Measurements Course 4600:483-001
Measurement Errors and Uncertainty Analysis
Information about this course can be found at http://gozips.uakron.edu/~dorfi/
Introduction
In the previous lecture we defined probabilistic measures such as the true mean and sample mean and
confidence intervals based on statistical distributions. We derived equations, which can be used to bracket
the data within a certain interval for a given level of confidence (always less than 100%). These concepts
will now be applied to uncertainty analysis. Uncertainty is the probable error, which can occur during the
measurement. It is not the true error, which is the difference between the true value and the measured value,
but it is an error range, which is most likely not exceeded. The likelihood is usually chosen to be 95% for
uncertainty analysis. In other words we are willing to accept a 5% chance (1 in 20) that the real error
exceeds the probable error as estimated from the uncertainty analysis.
When talking about measurement errors we do not include things such as:
•
•
•
misreading instruments
failing to apply known correction or calibration factors
recording incorrect data or misinterpreting the data
These are simply mistakes, which can and must be avoided.
Bias (Systematic) and Precision (Random) Errors
An error may contain either a bias
(systematic) or precision (random)
errors or both.
Recall the interpretation of bias,
precision and accuracy using the dart
board example. Precision is equivalent
to good repeatability (small standard
deviation for repeated measurements).
Precise but
off-center
(biased)
Not Precise but not
off-center (no bias)
Precise and on-center
(small bias) = ACCURATE
Dart Board Example: Precision, Bias and Accuracy
Accuracy requires both good
precision and small bias. An alternate terminology to describe precision and bias is
• precision error =
random error
• bias error
= systematic error
An error is considered a precision or random error, if it is obtained from a statistical estimation only. By
holding controlled inputs constant, the random error can be estimated through repeated measurements.
If the error cannot be estimated through statistical means it is considered a bias or systematic error. Bias
errors cannot be found through repeated measurements. To estimate a bias error, we need a reference for
comparison such as a calibration process, comparison of tests between laboratories, multiple methodologies
or experience.
With uncertainty analysis we establish error bands based on the precision and bias of the instruments and
transducers used in a particular experiment. This lecture gives an overview of uncertainty analysis. Further
examples of uncertainty analysis will be given throughout the semester.
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Measurements Course 4600:483-001
Measurements Errors
Envision an experiment, which is repeated several times. The errors can be identified as follows (see figure
below):
• The total error in any single
Measured
measurement is the sum of the bias and value
True Value (Mean)
x' Systematic
the precision error in that single
measurement.
• The total error in a set of
Bias Error
measurements can be described by an
average bias error and a statistical
Sample
x Mean
estimate of the precision errors.
• The average bias error shifts the sample
Probability
Error Band
mean away from the true mean by a
Distribution
(=
Precision
error
)
fixed amount.
#
of
• The precision or random errors create a
measurements
normal distribution or scatter around
Distribution of Errors upon Repeated Measurements
the sample mean.
Since the precision error is associated with the random variation during repeated measurements, it can be
determined from a statistical analysis of the variation in the repeat measurements. On the other hand, since
the bias error is a systematic error, it can only be determined or estimated through comparison to a
calibrated reference.
Since we desire to estimate the true value of the measurand, we make use of the statistical description
given in the previous lecture, which estimates the true mean based on the sample mean and a precision
range. This precision range is now the uncertainty ux,: The uncertainty ux can be interpreted as an error
band, which contains the true value with a certain probability (usually 95% for uncertainty analysis).
x′ = x ± u x
(P % probability)
Uncertainty analysis is the method used to quantify the ux term.
In the above example ux consists of both a contribution from the precision error and the bias error.
Therefore the next question that arises is how to add these two errors.
Combining Errors: Root Sum Squares (RSS) Method
Any experiment is composed of multiple instruments and sensors, all of which introduce errors and thus
uncertainties. A strategy to combine errors or uncertainties is the RSS method. It combines the errors by
adding the squares of each error and taking the square root of the sum. The RSS method is based on the
assumption that the errors are not correlated (errors combine randomly).
u x = ± u12 + u 22 + ... + u K2
( P % probability )
In the above equation it is important that all error terms are based on the same units and with respect to
the same reference. A common reference is to convert all errors to a percentage of the output signal.
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Measurements Course 4600:483-001
Example: Combining Errors
A 26lbm calibrated weight (accurate to within
26.4
0.1%) is placed on a scale. 20 repeated
26.2
measurements are performed and the data is shown
on the right.
26
A statistical analysis of the data gives a mean value
x of 25.72 and a sample standard deviation Sx =
0.34.
Estimate the uncertainty of the mass
measurement um.
Nominal Cal
Weight
25.8
Sample Mean
of data
25.6
25.4
0
5
10
15
20
Data Sample (N=20)
1) The mass of the calibration weight itself has some uncertainty. It is given as 0.1%. It can be assumed
that this percentage has a confidence level of 95%, in other words 95% of the calibration weights will be
within a 0.1% range of the nominal value (26lbm ).
Therefore
mcal = mcal − nominal ± ucal = 26 ± 0.026 lbm (Prob. = 95%)
and
ucal = 0.026 lbm
From the data we recognize that there is an offset between the measured mean value and the nominal cal
weight. This is a bias or systematic error in the measurement.
ubias = 26 − 25.72 = 0.28 lbm
Finally, the data indicates that there we have limited precision when the measurement is repeated due to the
scatter of the data around the mean. This is the precision or random error. We can also assign an
uncertainty to it based on the student-t distribution for 20 data points and a 95% probability.
u Pr ecision = tv , P S x = t19,95% S x = 2.093 *0.34 = 0.71 lbm
Combining all the uncertainties based on the RSS method we get the total uncertainty in the mass
measurement
2
2
um = ucal
+ ubias
+ uPr2 ecision = .026 2 + .282 + .712 = 0.77 lbm
The analysis shows that for the given measurement system we can only estimate the mass with an
uncertainty of 0.77lbm. The result also shows the major contributors to the uncertainty, the bias and the
precision error. The error in the cal weight is much smaller (as it should be for a good calibration
reference).
Using the calibration process, one can identify and thus remove the systematic bias error ubias from the data
by subtracting the average bias error. In that case the uncertainty in the mass measurement can be reduced
to
2
um = ucal
+ 0 + uPr2 ecision = .026 2 + 0 + .712 = 0.71 lbm
Note that the total uncertainty has not been significantly reduced due to the relative large precision error.
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Measurements Course 4600:483-001
Design Stage Uncertainty Analysis
Uncertainty analysis can be performed in different stages of the experiment (experimental design phase,
data collection phase, data processing phase). One important stage is the design stage, where the
experiment, the instrumentation and the data analysis are being defined.
Uncertainty Analysis of the design stage is particularly important when certain accuracy is critical to
obtaining meaningful results. Design state uncertainty analysis is also useful in selecting instrumentation
and measurement techniques and to identify potential sources of errors.
Different types of uncertainty are discussed below.
Zero-order Uncertainty: u0 (Instrument Resolution Error)
Even if all errors are otherwise zero, a value of uncertainty will be provided by the resolution of each
instrument. This uncertainty associated with the instrument resolution is called zero-order uncertainty and
denoted u0. Typically we assign the error u0 to be ½ of the resolution of the instrument at 95% probability:
1
u0 = ± resolution ( P = 95% probability )
2
Example: Resolution Error
A 12 bit data acquisition board is configured for a Voltage range of 0-10V. What is the uncertainty u0
associated with the board resolution?
Recall the resolution of a DAQ board as
Res =
VRange
2
Nbits
=
10V
= 2.44mV
212
The uncertainty due to the DAQ board resolution is therefore
u0 = ±
2.44mV
Res
=±
= ±1.22mV
2
2
Uncertainty associated with Instrument Error: uc
In addition to the error associated with the resolution, there are likely to be errors associated with the
linearity and repeatability of the instrument and possibly other error sources. The specification sheet for
each instrument usually gives estimates for these errors.
Typically the uncertainty associated with the instrument errors is established during a calibration process.
The calibration process may be performed by the instrument manufacturer and a calibration sheet is
provided with each instrument. The uncertainty associated with the total instrument error is denoted by uc .
As mentioned before it may be composed of multiple errors. The RSS method can again be used to
combine these errors into a single instrument uncertainty uc.
Example: Instrument Error
The specification for the 12-bit data acquisition board configured for a range of 0-10V lists the following
errors:
linearity error
: 0.01% of full range
repeatability error : 3 LSB(Least Significant Bits, digitzed number can be off by up to 3
bits)
offset error
: 1mV
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Measurements Course 4600:483-001
What is the uncertainty uc associated with this instrument?
Note that the errors have different units. In order to combine them, we have to convert to common units
and to the same reference. The most appropriate reference to convert to would be the signal level going into
the DAQ board in Volts.
The linearity error u1 is given as a percentage at full range, conversion to Volts yields
u1 = 0.01% *10V = 1mV
The repeatability error is given as a bit (LSB) error, convert to Volts using the board resolution, which
relates a bit change to a voltage change:
u 2 = 3bits
10V
= 7.3mV
4096bits
The offset error is already given in Volts, it is
u3 = 1mV
The total instrument error uc is found by combining the three errors with the RSS method:
uc = u12 + u22 + u32 = 7.32 + 12 + 12 mV = 7.4mV
For this instrument, the repeatability error clearly dominates the instrument and offset error.
Total Design Stage Uncertainty ud
The design stage uncertainty for an instrument is the combination of the zero-order uncertainty (the
resolution error) and the instrument error. If we make again use of the RSS method we get
ud = u02 + uc2
The design stage uncertainty can be used as a guide to select equipment. It is not the uncertainty associated
with an experiment, because at this point we have not accounted for any uncertainty in test conditions or
measurement procedure but simply for uncertainty in the instrument itself. For example, temperature or
supply power fluctuations during the experiment would increase the uncertainty associated with the
experiment. However, the design state uncertainty ud is the lowest level of uncertainty we can achieve
under perfect test conditions with this particular instrument.
Example: Total Design Stage Uncertainty
Calculate the total design state uncertainty for the 12-bit data acquisition board configured for a range of 010V using the zero-order uncertainty and the instrument error from the earlier examples:
ud = u02 + uc2 = 7.4 2 + 1.22 2 mV = 7.5mV
The total design stage uncertainty for this data acquisition board is 7.5mV. It is primarily due to the 3-bit
repeatability error of the board.
Sources of Errors
In general error sources can be divided into the following three groups:
• Calibration Errors:
• Bias and precision errors in the calibration standard (the calibration is only as good as the standard
used to calibrate the instrument)
• Application of the standard to the measurement system (how is the calibration performed)
• Data Acquisition Errors:
• Instrument errors (signal conditioning, digitization, see also earlier examples)
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Measurements Course 4600:483-001
• Uncontrolled variables such as environmental changes, power fluctuations, spatial variation,…
Data Reduction Errors:
• Truncation error of calculations
• Curve fit errors
When trying to identify errors, it is useful to first try to identify possible error sources based on the above
groupings.
•
Propagation of Error
Quite often we need to know, how an error in one instrument propagates throughout the measurement
system. Suppose we have a relationship y=y(x) between a dependent (output) variable y and a measured
variable x. How will the uncertainty ux associated with x be reflected in the uncertainty uy of y?
In equation form we can write the relationship between a nominal input x and its uncertainty ux and output y
and the uncertainty uy due to the propagation of ux as
y = f (x)
y ± u y = f (x ± ux )
Our objective is to determine uy, the uncertainty associated with the dependent variable y (see figure).
Expansion of the second equation into a Taylor series yields
Output y
y=f(x)
∂y
uy
y ± u ≈ f (x) ±
u
y
∂x
x
∂y
∂x
x= x
y = f (x ) yields the uncertainty uy as
∂y
uy =
ux
∂x x = x
y
Substitution of
x= x
ux
The uncertainty uy of the dependent variable y is therefore the
uncertainty ux associated with the measured variable x
multiplied by the sensitivity (slope) of y with respect x at the
operating point x .
Input x
x
Error Propagation
If the dependent variable y depends on several inputs xi use the RSS method to combine the uncertainties

∂y
u y = 
 ∂x1

2


 u 2 +  ∂y
 x1  ∂x2
x1 = x1 



 u 2 + ...
 x2

x2 = x2 

2
1
2
To estimate the relative uncertainty (percentage) associated with the operating point
the above equation with respect to

∂y
= 
y  ∂x1

uy
y normalize (divide)
y
2
 u x2  ∂y
 1 +
 y 2  ∂x2
x1 = x1 


 u x2
 2 + ...
 y2

x2 = x2 

2
1
2
The relative uncertainty, which is propagated from each independent variable xi to y is defined as ei:
ei =
1 ∂y
y ∂xi
u xi
xi = xi
therefore the total relative uncertainty ey is
e y = e12 + e22 + ... + e i2
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Measurements Course 4600:483-001
Uncertainty for dependent variables y, which are proportional to powers of xi
For functions, which are powers of the independent variables xi, the relative uncertainty can be quickly
calculated. Assume that y can be written as
y = f ( x1 ,..., xi ,...) = g ( x1 ,...) x ni
The relative uncertainty was defined earlier as
ei =
1 ∂y
y ∂xi
u xi
xi = xi
Differentiate y with respect to the variable xi .
∂y ∂ ( g () xin )
=
= g ()nx in −1 and substitute the result into the equation for ei
∂xi
∂xi
nx in−1
ux
1 ∂y
1
n −1
ei =
u xi =
g ()nx i u xi = n u xi = n i
n
y ∂xi
g () x i
xi
xi
In summary we find that if the variable xi is a power of n, its relative uncertainty ei is simply
ei = n
u xi
xi
Example:
A ruler is used to measure the dimensions of a cylinder. The volume is then calculated from the
cylinder dimensions. What is the uncertainty in the volume?
Measured cylinder dimensions are: Diameter d= 2”, Height h=10”. The resolution of the ruler is
1/16”.
π
d 2h
The cylinder volume is given by
V=
The total relative uncertainty in the volume can be written as
eV = ed2 + eh2
4
Cylinder
Volume ?
But since d and h are simple powers of V(d,h) , we can use
the equation
ei = n
u xi
xi
,
where n=2 for d (d2) and n=1 for h. Thus
ed = 2
ud
d
and
eh = 1
uh
h
2
The total relative uncertainty in the volume is thus
 u  u 
eV =  2 d  +  h 
 d  h
To go from relative to total uncertainty
uV = VeV
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Measurements Course 4600:483-001
The uncertainty in the length measurement of d and h
is the zero order uncertainty (instrument resolution error),
which is simply ½ the instrument resolution:
ud =
1 1
1
1
in = in and u h = in
2 16
32
32
2
2
 1 / 32   1 / 32 
Thus eV =  2
 +
 = 0.031 = 3.1%
2   10 

The volume uncertainty can now be determined by
multiplying the relative uncertainty with the measured
cylinder volume.
uV = VeV =
π
4
* 4in 2 *10in * 0.031 = 0.97in 3
With the given ruler resolution and the cylinder dimensions the uncertainty in the volume was found as
about 1 cubic inch or 3% of the measured volume.
Strategies for Uncertainty Calculations
When combining uncertainties from different sources, it is helpful to use the following strategy:
1.
2.
3.
4.
5.
6.
7.
Draw a diagram of the test system and label the different signals and variables in the test system.
Determine the nominal values of the signals and variables in the test system. Nominal may be the
max input or the nominal input.
Determine uncertainties for the test variables.
If appropriate, write an equation that determines the test result from the test variables.
Determine, how uncertainties propagate from the test variables to the test result.
Apply the known uncertainties to the propagation equation to calculate the propagation of the
uncertainties.
Add all uncertainties using the RSS equation into a total uncertainty value (can only be done, if all
uncertainties are referred to the same reference)
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