Unc
erta
inty
Ana
lysi
s
10Apr15
Contents
Cou
rse
2015
4`Sourceso
fSystematicEr
ror
`Re
ductionofSy
stem
aticEr
rors
9Calibratio
n9
SignalFilte
ring
`Calculationo
fOverallS
ystematicEr
ror
`Sourcesa
ndTreatm
ento
fRando
mEr
rors
`StatisticalAnalysisofM
easurementsSu
bjecttoR
ando
mEr
rors
9Estim
ationofRando
mEr
rorinaS
ingleM
easuremen
t9
Distrib
utionofM
anufacturin
gTolerances
`Ag
gregationofM
easurementSystemEr
rors
9Ag
gregationo
fErrorsfromSe
parateM
easuremen
tSystemCom
pone
nts
Introd
uctio
n
Cou
rse
2015
5Measuremente
rrorsa
reim
possibletoav
oid
9ariseduringt
hem
easurementp
rocess
9ariseduenoise
System
aticer
rorsde
scrib
eerrorsintheo
utpu
treadingso
fa
measurementsystemth
atar
econ
sistentlyononesideofthe
correctreading,thatis,eitherallerrorsa
repo
sitiveo
rareall
negative.Tw
omajorso
urceso
fsystematicer
rorsar
e9systemdisturbancedu
ringm
easurement
9thee
ffectofe
nviro
nmentalchanges
`Ra
ndomer
rorsar
eperturbationsofthem
easuremente
ither
side
ofth
etruev
alueca
usedbyra
ndoman
dun
pred
ictable
effects.
Sourceso
fSystematicEr
ror
Cou
rse
2015
6`disturbanceind
ucedbyth
eactofm
easurement
`effectofe
nviro
nmentaldisturbances
`du
etow
eara
ndag
ingininstrumentcom
pone
nts
`resistanceo
fcon
nectingleads
Disturbanceind
ucedbyt
heac
tofm
easurement
Cou
rse
2015
7`Thep
rocessofm
easurementa
lwaysd
isturbsth
esystembe
ing
measured.W
ayso
fminim
izingdisturbanceo
fmeasured
system
sareim
portantcon
siderationsininstrumentd
esign
Inacaseofm
easurin
gwaterte
mpe
raturewithamercury
inglassth
ermom
eter,ahe
attransferwou
ldta
keplace
betw
eenthew
ateran
dthet
herm
ometer.Thish
eattransfer
wou
ldch
angeth
etem
peratureofthewater.
`Measurementsinelectriccircuit
circuitinwhichth
evoltage
acrossR5istobem
easured
Errord
uetom
easurementd
evice
Cou
rse
2015
8
`volta
geac
rossre
sistorR
5istobem
easuredbya
voltm
eterwithre
sistance
R m
`eq
uivalentcircuitb
yThven
instheo
rem
`circuitu
sedtofindth
eeq
uivalentsingle
Measurementsinelectriccircuits
Cou
rse
2015
9`Th
edesignstrategysh
ouldbet
omakeR
mashighaspossible
tom
inim
izedisturba
nceo
fthem
easuredsystem
.`Bridgecircuitsfo
rmeasurin
gresistanceva
luesar
eafu
rthe
rexam
pleo
fthene
edfo
rcarefuldesignofth
emeasurement
system
.
Exam
ple:m
ovingc
oilm
eter
Cou
rse
2015
10`Toincreaseth
enum
bero
fturnsin
thec
oilcanac
hieveahighinternal
resistanceinthed
esignofa
movingcoilv
oltm
eter,butth
atwill
decreasesthecu
rrentflowingin
thec
oilandde
creasin
gthe
measurementsen
sitivity
`An
yattem
pttoim
provet
he
perfo
rmanceofaninstrum
entin
oner
espe
ctge
nerallydecreases
thep
erform
anceinso
meo
ther
aspe
ct.
Exam
ple:Eq
uivalentcircuit
Step
0: T
he o
rigi
nal c
ircui
tSt
ep 1
: Cal
cula
ting
the
equi
vale
nt o
utpu
t vo
ltage
Step
2: C
alcu
latin
g th
e eq
uiva
lent
res
ista
nce
Step
3: T
he e
quiv
alen
t ci
rcui
t
kRV
Ieqeq
SC25.7
Exam
ple
Cou
rse
2015
12
Solutio
n
Cou
rse
2015
13
System
aticEr
rors
Cou
rse
2015
14`ErrorsduetoEn
vironm
entalIn
puts
9Anen
vironm
entalin
putisd
efinedasanap
parentlyre
al
inpu
ttoam
easurementsystemth
atisac
tuallyca
usedbya
changeinth
eenviro
nmentalcon
ditio
nssu
rrou
ndingt
he
measurementsystem.
9Itcausesasensitivitydr
iftan
d/orze
rodrift
9Them
agnitudeofan
yenviro
nmentalin
putm
ustb
emeasuredbeforeth
evalueofthem
easuredqu
antity(the
realinpu
t)ca
nbed
eterminedfromth
eoutpu
treadingofan
instrument.
Mea
sure
men
t ou
tput
r
eal +
envi
ronm
enta
l inp
uts
System
aticEr
rors
Cou
rse
2015
15`WearinInstrumentC
ompo
nents
9System
aticer
rorsca
nfrequ
entlydevelopoverape
riodof
timeb
ecauseofw
earininstrum
entcom
pone
nts.
Recalibratio
nofte
nprovidesafulls
olutiont
othisp
roblem
.`Co
nnectin
gLeads
9Thefailuretota
kepr
operac
coun
tofthere
sistanceo
fconn
ectin
gleads.Ifth
eyar
etho
ughtlik
elytob
esub
jectto
electricalorm
agne
ticfieldsth
atco
uldo
therwiseca
use
indu
cedno
ise.
9Exam
ple:th
ethe
rmom
eterisse
paratedf
romot
herp
artsofth
emeasuremen
tsystembyperhaps10
0meters.Th
eresistanceof
suchalengthof20gaugeco
pperwireis7:,an
dthe
reisa
furthe
rcom
plicationt
hatsuchw
irehasatempe
ratureco
efficient
of1m:
/qC.
`SignalFilte
ring:ba
ndstopfiltercanbeap
pliedtore
duce
perio
dicn
oiseco
rrup
tionintrod
ucedbycu
rrentc
arrying
cables,m
echanicalvibratio
n.
`IntelligentInstruments
Intelligentinstrumentsco
ntaine
xtrase
nsorsthatm
easureth
evalueo
fenviro
nmentalin
putsan
dautomaticallyco
mpe
nsate
thev
alueoftheoutpu
treading.
Tradeoffb
etweenpe
rform
ancean
dcost
Redu
ctionofSy
stem
aticEr
rors Co
urse
201
517
Quantificatio
nofSy
stem
aticEr
rors
Cou
rse
2015
18`Onces
ystematicer
rorshaveb
eenredu
cedasfara
sreasonably
possiblete
chnically,ase
nsibleap
proachtoes
timatet
he
vario
uskind
sofrem
ainingsy
stem
aticer
rorw
ouldbe
9Environm
entalcon
ditio
nerrors
9Calibratio
nerrors
9Systemdisturbancee
rrors
9Measurementsystemloadinge
rrors
Env
ironm
enta
l con
ditio
n er
rors
Cou
rse
2015
19Environm
entalcon
ditio
nerrors
9subjecttou
npredictableen
vironm
entalcon
ditio
ns9Asystem
aticer
rorrathe
rthanara
ndomer
ror
9toas
sumem
idpo
inte
nviro
nmentalcon
ditio
nsan
dspecifyth
emaxim
umm
easuremente
rrorasx%ofth
eoutpu
treading
Calibratio
nerrors
9Them
axim
umer
rorjustb
eforet
heinstrumentisd
uefo
rrecalibratio
nbecom
esth
ebasisfore
stim
atingt
hem
axim
um
likely.
9Them
axim
umm
easuremente
rrorbetweenwhe
nthe
instrumenth
asjustbeencalibrateda
ndtimejustb
eforet
he
nextca
libratio
nisd
ueca
nthe
nbeex
pressedasx%ofth
eou
tputre
ading.
CalculationofOverallS
ystematicEr
ror
Cou
rse
2015
21`Aworstca
sepredictionofm
axim
umer
rorw
ouldbetosimply
addupea
chse
paratesy
stem
aticer
ror.
`Ap
plyingth
ismetho
dfornsystem
aticco
mpo
nenter
rorsof
magnitudex 1%,x
2%,x
3%,x
n%,thebe
stpredictionoflik
ely
maxim
umsy
stem
aticer
rorb
ythero
otsu
msq
uaresm
etho
dis
`Measurementu
ncertaintyorinaccuracyv
aluequo
tedinth
edatash
eetsindicatesthepe
rform
ancewhe
nitisne
w,us
ed
unde
rspe
cifie
dcond
ition
s,an
drecalibratedatth
erecommen
dedfreq
uency.
Exam
ple
Cou
rse
2015
22
Rand
omEr
rors
Cou
rse
2015
23`Ra
ndomer
rorsinm
easurementsar
ecausedbyun
pred
ictable
varia
tionsinth
emeasurementsystem.
`Alsoca
lledasPr
ecisioner
rors
`Sourceso
fran
domer
rors
9measurementsta
kenb
yhum
an
observationofanan
aloguem
eter,
espe
ciallywhe
reth
isinvolvesinterpolation
betw
eenscalep
oints.
9rand
omen
vironm
entalchanges,
fore
xample,su
ddendraughtofair.
9electricalno
ise.
Treatm
ento
fRando
mEr
rors
Cou
rse
2015
24`Sm
allperturbationsofthem
easurementd
uetora
ndomer
rors
appe
arso
neith
ersideoftheco
rrectvaluean
dcanb
elargely
beelim
inatedbyca
lculatingt
heav
erageo
fanu
mbe
rof
repe
atedm
easurements.
MeanandMed
ianValues
Cou
rse
2015
25AMeaniscompu
tedb
yadd
ingu
pallt
heva
luesan
ddividingt
hat
scoreb
ythenum
bero
fvalue
s.Th
earith
meticmeanofasample
x 1,x
2,,x
n,isthes
umth
esam
pledva
luesdivide
dbyth
enum
ber
ofitem
sint
hesa
mple:
TheMed
ianisthen
umbe
rfou
ndatth
eexactm
iddleo
fthese
tof
values.Am
edianc
anbeco
mpu
tedb
ylistinga
llnum
bersin
ascend
ingo
rderan
dthe
nlocatingt
henu
mbe
rinthec
entero
fthat
distrib
ution.Th
isisapplicabletoanod
dnum
berlist;inca
seofan
evennu
mbe
r ofo
bservatio
ns,the
reisnosin
glem
iddlev
alue
,soitis
ausualpr
acticet
otaketh
emeano
fthetw
omiddlev
alue
s.
Exam
ple
Cou
rse
2015
26`Aninest
udentclassre
sultssc
oresonat
est:2,4,5,7,8,10
,12,
13,83.
`Thea
veragesc
ore(
orth
emean)isth
esumofallthesc
ores
divide
dbynine,14
4/9=16
.Notet
hate
venthou
gh16isth
earith
meticav
erage,itisdistortedbyth
eunu
suallyhighs
core
of83co
mparedt
oothersc
ores.A
lmosta
llofthestud
ents'
scoresar
ebelowthea
verage.The
refore,inth
iscaseth
emean
isno
tago
odre
presentativeo
fthe
centralten
dencyofth
issample.
`Themed
ian,ont
heotherhand,isth
evaluewhichissu
chth
at
halfthes
coresa
reab
oveita
ndhalfthesc
oresbelow
.Soin
thisexam
ple,th
emed
ianis8.Th
erea
refo
ursc
oresbe
lowan
dfourab
ovet
heva
lue8
.So8re
presentsth
emidpo
into
rthe
centralten
dencyo
fthesa
mple.
Exam
ple
Cou
rse
2015
27`Supp
oset
hatthelengthofasteelbarism
easuredb
yanu
mbe
rof
diffe
rentob
serversa
ndth
efollowings
etof1
1measuremen
tsar
erecorded(u
nitsm
illim
eter).Wew
illca
llthism
easuremen
tsetA.
3984
2039
441
640440
840042
039641
3430
mean=40
9.an
dmed
ian=40
8.`Abe
tterm
easurin
grulep
rodu
cesthefo
llowingm
easuremen
tsetB:
40940
6402
4074
0540
440740
440740
7408
mean=40
6andm
edian=40
7Re
mark:
SetBism
orer
eliableb
ecauseth
emeasuremen
tarem
uchm
orec
loser
together.thesm
allerthesp
readofthem
easuremen
ts,them
ore
confiden
ceweh
aveint
hem
eanorm
edianv
alueca
lculated
.
Cou
rse
2015
28`Asth
enum
bero
fmeasurementsincreases,th
edifferen
ce
betw
eenmeana
ndm
edianvaluesbecom
esve
rysm
all.
`Byex
tend
ingm
easurementsetBto23m
easurements
4094
0640
240
740540
440740
440740
740840
6410
406
4054
0840
640940
640540
940640
7mean=40
6.5a
ndm
edian
StandardDeviatio
nandVaria
nce
Cou
rse
2015
29`Measurementx
i
`De
viation(
error)
`Varia
nce
`Standardde
viation
`Unfortunately,th
esef
ormaldefinition
sforth
evariancea
nd
standarddeviatio
nofda
taar
emadew
ithre
specttoan
infin
itepop
ulationofdatav
alue
swhe
reas,inallpractical
situatio
ns,w
ecanon
lyha
veafin
itese
tofm
easurements.
`truem
ean:Pf
orinfin
itepo
pulatio
n
Determ
ineVan
dVf
orFiniteDataS
et
Cou
rse
2015
30`Abe
tterpredictionofth
evarianceo
ftheinfin
itepo
pulatio
ncanb
eobtaine
dbyap
plyingth
eBesselcorrectionf
actor(n/n1
)Thatis
`Thisleadstoasimilarb
etterp
redictionofth
estand
ard
deviationas
Samplev
ersusP
opulation
`Po
pulatio
n:th
ecom
pletec
ollectiono
fallm
embe
rsre
levantto
aparticularissue
`Sample:asubsetofthatp
opulation,whichisob
tainedbya
processo
frando
mse
lectionwitheq
ualprobability.
Statisticso
fthesa
mple
providessa
mplem
ean
value()andsa
mple
varia
nce,w
hichca
nbe
usedtoes
timatet
rue
meanv
alue(x)andtrue
varia
nceV
2through
statisticalinference.
x2 xS
Exercise
Cou
rse
2015
32`Calculatesan
dVform
easurementsetsA
,B,andC.
`A Measurement3
9842
039441
640440
840042
039641
3430
`B Measurement4
0940
640240
740540
440740
440740
7408
`C Measurement4
0940
640240
740540
440740
440740
7408
4064
1040
640540
840640
940640
540940
6407
Solutio
n
Cou
rse
2015
33
Mea
nn
Var
ianc
eSt
anda
rd d
evia
tion
A40
911
137
11.7
B40
611
4.2
2.05
C40
6.5
233.
531.
88
V a
nd V
Con
fiden
ceR
ando
m E
rror
Num
ber
of
mea
sure
men
ts K
LK
L
Aswec
anon
lym
akeafinitenu
mbe
rofm
easurementsina
practicalsituation,th
eaverageva
luew
illstillha
veso
mee
rror.
Thiserrorcanbequantifiedasth
estand
arderroro
fthem
ean.
Plottin
gStatisticalInform
ation
`Signal:am
easurand
smagnitudewithre
specttot
imeo
rspace
`An
alog:con
tinuo
usinbo
thm
agnitudean
dtim
eorspace
`Discrete:con
tinuo
usinm
agnitudebu
tatspe
cifictimeo
rspace
`Digital:h
avings
pecific,fixe
dintervalva
luesinbo
th
magnitudean
dtim
eorspace
`Histogram:plottop
rovidece
ntral
tend
encyofthesignalan
dthe
freq
uencyo
foccurrenceo
fdata
`Exam
ple:
Ther
esolutionofth
edigitizatio
nprocessforth
iscaseis0.5V.
GraphicalD
ataA
nalysisTe
chniqu
esFreq
uency
Distrib
utions
Cou
rse
2015
35`Graphicalanalysis:R
ando
mm
easuremente
rrorsd
istrib
ution
`Histogram
Histogram
`Equalp
robabilityintervalhistoram
s:classintervalsofva
riablew
idth
eachco
ntainingth
esam
enum
ber
ofoccurrences
`Equalw
idthintervalhistotam
s:classintervalsoffixedwidth
eachpossib
lyco
ntainingadiffe
rent
numbe
rofo
ccurrences
freq
uentlyusedtosh
owbo
thth
efreq
uencya
ndth
edistrib
utationof
occurren
ces
keyp
rameters:nu
mbe
rofinterval,
intervalorig
in.
How
to
choo
se t
he n
umbe
r of
inte
rval
s?
Num
bero
fIntervals
`Toofewortoom
anyintervalswou
ldno
treflectth
edistrib
utionofth
epop
ulation.
K=N
1/2usedbyEx
celhistog
rams ov
erestim
atedop
timal
Man
n &
Wal
d
Scot
t
EqualP
robabilityIntervalH
istogram
`Theintervalshavedifferentw
idths.
`Thew
idthsa
rety
picallyde
term
inedsu
chth
atth
eprobability
ofanintervaleq
uals1/K,whe
reKisthen
umbe
rofintervals.
`Exam
ple:
25stud
entsinatestar
edividedinto5categorie
switheq
ual
prob
abilityasth
efollowingt
ables(i.e.5stud
entsinea
ch
interval).Plotth
eequ
alprobabilityh
istogram.
cate
gory
(sc
ore
rang
e)N
umbe
r of
st
uden
ts
1 (0
-50)
5
2 (5
0-60
)5
3 (6
0-65
)5
4 (6
5-75
)5
5(7
5-10
0)5
Equalw
idthintervalhistotam
s Cours
e 20
1541`Ba
nds(databins)o
fequ
alwidthac
rossth
erangeof
measurementvalue
sarede
finedan
dthen
umbe
rof
measurementswith
inea
chba
ndisco
unted.
`Exam
ple:Drawahistogramfo
rthe23m
easurementsinse
tCMeasurement4
0940
6402
4074
0540
440740
440740
7408
4064
1040
640540
840640
940640
540940
6407
num
ber
of b
ands
: 1+
3.3
log 1
0(23
)=5.
49.
first
ban
d as
401
.5 t
o 40
3.5
no m
easu
rem
ents
sho
uld
fall
on t
he b
ound
ary
betw
een
diffe
rent
ba
nds
and
caus
e am
bigu
ity a
bout
w
hich
ban
d to
put
the
m in
.
Prob
abilityDen
sityF
unction
Cou
rse
2015
43
freq
uenc
y di
stri
butio
n cu
rve
prob
abili
ty c
urve
prob
abili
ty d
ensi
ty fu
nctio
n (p
.d.f.
)
Nor
mal
izat
ion
cum
ulat
ive
dist
ribu
tion
func
tion
(c.d
.f.)
Dp
: the
val
ue o
f dev
iatio
n th
at h
as t
he g
reat
est
prob
abili
ty.
Dp
indi
cate
s sy
stem
atic
err
ors
know
n as
bias
, th
at c
an b
e re
mov
able
by
reca
libra
tion.
Dp
= 0
: en
tirel
y ra
ndom
in n
atur
e
Gaussia
n(Normal)D
istrib
ution Cou
rse
2015
44`Measurementsetsthato
nlyc
ontainra
ndomer
rorsusually
conformtoadistrib
utionwithaparticularsh
apet
hatisc
alled
Gaussia
n.`Alternativen
amesfo
rtheGaussiandistrib
utionaren
ormal
distrib
utionorbe
llshapeddistrib
ution.
`misth
emeanv
alue
D=xm
`Asm
allerV
correspo
ndsw
ithsm
allerd
eviatio
nsof
measurementsfromth
emeanv
alue
Prob
abilityofaparticularm
easurementinaG
aussian
datase
t
Cou
rse
2015
45
`thep
robabilityt
hattheer
rorliesin
abandbe
tweenerrorlevelsD
1and
D 2canb
eexpressedas
The
stan
dard
Gau
ssia
ncu
rve
has
ast
anda
rdde
viat
ion
ofon
e(V
=1)
and
am
ean
ofze
ro
StandardGaussianTables
Cou
rse
2015
46
z
V= 1
`ForV
,z=1.0
`P(1)=1P
(1)=10
.8413=0.15
87
=10.15870.1587
=0.6826
`32
%ofthem
easuremen
tslieoutsid
ethe
rVbo
undarie
s,th
en68
%ofthe
measuremen
tslieinsid
e.`bo
undarie
sofr
2Vcontain95.4%ofdatap
oints
`r3
Vbou
ndariese
ncom
passes99
.7%ofdatap
oints
Howm
anym
easurementshaveade
viationgreater
than|V
|
Cou
rse
2015
47
68%
StandardEr
roro
ftheM
ean
Cou
rse
2015
48`Averaginganu
mbe
rofm
easurementswillon
lyyieldthet
rue
valueifthenu
mbe
rofm
easurementsisinfin
ite.
`Standarder
roro
fthem
ean
`Dt
endsto
wardz
eroasth
enum
bero
fmeasurements(n
)in
thed
atas
etex
pand
stow
ardinfinity.
`Proced
ure
nsubsetsa
reta
kenf
romaninfin
iteda
tapo
pulatio
nthem
eansofthesu
bsetsw
illfo
rmaGa
ussia
ndistrib
ution
withacorrespo
ndings
tand
ardde
viationV
MeasurementswithCon
fiden
ceLimit
Cou
rse
2015
49`Ho
wtopr
edicttheer
rorb
etweenthec
alculatedm
eano
fa
finitese
tofm
easurementsan
dthem
eano
ftheinfin
ite
popu
latio
nusingt
hestandarder
roro
fthem
ean?
`Them
easurementvalueobtaine
dbyca
lculatingt
hem
eano
fa
seto
fnm
easurementsca
nbee
xpressedas
with68
%ce
rtainty
with95
.4%ce
rtainty
and99
.7%fo
rr3D
`Forn=23
,V=1
.88,an
dD=0.39
Them
easurementvalueca
nbee
xpressedas
406.5r
0.4(
68%co
nfiden
celim
it).
Mea
n of
mea
sure
men
ts :
409
406
402
407
405
404
407
404
407
407
408
406
410
406
405
408
406
409
406
405
409
406
407
Estim
ationofRando
mEr
rorinaS
ingleM
easurement
Cou
rse
2015
50`Estim
atet
helik
elym
agnitudeofer
roro
fam
easurements,if
onlyon
emeasurementcanbem
ade.
`Then
ormalap
proachtoth
isistoca
lculatet
heer
rorw
ithin
95%co
nfiden
celim
its.The
selim
itsco
rrespo
ndtoade
viation
of1.96V
Vf
or
`Them
axim
umlik
elye
rrorinasin
glem
easurementcanbe
expressedas
whichinclud
esdeviatio
nofth
emeasurementfromth
ecalculatedm
eana
ndth
estand
arderroro
fthem
ean
For
95%
con
fiden
ce li
mits
Exam
ple
Cou
rse
2015
51`Supp
oset
hatastandardm
assism
easured30timeswithth
esameinstrum
enttoc
reateare
ferenced
atas
et,andth
ecalculatedva
luesofV
andDa
reV
=0.46a
ndD
=0.08.Ifth
einstrumentisthe
nusedtom
easureanun
know
nmassa
ndth
ereadingis1
05.6kg
,howsh
ouldth
emassv
aluebee
xpressed
?
`Solutio
n:`1.96
(VD
)=1.06
.Them
assv
aluesh
ouldth
ereforeb
eexpressedas10
5.6r
1.1k
g.
Endofth
eChapter
Cou
rse
2015
79