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Uncertainties
Using & Calculating
Uncertainties for Electrical
Measurement
Expressing Uncertainty of
Measurement
All measurements, even the most accurate,
have an unknown inaccuracy or doubt.
The is known as the UNCERTAINTY
As there is always an uncertainty with any
measurement we need to estimate this
amount. We also need to calculate our
confidence in the estimate of uncertainty, which is
how sure we are that the true value is within the
uncertainty we have estimated.
Expressing Uncertainty of
Measurement
As an example we may measure 1 Volt and
be 95% confident that we are within 10uV
Expressing Uncertainty of
Measurement
UNCERTAINTY vs. ACCURACY
There is no connection between these terms.
 Uncertainty is purely the unknown in any
measurement.
 Accuracy or Tolerance is the difference
between the desired value and the actual
measured value.
Expressing Uncertainty of
Measurement
Example
With a digital watch with 1 second resolution
this resolution will limit the best uncertainty
to which you can tell the time (ie. 1 second),
however the watch itself may only be
accurate to a few minutes.
Expressing Uncertainty of
Measurement
There are many sources of uncertainty in any
measurement which need to be combined
using statistical techniques to give a total.
Different types of uncertainty need to be
treated differently to obtain an accurate
estimation.
Expressing Uncertainty of
Measurement
To calculate uncertainty you must first
identify all the sources of error, estimate the
size of the contribution from each source and
also decide on the type of uncertainty for
each source.
There are two types of uncertainty
 Type A – Based on using statistics
e.g. repeated readings
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Type B – Based on other factors
e.g. manufacturers specifications
Sources of Uncertainty in
Electrical Measurements.
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Imported Uncertainty
Drift of reference instrument
Temperature effects
Lead and thermal errors (DC volts)
Rounding errors due to resolution
Repeatability
Noise
Self heating of high current shunts
Imported Uncertainty

Imported uncertainty is taken directly from the
certificate issued by the laboratory which
calibrated the reference instrument. The
probability distribution is ‘NORMAL’
Drift of reference instrument
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Drift of reference instrument can be either taken form
historical data on the instrument or from the manufacture
specification for stability. If the drift with time can be
predicted it is possible to use a corrected figure for the
actual value of the reference with a reduced figure for
drift. However it is more normal to use an ‘un corrected’
figure.
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The probability distribution is ‘Rectangular’.
Temperature Effects
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The effect of temperature on many modern instruments is
often very small, and in many cases the instruments
specification covers a band of temperature without any
further addition.
Some reference standards for example resistors the
Temperature coefficient may be quite important. The
figure for TC can be taken from the manufactures spec or
measured.
The probability distribution is ‘Rectangular’.
Lead and thermal errors
(DC volts)
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Thermal emf can be difficult to evaluate. With a
little care and correct leads it is normal for
thermal EMF to be less than 1uV, or even 0.5uV
which is a figure often used in calculations.
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The probability distribution is ‘Rectangular’.
Resolution of Measurement
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It is firstly important to understand that there is a big difference
between the resolution of a measuring instrument and that of a
reference source.
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A source, such as a standard resistor may have no resolution at
all, but can still be very accurate, while for a measuring
instrument resolution is essential to achieve accuracy.
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It is only necessary to include measurement resolution in the
uncertainty calculation. Note if a DMM is used to compare say
two resistors then the resolution must be entered twice.
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The probability distribution is ‘Rectangular’
Combining Uncertainties
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It is normal these days to use a spread sheet, in Excel, taking a
template from M3003.
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Use column 1 for a description of source of uncertainty
Use column 2 for the value of uncertainty usually in ppm
Use column 3 for a description of type of Probability distribution
Use column 4 for the divisor, 2 for a normal dist, 1.732 for rect
Use column 5 for a coeficient used to convert say millvolts to microvolts
Use column 6 = column2(value) x column4(divisor) x column5(coeff)
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Use the sum of square to total column 6 to get the Combined Standard
Uncertainty
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To obtain the Expanded uncertainty (K=2, 95% Confidance) multiply the
result above by 2. Then round to 2 significant places.
Expressing Uncertainty of
Measurement
Example
Using ProCal to calculate
Uncertainties
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ProCal use three key elements to dynamically calculate
uncertainties as the test is run
1) A table with imported uncertainties and calibrator specifications
2) A laboratory procedure incorporating additional factors such as
lead /connection errors etc.
3) The noise / flicker which can be input at the time of test
Set Up Instrument Spec and
Imported Uncertainties
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Use Proset, select instrument Traceability in the file menu.
Select the Instrument required
Select the ‘uncertainties button to access the table
Enter the data, note default table exists for 2000 Series
Set Up Procedure Template
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Use ProSet, select ‘laboratory procedures’ in the file menu.
Select the Instrument required
Select the procedure spreadsheet template say for DC voltage
Enter the parameters to be calculated, note the imported and
reference specification will always be added automatically.
Note the procedures for the main functions of the calibrator are
Installed as default
Set Up Calibration Procedure
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Use ProEdit, edit a procedure and go to the ‘Instruments’ tab.
Check the calibration instrument, lab. Procedure and
uncertainty line.
If a Transmille calibrator is in use these items will be set
automatically, without having to be selected.
Calibrating An Instrument
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Use ProCal, run a calibration
Input the reading and select / enter the noise / flicker in the drop
down box displayed.
If required, click on the UNCERT button to view the uncertainty
calculation.
Where to Get More Information
UKAS (www.UKAS.co.uk)
 National Physical Laboratory
 Transmille Ltd (www.transmille.com)
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