5 errors s16
TRANSCRIPT
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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
Combining uncertainties
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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
Combining uncertainties
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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 =
Combining uncertainties
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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 + $$.
hen /+'(!e)iations2*n 1$
+$&,0$0+$&, + 2 33'4
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(et - )mean * $06,
Frm this data" usin. same anal)sis"V 2 5'6 and 1 2 6'78'
( ) $06 5,
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(et )mean * $06.5,
Frm this data" usin. same anal)sis" V 2 9'89
and 1 2 3'::
N h h ll l fSummary
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V 1
Set A 394 33'4Set B 5'6 6'78
Set C 9'89 3'::
Nte that the smaller !alues f
V and 1 fr measurement set B
cmpared $ith A crrespnd
$ith the respecti!e si;e f thespread in the ran.e #et$een
ma-imum and minimum !alues
fr the t$ sets'/hus as and decrease for a measurement set we are ale to
e&ress greater confidence that the calculated mean or median
!alue is close to the true !alue i.e. that the a!eraging &rocess has
reduced the random error !alue close to 3ero.omå and for measurement sets - and and get
smaller as the numer of measurements increases confirming
that confidence in the mean !alue increases as the numer of
measurements increases.
Summary
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3raphical !ata analysis techni4ues H"re4uency !istributions
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3raphical !ata analysis techni4ues H"re4uency !istributions
an! 7nter)al8 90$.5190&.9 90&.51905.5 905.5190,.5 90,.5190:.5 90:.519$$.5
;umber o"
measurements$ 5 $$ 5 $
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3raphical !ata analysis techni4ues H"re4uency !istributions
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
3raphical !ata analysis techni4ues H "re4uency !istributions
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3raphical !ata analysis techni4ues "re4uency !istributions
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/
3raphical !ata analysis techni4ues H "re4uency !istributions
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3raphical !ata analysis techni4ues "re4uency !istributions
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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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
$ith a lar.e errr'Alternati!e names fr the
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4aussian distriutionA
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4aussian distriution
If the de!iatins >= 2 -?m@
su#stituted in the e0uatin"
then
4 i di t i ti
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4aussian distriution
The cur!e f de!iatin fre0uenc) F%=& pltted a.ainst
de!iatin ma.nitude = is a
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4aussian distriutionSlutin f this e-pressin is simplified #) the
su#stitutin/ ? + > + x1mThe effect f this is t chan.e the errr distri#utin
cur!e int a ne$
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4aussian distriution
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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4aussian distriution
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 ;"
$here F%;& is .i!en #)/
4aussian distriution
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4aussian distriution
=91'= H
91%=
(tandard 4aussian tales
The ;3
and ;6@can #ee-pressed as/
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EA+DLE
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EA+DLE/
Findin. the area under the
standard nrmal cur!e t theleft f 3'69
EA+DLE/ Findin. the area under the standard
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.
nrmal cur!e t the ri.ht f 7'4
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EADQPE:7BBU T?E A6EA )BE6 T?ETABA6 BEEB .KM AB 8.M2
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SUMMARY
Example:
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Example:
,$ 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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78 9 ;5