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MEASUREMENT UNCERTAINTY
Introduction
The goal of experimentation is to determine the behavior of
physical systems.
In conducting experiments we utilize measurement systems that
interact with the physical system being studied such that we are
able to estimate the value of physical parameters of interest.
We then use the measurement results to test hypotheses about the
physical systems.
In order to test hypotheses and draw conclusions we must be able
to assess the quality of the measurements, i.e., we must be able to
estimate the extent of any error in the measurements.
All experimental measurements contain some error.
Fixed (systematic) errors are associated with repeated
readings that are in error by the same amount.
Random errors usually follow some statistical
distribution.
Gross (illegitimate) errors are associated with invalid
data.
In general, it can be difficult to distinguish between fixed and
random errors.
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Classification of Measurement Errors
Systematic or Fixed Errors
 Calibration errors
 Consistently occurring human error
 Errors of the measurement instrument
 Consistent calculation errors
 Uncorrected errors due to the presence of the measurement
instrument
Random Errors
 Errors due to uncontrolled environmental variations
 Inconsistently occurring human errors
 Errors derived from a lack of sensitivity
Illegitimate (Gross) Errors
 Mistakes
 Computational errors
 Extreme events (failure)
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Important Concepts
The quality of measurements is characterized by the concepts of
precision, accuracy and bias.
Precision is the closeness of repeated measurements of the
same quantity.
A measurement with small random errors is said to have
high precision.
Accuracy is the closeness of a measured value to its true
value.
A measurement with small error of any kind is said to
have high accuracy.
A measurement with small systematic errors is said to be
unbiased.
Calibration is the process followed to remove bias and define
precision.
Calibration is performed by employing a system that produces
a known value of a given parameter (standard).
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Consequences
A biased but sensitive scale might yield inaccurate but very precise
weight.
An insensitive scale might result in an accurate although imprecise
reading since a repeated weighing would be unlikely to yield an
equally accurate weight.
Unless there is bias in a measuring instrument increasing precision
will lead to an increase in accuracy.
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Significant Figures
The last digit of an approximate number should always be
significant, i.e., it should imply a range for the true measurement
corresponding to a "unit step" above and below the recorded value.
For example,
193
implies
192.5 - 193.5
192.8
implies
192.75 - 192.85
192.76 implies
192.755 - 192.765
Note that these three numbers are given in order of increasing
accuracy.
This scheme applies to all digits including zero.
For example, 7.80 implies the limits 7.795 - 7.805; if 7.75 7.85 is implied, then the measurement should be recorded as
7.8.
Zeros should not be written at the end of approximate numbers to
the right of the decimal point unless they are meant to be
significant digits.
Rounding
This is a process to reduce the number of significant figures.
The general rules result in:
Original Number
Significant Figures Desired
Result
26.68
2
27
133.7137
5
133.71
0.03725
3
0.0373
0.03715
3
0.0372
18316
2
18,000
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Expressing Uncertainty in Measurements
Suppose we are to perform a measurement of a quantity, y.
In all experiments there is some imprecision, and thus the
measurement could be in error.
The uncertainty is our statement as to the probability of any single
measurement being in error.
The statement is generally made in the form,
yw
@ 1     100%
where y is the instrument reading, w is half of the confidence
interval (symmetric), and 1- is the confidence level usually
expressed in percent.
If we knew the distribution function that described variability
in the instrument, we could be precise about w for a given
confidence level.
Ideally the confidence interval is obtained from calibration.
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In a calibration experiment, we attach the instrument to a known,
stationary source (standard).
A sequence of readings from the instrument is then taken, and the
statistical parameters of the distribution are calculated.
The sample mean is used to correct the measurement for bias
and the sample variance is used to estimate the confidence
interval of a single measurement.
The bias-corrected result is then presented as,
y  t
2,
S
n
@ 1     100%
where S is the calibration standard deviation and n = 1 is used for a
single reading.
Note that t/2, depends upon the confidence level and the
number of readings used for S.
Typically an  of 0.05 is used leading to a confidence level of
95%.
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Measured Value
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Perfect
Calibration
True Value
Measured Value
Systematic
Error
One-Point
Calibration
Correction
True Value
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Measured Value
Systematic
Error
(Linear)
Two-Point
Calibration
Correction
True Value
Measured Value
Systematic
Error
(Non-Linear)
Multiple-Point
Calibration
Correction
True Value
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If uncertainty is limited by resolution rather than by variability,
then a reasonable way to express the result of a measurement is,
y  Rm
@ 1     100%
where Rm is the resolution with which a 1-  confidence level can
be maintained.
A good example of a resolution-limited instrument is a ruler
with coarse markings.
Guessing is the last resort for determining instrument uncertainty.
In practice, one is usually forced to make a conservative
estimate for w rather than risk drawing inappropriate
conclusions from the data.
Results should be reported as,
yw
@ 1     100%
and a statement made that the uncertainty has been estimated.
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Propagation of Uncertainty
The above discussion applies to the description of uncertainty in a
measured quantity.
In practice, directly measured quantities are usually used to
calculate a secondary quantity.
If the uncertainty in a directly measured quantity is known, how is
the resulting uncertainty in the calculated quantity determined?
Suppose we calculate a result R that is a function of individual
measurements of different quantities x1, x2, ……. xn,
Then,
R  Rx1, x2 ,....xn 
and statistical analysis can be used to show that if the x’s are each
independent and normally distributed,
2

n  R

wR   
 w 
j
j 1  x
j


Here, wR is the uncertainty in the calculated quantity.
In some circumstances the fractional uncertainty, wR/R, is more
useful and can be represented by,
wR

R
  ln R w j 
 
 
j 1   ln x j x j 

n
2
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Once the uncertainty for the calculated result is known, the value is
then expressed as,
R  wR
@ 1     100%
and for fractional uncertainty,
R
wR
R
@ 1     100%
It is important to keep in mind that the uncertainty analysis results
in a statement regarding the variability introduced by the
measurement system due to “noise” or resolution limits not
variability in the sample or sample population under study.
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EXAMPLE
A resistor has a nominal stated value of 10  1 percent. A
voltage is impressed on the resistor, and the power dissipation is to
be calculated in two different ways: (1) from P = E2/R and (2)
from P = EI. In (1) only a voltage measurement will be made,
while both current and voltage will be measured in (2). Calculate
the uncertainty in the power determination in each case when the
measured values of E and I are:
E = 100 V  1 % (for both cases)
I = 10 A  1 %
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SOLUTION
Using the schematic shown, for the first case we have:
P 2 E

;
E R
P
E2
 2
R
R
and applying the basic equation,


2
2 2

2
E
E


2
2
wP  
 wE    2  wR 
 R 

 R 


12
Dividing by P = E2/R gives
2
2
wP   wE   wE  
 4
 
 
P   E   R  
12
Inserting the numerical values for uncertainty,


12
wP
 40.012  0.012
 2.236%
P
For the second case we have,
P
 I;
E
P
E
I
and after appropriate algebraic manipulation, we obtain
2
2
wP  wE   wI  
 
   
P  E   I  
12
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Once again, inserting the numerical values for uncertainty,


wP
2
212
 0.01  0.01
 1.414%
P
Thus, the second method of power determination provides
considerably less uncertainty than the first method, even though
the primary uncertainties in each quantity are the same. In this
example the utility of the uncertainty analysis is that it affords the
individual a basis for selection of a measurement method to
produce a result with less uncertainty.