• Instructor-Led Classroom Training

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More information:

www.flukecal.com/training

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©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU !

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