30
UncertaintyUncertainty Analysis – What it is Analysis – What it is
There is no such thing as a perfect measurements. All measurements of a variable contain inaccuracies.
The analysis of the uncertainties in experimental measurements and results is a powerful tool, particularly when it sis used in the planning and design of experiments
Although it may be possible to an uncertainty by improved experimental method or the careful use of statistical technique to reduce the uncertainty, it can never be eliminated
Issues of AnalysisIssues of Analysis
Systematic and Random Uncertainties
Issues of AnalysisIssues of AnalysisSystematic Uncertainties
Offset uncertainty
Clearly there is a problem here: the boiling point of water should be very close to 100.0 oC
while the melting point should be very close to 0.0 oC There is an offset uncertainty with the temperature
measuring system of about 7.5 oC Possible causes are inherent to measurement device (such
as low battery, malfunctioning digital meter, incorrect type of thermocouple, etc)
Issues of AnalysisIssues of AnalysisSystematic Uncertainties
Gain uncertainty
(mb - mc) versus mc
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
0.00 20.00 40.00 60.00 80.00 100.00 120.00
mc (g)
mb
- m
c
(g
)
Issues of AnalysisIssues of AnalysisRandom Uncertainties
Random uncertainties produce scatter in observed values. The cause :
o limitation in the scale of the instrument resolution uncertainty due to rounding up of measured value
o reading uncertaintyo random uncertainty due to environmental factor
(electrical interference, vibration, power supply fluctuation, etc)
Use statistical technique to get an estimate of the probable uncertainty and to allow us to calculate the effect of combining uncertainties
Issues of AnalysisIssues of AnalysisTrue Value, Accuracy and Precision
Issues of AnalysisIssues of AnalysisSelection and Rejection of Data
A sensitive subject and one that can bring out strong feeling amongst experimenters:
o One argue : All data are equal no circumstances in which the rejection of data can be justified
o Another argue : there as those that ‘know’ that a set of data is spurious and reject it without a second thought
Expert judgment confidence level Statistical test :
o Chauvenet’s criterion P = 1 – 1/(2N) - σ criterion = 2, 3, …
Issues of AnalysisIssues of AnalysisQuoting the Uncertainty
After making repeated measurement of a quantity, there are four important steps to take in quoting the value of the quantity:
1. Calculate the mean of the measured values2. Calculate the uncertainty in the quantity, making clear
the method used. Round the uncertainty to one significant figure (or two if the first figure is a ‘1’)
3. Quote the mean and uncertainty to the appropriate number of figures
4. State the units of the quantity
Issues of AnalysisIssues of Analysis
Uncertainty statement
Absolute uncertaintyo With unit of the quantity
) of(unit XUX x
Fractional uncertaintyo no unit
X
U Xy uncertaint fractional
Percentage uncertaintyo no unit
%100y uncertaint percentage X
U X
Determining Uncertainty – Single Determining Uncertainty – Single QuantityQuantity
Simple Method
n
XX minmax
tmeasuremen ofnumber
range mean in y uncertaint
Example:
)/( 8.341mean smc (m/s) 75.3385.345range cR(m/s) 875.08/7yuncertaint cU
(m/s) 100.0093.418
(m/s) 9.08.3412
c
Determining Uncertainty – Single Determining Uncertainty – Single QuantityQuantity
Statistical Approach to variability in data
mean theoferror tandard mean in y uncertaint s
nx
xxxx kUU limits confidence
Determining Uncertainty – Single Determining Uncertainty – Single QuantityQuantity
(s) 006.050
043.0 mean, theoferror Standard
(s) 043.0 deviation, Standard
(s) 604.0 Mean,
t
t
t
With 95 % level of confidence : (s) 012.02 ttU
Then : (s) 012.0604.0 t
Combining Uncertainty – Uncertainty Combining Uncertainty – Uncertainty PropagationPropagation
An experiment may require the determination of several quantities which are later to be inserted into an equation.
The uncertainties in the measured quantities combine to give an uncertainty of the calculated value
The combination of these uncertainties is sometimes called the propagation of uncertainty or error propagation
V
m
measured quantity with uncertainty
measured quantity with uncertainty
calculated quantity with propagation of uncertainty
Combining Uncertainty – Simple Combining Uncertainty – Simple MethodMethod Most straightforward method and requires only simple
arithmetic. Each quantity in the formula is modified by an amount equal
to the uncertainty in the quantity to produce the largest value and the smallest value
Example :In an electrical experiment, the current through a resistor was found to be (2.5 ± 0.1) mA and the voltage across the resistor (5.5 ± 0.3) V. Determine the resistance of the resistor R and the uncertainty UR !
1020.2A 102.5
V 5.5 33-I
VR
1042.2A 102.4
V 8.5 33-maxR
1000.2A 102.6
V 2.5 33-minR
10210.02
3minmax RR
102.02.2 3R
Combining Uncertainty – Partial Combining Uncertainty – Partial Differentiation Differentiation Based on differentiation of
function of several variables
bb
Va
a
VV
baVV
),(
baV Ub
VU
a
VU
bb
Va
a
VV
Uncertainty propagation:
Properties:
baVbaV
baVbaV
b
b
a
a
V
VabV
b
b
a
a
V
VbaV
/
Sum:
Difference:
Product:
Quotient:
Combining Uncertainty – Partial Combining Uncertainty – Partial Differentiation Differentiation
Example :The temperature of (3.0 ±0.2) x 10-1 kg of water is raised by (5.5 ± 0.5) oC by heating element placed in the water. Calculate the amount of heat transferred to the water to cause this temperature rise !
J 1088.4
)(0.5)(0.3)(4186 )(0.02)(4186)(5.5
and
J 6907)5.5)( )(4186 (0.3
mcUUcU
mcQ
cm
Q
UQ
Um
QU
mcQ
mQ
mQ
The value of c = 4186 J kg-1 oC-1 is assumed to be constant (neglecting its uncertainty)
J 101.19.6 3Q
Combining Uncertainty – Statistical Combining Uncertainty – Statistical ApproachApproach
22
22
22
22
2
),(
baV
baV
b
V
a
V
b
V
a
V
baVV
Taking uncertainty of the mean to relate to standard error of the mean and partial differential principle
General Uncertainty AnalysisGeneral Uncertainty Analysis
Consider a general case in which an experimental result, r, is a function of J measured variable Xi
JXXXrr ,...,, 21Then, the uncertainty in the result is given by :
2
1
2
2
2
2
2
2
2
2
1
2 ...21 i
J
i iX
JXXr U
X
rU
X
rU
X
rU
X
rU
J
iX
ii
XU
X
r
i variablemeasured in they uncertaint
tcoefficieny sensitivit absolute
Note : all absolute uncertainties (UX) should be expressed with the same level of confidence
General Uncertainty AnalysisGeneral Uncertainty AnalysisNondimensionalized forms:
222
2
2
2
2
2
1
2
1
12
2
...21
j
X
j
jXXr
X
U
X
r
r
X
X
U
X
r
r
X
X
U
X
r
r
X
r
U j
Note : factorion magnificaty uncertaint
ii
i UMFX
r
r
X
oncontributi percentagey uncertaint12
22
UPC
rU
X
U
Xr
rX
r
i
X
j
i i
General Uncertainty AnalysisGeneral Uncertainty AnalysisExample:
A pressurized air tank is nominally at ambient temperature (25 oC). Using ideal gas law, how accurately can the density be determined if the temperature is measured with an uncertainty of 2 oC and the tank pressure is measured with a relative uncertainty of 1%?
General Uncertainty AnalysisGeneral Uncertainty AnalysisUncertainty analysis:
222
2222
2
2
2222222
zero is that assuming
1
1
1
,,
T
U
p
UU
UT
U
R
U
p
UU
RT
p
RT
pT
T
T
RT
p
TR
pR
R
R
RT
p
p
p
T
U
T
T
R
U
R
R
p
U
p
pU
TRpRTp
Tp
RTRp
TRp
General Uncertainty AnalysisGeneral Uncertainty AnalysisUncertainty analysis:
%2.1012.0)298/2(01.0
01.0
298
2
29827325
2 2
22
2
UU
p
UT
U
KT
KCU
p
T
oT
Detailed Uncertainty AnalysisDetailed Uncertainty Analysis
X1
B1,P1
X2
B2,P2
Xj
Bj,Pj
1 2 j….
….
r=r(X1,X2, …, Xj)
rBr,Pr
Elemental error sources
Individual measurement system
Measurement of individual variables
Equation of result
Experimental result
B = bias (systematic uncertainty)P = precision (random) uncertainty
Detailed Uncertainty AnalysisDetailed Uncertainty Analysis
The uncertainty in the result is:222rrr PBU
Systematic (bias) uncertainty:
1
1 11
222 2J
i
J
ikikki
J
iiir BBB
J
iiir PP
1
222
Precision (random) uncertainty:
Correlated systematic uncertainty
Systematic UncertaintySystematic Uncertainty Systematic error can be determined and eliminated by
calibration only to a certain degree (A certain bias will remain in the output of the instrument that is calibrated)
In the design phase of an experiment, estimate of systematic uncertainty may be based on manufacturer’s specifications, analytical estimates and previous experience
As the experiment progress, the estimate can be updated by considering the sources of elemental error:
o Calibration error: some bias always remains as a result of calibration since no standard is perfect and no calibration process is perfect
o Data acquisition error: there are potential biases due to environmental and installation effects on the transducer as well as the biases in the system that acquires, conditions and stores the output of the transducer
o Data reduction errors: biases arise due to replacing data with a curve fit, computational resolution and so on
Random Uncertainty AnalysisRandom Uncertainty Analysis Random uncertainty can be determined with various ways
depending on particular experiment:o Previous experience of others using the same/similar
type of apparatus/instrumento Previous measurement results using the same
apparatus/instrumento Make repeated measurement
When making repeated measurement, care should be taken to the time frame that required to make the measurement:
o Data sets should be acquired over a time period that is large relative to the time scale of the factors that have a significant influence on the data and that contribute to the random errors
o Be careful of using a data acquisition system
Random Uncertainty AnalysisRandom Uncertainty Analysis
∆t
Time, t
Y
Failure to determine random uncertainty due to inappropriate data acquisition
Some Detail Approach/GuidelinesSome Detail Approach/Guidelines Abernethy approach (1970-1980):
o Adapted in SAE, ISA, JANNAF, NRC, USAF, NATO
estimate confidence 99%for
estimate confidence 95%for 2/122
rrADD
rrRSS
tSBU
tSBU
Coleman and Steele approach (1989 renewed 1998):o Adapted in AIAA, AGARD, ANSI/ASME
222rrr PBU
1
1 11
222 2J
i
J
ikkiikki
J
iiir BBBB
1
1 11
222 2J
i
J
ikkiSikki
J
iiir PPPP
Some Detail Approach/GuidelinesSome Detail Approach/Guidelines ISO Guide approach (1993):
o Adapted by BIPM, IEC, IFCC, IUPAC, IUPAP, IOLMo Using a “standard uncertainty”o Instead of categorizing uncertainty as systematic and
random, the “standard uncertainty”values are divided into type A standard uncertainty and type B standard uncertainty
o Type A uncertainties are those evaluated “by the statistical analysis of series of observations”
o Type B uncertainties are those evaluated “by means other than the statistical analysis of series of observations”
NIST Approach (1994):o Use “expanded uncertainty” U to report of all NIST
measurement other than those for which Uc has traditionally been employed
o The value of k = 2 should be used. The values of k other than 2 are only to be used for specific application
