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Ch.5 Errors

during themeasuremen

t process

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IntroductionError is the diference between the

measured value and the true value:Absolute error|Error| = | measured value true value |

= | E A |

Percent Error

Problem: The true value is veryseldom known

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ExampleAn object is known to weigh 25.0 grams. You weight the

object as 26.2 grams. What is the accuracy inaccuracy

error an! percentage error o" your measurement#

The instrument is 95.2 accurate

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Introductionncertainty

An! measured "uantit! should besub#ected to uncertaint! $can%t beavoided& de'ends on instrument

resolution( )ncertaint! = we are estimatin* the'robable error& *ivin* us an intervalabout the measured value in which we

believe the true value must +all. )ncertaint! Anal!sis

, 'rocess o+ identi+!in* and "uali+!in*

errors.

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Introduction

/ Con!dence interval:The ran*e o+'robable values o+ an e'eriment

ncertainty

Error is 'rimaril! a theoretical conce't&because its value is un0nowable

)ncertaint! is a more 'ractical conce't

Evaluatin* uncertaint! allows !ou to'lace a bound on the li0el! si1e o+ theerror

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E""#"$process

'& Corruption during trans(er o( the signal(rom the point o( measurement to some

other point due to noise.#nly !rst type will be discussed here whichdivided to:Systematic errors: Errors that are

consistently on one side o( the correctreading) i.e. either all the errors arepositive or they are all negative.Random errors: Errors on either side o( thetrue value caused by random andunpredictable e*ects."andom errors o(ten arise when

measurements are taken by humanobservation o( an analo ue meter

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$ources o( systematic

error%& E*ect o(environmental

disturbances) wear) dust)and (re+uent use

'& ,isturbance o( themeasured system by the

act o( measurement. $ di b d

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$ystem disturbance due tomeasurement processhot water with a mercury&in&glassthermometer. The thermometer is a coldmass) causes to lower the temperature o(

the water.-easure ow rate o( a uid in a pipe usingori!ce plate) which is a (unction o( thepressure drop due to ori!ce and the ori!ce)

causes a pressure loss in the owing uid./eneral rule) the process o( measurementalways disturbs the system beingmeasured.The magnitude o( the disturbance varies(rom one measurement system to otherdepends on the type o( instrument used (or

measurement.0a s o( minimi1in disturbance o(

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Measurements in electric circuits2ridge circuits (or measuringresistance values are a (urther

e3ample o( the need (or care(uldesign o( the measurementsystem. The impedance o( the

instrument measuring the bridgeoutput voltage must be very largein comparison with the

component resistances in thebridge circuit. #therwise) themeasuring instrument will load

the circuit and draw current (rom

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Errors due to environmental inputsThe static and d!namic characteristicss'ecied +or measurin* instruments

are onl! valid +or 'articularenvironmental conditions 3e.*. o+tem'erature and 'ressure4& awa! +rom

the s'ecied calibration conditions&the characteristics o+ measurin*instruments var! to some etent and

cause measurement errors. Thema*nitude o+ this environmentvariation is "uantied b! the two

constants 0nown as sensitivit! dri+t

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0ear in instrument components

!stematic errors can +re"uentl!develo' over a 'eriod o+ time

because o+ wear in instrumentcom'onents. 6ecalibration o+ten'rovides a +ull solution to this'roblem.

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Connecting leads

The resistance o+ connectin* leads inelectrical measurement s!stems 3or'i'es in the case o+ 'neumaticall! or

h!draulicall! actuated measurements!stems4& are a common source o+error. 7or instance& in a resistance

thermometer that se'arated 8 m+rom other 'arts o+ the measurements!stem. The resistance o+ such a

len*th o+ 2 *au*e co''er wire is .

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"andom errors;aused b! un'redictable variations.

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)ncertaint! Anal!sis

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>?@ T?E6E )B;E6TABT@CMeasurements are performed with

instruments, and no instrument can read toan infinite number of decimal placesUncertainty in measurementdepends on

the scale of the apparatus?

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)B;E6TABT@ B DEA)6EDEBT

A reading uncertaintyis how accurately aninstruments scalecan be read.

Analogue Scales

Where the divisionsarefairly large, theuncertainty is taken as:

half the smallest scaledivision

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the smallest scaledivision

Where the divisions are small, the uncertainty istaken as:

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Digital Scale

For a digital scale, the uncertaintyis taken as:

the smallest scale reading

e.g.voltage = 29.7 mV 0.1 mV

This means the actual readingcould be anywhere from

29. to 29.!

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Measuring Length with meter: smallest scale is 0.1 mm

so uncertainty is 0.05 mm

Example $%

length of a marker is 12.6 cm uncertainty is 0.05mm

Indicating the length of the marker could e 12.65 cm or

12.55

Example 2 %

Measuring !olume li"uid in a cylinder: smallest scale is2 ml so uncertainty is 1 ml.

Measured !olume is #$.$ ml uncertainty is 1 ml

so !alue can e #5.$ ml or ##.$ ml

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Example &:

Measuring a mass with a digital weigh scale

smallest measurement is 0.1 g so uncertainty

is 0.1 g

% measurement of a sam&le is 2#.# g

'ritten as 2#.# 0.1 g

so the !alue can e 2#.2 or 2#.$ g

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2A$IC "4E$ #" 6CE"TAI6T7CA4C4ATI#6$

"##reading

yuncertaintabsolute

yuncertaint$ =

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Eam'le

m 8 9. ; .%%

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Combining uncertainties

%= Addition and $ubtraction: A theAbsolute )ncertainties"ule: 3A F GA4 H 3I F GI4 = 3A H I4 F

3GA H GI43A F GA4 J 3I F GI4 = 3A J I4 F 3GA H GI4;onsider the numbers: 3K.5 F .54 m and

3L.L F .84 mAdd: 3K.5 F .54 m H 3L.L F .84 m = 39.MF .K4 m

ubtract: 3K.5 F .54 m J 3L.L F .84 m =

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'= -ultiplication and ,ivision: A the6elative )ncertainties"ule: 3A F NA4 3I F NI4 = 3A I4 F 3NAH

NI4 3A F NA4 3I F NI4 = 3A I4 F 3NAH

NI4

;onsider the numbers:35. m F O.4 and 3L. s F L.L4Dulti'l!:35. m F O.4 3L. s F L.L4 = 385. m/s

F .L4


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= or a number raised to a power&+ractional or not& the rule is sim'l! toD)PTQP@ the 6elative )ncertaint! b! the

'ower."ule: 3A F NA4

n= 3AnF nNA4

;onsider the number: 32. m F 8.4

;ube: 32. m F 8.4

= 3M. m

F L.4"uare 6oot: 32. m F 8.4%'= 38.O m%'F .54


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= or multiplying a number by aconstantthere are two diferent rules de'endin* onwhich t!'e o+ uncertaint! !ou are wor0in*

with at the time."ule & Absolute ncertainty: c3A F GA4= cA F c3GA4

;onsider: 8.532. F .24 m = 3L. F .L4 mBote that the Absolute )ncertaint! ismulti'lied b! the constant."ule & "elative ncertainty:

c3A F NA4 =


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%se the following data to calculate thes&eed, and the uncertainty in s&eed, of amoving ob'ect.

t

dv =

cm#.(cm"d =

s#.(s2t =

)v=

Eam!le" #alculation of S!eed

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rtainties(or Calculations Involving un

FUNCTIONS OF ONE VARIABLE

If the calculated parameterR is a functin f the measured !aluex"

thenR is said t #e a functin fx" and it is ften $ritten asR%x&'

(hen this is the case" the uncertaint) assciated $ithR is #tained #)

$here

*x is the uncertaint) in the measurement fx'

is the a#slute !alue f the deri!ati!e fR $ith respect tx

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t i ti ( C l l ti I l i

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ertainties(or Calculations Involving unFUNCTIONS OF +ORE T,AN ONE VARIABLE

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al analysis o( measurements subDect to randoan and median valuesThe avera*e value o+ a set o+ measurements o+ aconstant "uantit! can be e'ressed as either the mean

value or the median value.As the number o+ measurements increases& thediference between the mean value and median valuesbecomes ver! small.

7or an! set o+ n measurements 8& 2& RR& n themean *iven b!:

The median is the middle value when themeasurements in the data set are written down inascendin* order o+ ma*nitudeS the median value is*iven b!:

an and median values

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7or a set o+ 9 measurements 8& 2& R..&9 themedian = 35.7or 8 measurements 8&R..&8& the median =

an and median values

$uppose that the length o( a steel bar is measuredby a number o( di*erent observers and the

(ollowing sets o( %% measurements are recorded9units mm=. 0e will call this measurement set A.9-easurement set A=F & '< & & %> &

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an and median values>hich o+ the two measurement sets A and I& should wehave most condence inC.et I as bein* more reliable since the measurements are

much closer to*ether. n set A& the s'read between thesmallest 3L9K4 and lar*est 3OL4 value is LO& whilst in set I&the s'read is onl! K.Thus, the smaller the spread of the measurements,the more condence we have in the mean or median

value calculated.Pet us now see what ha''ens i+ we increase the number o+measurements b! etendin* measurement set I to 2Lmeasurements. >e will call this measurement set ;.9-easurement set C=

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(tandard de!iation and !ariance

E-pressin. the spread f measurements as the

ran.e #et$een the lar.est and smallest !alue isnt a !er) .d $a) f e-aminin. h$ the

measurement !alues are distri#uted a#ut the

mean !alue' A much #etter $a) f e-pressin.the distri#utin is t calculate the !ariance r

standard de!iatin f the measurements'

The variance %V& is then .i!en #)/

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(here/ d is the de!iatin frm the mean

Thestandard deviation is simpl) the s0uare rtf the !ariance' Thus/

Example:

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Example:

Calculate 1 and V fr measurement sets A" B and C

Solution:(et % )mean * $0+,

'(!e)iations*2+ $&,0-n + number o" measurements + $$.

hen /+'(!e)iations2*n 1$

+$&,0$0+$&, + 2 33'4

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(et - )mean * $06,

Frm this data" usin. same anal)sis"V 2 5'6 and 1 2 6'78'

( ) $06 5,

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(et )mean * $06.5,

Frm this data" usin. same anal)sis" V 2 9'89

and 1 2 3'::

N h h ll l fSummary

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V 1

Set A 394 33'4Set B 5'6 6'78

Set C 9'89 3'::

Nte that the smaller !alues f

V and 1 fr measurement set B

cmpared $ith A crrespnd

$ith the respecti!e si;e f thespread in the ran.e #et$een

ma-imum and minimum !alues

fr the t$ sets'/hus as and decrease for a measurement set we are ale to

e&ress greater confidence that the calculated mean or median

!alue is close to the true !alue i.e. that the a!eraging &rocess has

reduced the random error !alue close to 3ero.om&aring and for measurement sets - and and get

smaller as the numer of measurements increases confirming

that confidence in the mean !alue increases as the numer of

measurements increases.

Summary

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3raphical !ata analysis techni4ues H"re4uency !istributions

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an! 7nter)al8 90$.5190&.9 90&.51905.5 905.5190,.5 90,.5190:.5 90:.519$$.5

;umber o"

measurements$ 5 $$ 5 $

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igure A: istogram o( measurements anddeviations.As the number o( measurementsapproaches in!nity) the histogram becomes

a smooth curve known as a frequency

igure A igure 2

3raphical !ata analysis techni4ues H "re4uency !istributions

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3raphical !ata analysis techni4ues "re4uency !istributions

The rdinate f this

cur!e is the fre0uenc)

f ccurrence f each

de!iatin !alue" F %=&"

and the a#scissa is thede!iatin" ='

I( the (re+uency distribution curve isnormali1ed such that the area under it isunity) then the curve is known as a

probability curve) and 9,= at any given


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The cnditin that the

area under the cur!e is

unit) can #e e-pressed

mathematicall) as/

The pr#a#ilit) that the errr in an) measurement lies #et$een

t$ le!els =3 and =6 2 the area under the cur!e cntained

#et$een t$ !ertical lines dra$n thru.h =3 and =6' This can

#e e-pressed mathematicall) as/


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Thecumulativedistributionfunction9c.d.(.= This is

de!ned astheprobability o(

observing avalue lessthan or e+ual

to ,

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4aussian distriution

Fr measurement f randm errrs nl)The fre0uenc) f small de!iatins frm the

mean !alue is much .reater than the fre0uenc)

f lar.e de!iatins'The num#er f measurements $ith a small errr

is much lar.er than the num#er f measurements

$ith a lar.e errr'Alternati!e names fr the

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Fr measurement f randm errrs nl)The fre0uenc) f small de!iatins frm the

mean !alue is much .reater than the fre0uenc)

f lar.e de!iatins'The num#er f measurements $ith a small errr

is much lar.er than the num#er f measurements

$ith a lar.e errr'Alternati!e names fr the

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4aussian distriutionA

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If the de!iatins >= 2 -?m@

su#stituted in the e0uatin"

then

4 i di t i ti

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The cur!e f de!iatin fre0uenc) F%=& pltted a.ainst

de!iatin ma.nitude = is a

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4aussian distriutionSlutin f this e-pressin is simplified #) the

su#stitutin/ ? + > + x1mThe effect f this is t chan.e the errr distri#utin

cur!e int a ne$

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Unfrtunatel)" this e0uatin cant #e sl!ed anal)ticall) usin.ta#les f standard inte.rals" and numerical inte.ratin

pr!ides the nl) methd f slutin'

,$e!er" standard

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Thus" F%;& .i!es the prprtin f data !alues that are

less than r e0ual t ;' This prprtin is the area

under the cur!e f F%;& a.ainst ; that is t the left f ;'

(tandard 4aussian tales

It ta#ulates F%;& fr

!arius !alues f ;"

$here F%;& is .i!en #)/


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=91'= H

91%=

(tandard 4aussian tales

The ;3

and ;6@can #ee-pressed as/

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EA+DLE

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EA+DLE/

Findin. the area under the

standard nrmal cur!e t theleft f 3'69

EA+DLE/ Findin. the area under the standard

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.

nrmal cur!e t the ri.ht f 7'4

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EADQPE:7BBU T?E A6EA )BE6 T?ETABA6 BEEB .KM AB 8.M2

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SUMMARY

Example:

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Example:

,$ man) measurements in a data set su#ect t

randm errrs lie utside de!iatin #undaries f G1

and ?1Solution

7or E = FW& X = F8

The re0uired num#er isrepresented #) the sum f the

t$ shaded areas in Fi.ure =

This can #e e-pressedmathematicall) as/

D%E H ?1 r E G1&2 D%J H ?3& G D%J G3&

D%E E G & D%J 3& G D%J G3&

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D%E H ?1 r E G1&2 D%J H ?3& G D%J G3&

@sing able(B C 1$* + 0.$5D,

(B F$* +$1 0.D9$&

+0.$5D,

(E C 1 or E F* + 0.$5D, F 0.$5D, + 0.&$,9 G &2H

i.e. &2H o" the measurements lie outsi!e the I

boun!aries then 6DH o" the measurements lie insi!e.

Similar anal)sis sh$s that #undaries f K61 cntain

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Similar anal)sis sh$s that #undaries f K61 cntain

8'5M f data pints" and e-tendin. the #undaries t

K91 encmpasses '4M f data pints'

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(tandard error of the mean

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f

The pre!ius anal)sis measurements $ith randm

errrs are distri#uted a#ut the mean !alue' ,$e!er"

sme errr remains #et$een the mean !alue f a set f

measurements and the true !alue"

i.e. a)eraging a number o" measurements will only

yiel! the true )alue i" the number o" measurementsis in"inite.

The errr #et$een the mean f a finite data set and the

true measurement !alue %mean f the infinite data set&is defined as thestandard error of the mean" ' This is

calculated as/

(tandard error of the mean

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f

tends t$ards ;er as the num#er f measurementse-pands t$ards infinit)' The measurement !alue

#tained frm a set f n measurements" -3" -

6"RRR

-n" can then #e e-pressed as/ - 2 -meanK Fr the data set C " n 2 69" 1 2 3':: and 2 7'9'

The len.th can therefre #e e-pressed as 57'8 K 7'5

%:M cnfidence limit&' ,$e!er" it is mre usual te-press measurements $ith 8M cnfidence limits

%K61 #undaries&' In this case" 61 2 9'4" 6 2 7'4:

and the len.th can #e e-pressed as 57'8 K 7': %8M

cnfidence limits '

7 i i f d i i l

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7stimation of random error in a single measurement/

Errr 2 K%3'1 G &

Example 3.4Suppse that a standard mass is measured 97 times"

and the calculated !alues f 1 and are 1 2 7'59 and

2 7'7:' If the instrument is then used t measure anunn$n mass and the readin. is 378' ." h$

shuld the mass !alue #e e-pressedP

Solution

Errr 2 3'1 G 2 7'6' The mass !alue shuld

therefre #e e-pressed as/

378' K 7' .'

E3ample:The +ollowin* 8 measurements were made o+ out'ut volta*e +rom a hi*hJ*ain

li t i t d d t i Y t ti

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am'lier contaminated due to noise Yuctuations:1.53,1.57,1.54,1.54,1.50,1.51,1.55,1.54,1.56,1.53

Estimate the accurac! to which the mean value is determined +rom these 8measurements.

+ 8measurements were ta0en& instead o+ 8& but W remained the same& b! howmuch would the accurac! o+ the calculated mean value be im'rovedC >hat is the error in the 8.58 readin*& and write it.$olution:

8istriution of manufacturing tolerances

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+anufacturin. prcesses are su#ect t randm

!ariatins that cause randm errrs in measurements' In

mst cases" these randm !ariatins in manufacturin."$hich are n$n as tolerancesfit a

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$:.DCJ C 20.28 + 10.$C ? C F0.$8 + ? C 0.$8 1 ? C 10.$8

Krom tables ? C 0 $8 0 5&:D

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Krom tables ? C 0.$8 + 0.5&:D

? C 10.$8 + 0.9602

Lence ? C 0.$8 1 ? C 10.$8 + 0.5&:D 1 0.9602 + 0.0,:6

hus 0.0,:6 M $05

+ ,:60 transistors ha)e a current gain in the range "rom$:.D to 20.2.

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4oodness of fit to a 4aussian distriution

All f the anal)sis f randm de!iatins presented s

far nl) applies $hen the data #ein. anal);ed #eln.st a

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5 Errors S16

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