Uncertainty Lecture S12

8/13/2019 Uncertainty Lecture S12

http://slidepdf.com/reader/full/uncertainty-lecture-s12 1/31

uncertainty andstatistics

Physics 6510 / 4410Spring 2012

train derailment at GareMontparnasseParis 1895

nders !yd

than"s to #yle Shen$ Matthias %iepe$ Georg&o''staetter$ and others



(hy do (e care a)out uncertainties*

it pro+ides in'ormation a)out the precision o' the

measurement$ and can distinguish )et(een a lessprecise measurement and a ,o)el pri-e (orthydisco+ery

%et.s say you per'orm the a+endish alance

periment 3G10$ and o)tain a +alue 'or thegra+itational constant as

(here the epected +alue isesta)lished as

(ould )e a result in good agreement (ith the epected

+alue



accuracy +s7 precision

accurate$

not precise

precise$

not accurate

• uncertainty has to do (ith the precision 3i7e7reproduci)ility o' a gi+en measurement

• error has to do (ith the de+iation 'rom the accepted+alue 3not a mista"e or a )lunder

● (e are generally interested in the uncertainty )ecause

in science (e typically do not "no( the true +alue



di''erent "inds o' uncertainties anderror

!andom / Statisticalaused )y inherent$ unpredicta)le 'luctuations either inthe measurement apparatus$ the eperimenter.sinterpretation thereo'$ and/or counting statistics7

Systematicould originate 'rom imper'ect cali)ration o'instruments or inter'erence 'rom en+ironment7 :hese(ill a''ect the results o' an eperiment in a predicta)le$reproduci)le 'ashion7

:heoretical;n o)taining a +alue$ theoretical assumptions may needto )e made (hich may ha+e their o(n associateduncertainties 3muon li'etime eperiments



<uoting uncertainties in yourmeasurements

=hen possi)le$ <uote these uncertainties separately3particularly 'or the nuclear / particle eperiments7>'ten this is not possi)le7

an alternate (ay o' epressing the same in'ormation isas

ontrary to (hat you may ha+e )een taught in otherclasses$ (e do not (ant a separate rror nalysis

section in your la) report? @iscussion o' uncertaintiesshould )e em)edded throughout



rror analysis in your reportGood ad

onsider this a guideline ,ot all la)s 'its into this mold A discuss (ith yourinstructor

;ncluding error analysis



1. V F 07150 mC

2. V F 07180 mC

3. V F 072D6 mC

4. V F 074E2 mC

D704D$ D7426$ D726E$ 27954$ D7D2E$ D714D$ D7D94



Bou need to determine the +oltage drop across acircuit in your la)7 Bou ta"e a series o' +oltagereadings$ recording the +alue 3in mC on the

+oltmeter e+ery 5 secondsD704D$ D7426$ D726E$ 27954$ D7D2E$ D714D$ D7D94

What is the uncertainty in the mean voltage?

=hich o' the outcomes (ould you epect*

1. σV F 07180 mC

2. σV F 07090 mC

3. σV F 07068 mC

4. σV F 07026 mC



1. σV F 07180 mC

2. σV F 07090 mC

3. σV F 07068 mC

4. σV F 07026 mC

:he standard de+iation tells us a)out the +ariations'rom measurement to measurement7 &o(e+er$ i' (eta"e many measurements$ (e can determine the

mean to much )etter accuracy than the standardde+iation7

ta"ing more data allo(s us to moreaccurately determine the mean

+alue o' C )y a 'actor o' ,1/2

3eplanation later



Coltage

3mC5

Measurement ,um)er 3,

%et.s assume you made D00 di''erentmeasurements777

the rolling a+erage

the amplitude o' the (iggles in the a+erage3red line die out roughly as ,1/2



:he Gaussian 3or ,ormal @istri)ution

●

(e typically <uote uncertainty as H 1σ 3i7e7 the +aluelies in a )and (here it is 68I li"ely to 'all (ithin themean +alue● 2σ di''ers 'rom the 'ull (idth at hal' maimum3J=&M$ another common (ay o' gi+ing uncertainty



although the normal distri)ution is commonlyencountered 3central limit theorem$ it isimportant to note that many distri)utions are notnormal$ )oth in the la) as (ell as in the real(orld777

Percentageo'

:otal=

ealth

>(ned

3I5

Percentile 3I )ins are 1I (ide

@istri)ution o' (ealth 'or marriedhouseholds in K7S7 'or 6069 age )rac"et2004 Sur+ey o' onsumer Jinances

“part of the problem is today, only 53 percent pay any federalincome tax at all....47 percent pay nothing”

- Michelle Bachmann



counting eperiment measures e+ents caused)y a random process 3radioacti+e decay$ cosmicrays collisions (ith molecules in the outer

atmosphere7 Bou set up your eperiment and do10 separate measurements$ counting 'or 100seconds each instance7

=hich o' the outcomes should )e most plausi)le*

1. 145, 136, 150, 161, 146, 145, 1, 155, 165, 122

2. 150, 140, 151, 144, 13!, 14, 141, 156, 141, 140

3. 205, 1"!, "!, 113, 145, 153, 112, 163, 1!5, 160

4. 143, 14!, 143, 144, 145, 143, 143, 145, 146, 143



:he Poisson @istri)ution

P is the pro)a)ility that eactly ,e+ents occur i' the mean +alue o'e+ents is λ.

+ents o)ser+ed 3,

Pro)a)

ility

λ F 4● indi+idual e+ents are

uncorrelated and occurrandomly$ although the meanrate$ λ$ is "no(n



'irst use o' the Poisson distri)ution

nalysis o' people "illed )yhorse "ic"s in the Prussiancal+ary

Jatalities per unit per year



+ents o)ser+ed 3,

Pro)a)ility

λ F 4

+ents o)ser+ed 3,

Pro)a)ility

3104D5 λ F 50

;n the limit o' large λ 3λ > 10$ the Poissondistri)ution closely approimates a gaussian

:he standard de+iation o' the distri)ution=



4. 143, 14!, 143, 144, 145, 143, 143, 145, 146, 143

Someone claims that they actually measured thea)o+e pro)a)ility distri)ution7 =hat is theli"elihood that they are in 'act telling the truth*

1. 5I

2. 071I

3. 0700001I

4. 070000000001I

20I

307210 L 10E



&ome(or"

ll

Jor the Poisson distri)ution

Sho( that the mean isλ and the standard de+iation is

;' you do an counting eperiment$ e7g7 ,1$ ,2$ ,4$

,15$ ,16$ ,1E

!ecord a 'e( short data sets and eperimentally con'irmthat

=

=



1. 2500 # 100

2. 2500 # 300

3. 2500 # 500

4. 2500 # !00

Bou need to determine the +alue o' a resistor inyour eperiment7 Bou measure the +oltage dropacross the resistor as )eing D70 H 07D C and the

current across it as 172 H 072 n7

What is the value $ uncertainty o% theresistance?

=hich o' the outcomes (ould you epect*



ssumptions*• Caria)les a$)$c are uncorrelated• σₐ / a N 1 3i7e7 σₐ can )e considered as adi''erential

(here the deri+ati+es are e+aluated at their mean +alues

onsider the 'unction



;n an eperiment$ you plot the pea" +oltagemeasured on an oscilloscope as a 'unction o' delay

time that you can control7 Bou (ant to o)tain the +oltage as a 'unction o' thedelay$ )elie+ing that the +oltage should dependlinearly on the delay

@elay 3msPea" Coltage

3mC

171 276 H 175

D75 875 H 275

E78 1570 H 470

1274 2478 H 572

167D 2978 H 472

1872 D672 H D71



least s<uares 'itting

:he o)Oecti+e is tominimi-e S$ the sum o' thes<uared residuals7 :his isaccomplished )y

minimi-ing S (ith respectto the di''erentparameters 3 and in this eample



i' the signal is real$our con'idenceshould gro( 'rom 2

sigma to a)out 2sigma 21/2 L 278sigma$ since ouruncertainty typically

goes do(n )y ,1/2 (here , is theamount o' datacollected777



Jollo(up lecture 3Je)7 2E* (ill loo" atiasentral limit theorem%i"elihood%imit setting

Uncertainty Lecture S12

Documents

Transcript of Uncertainty Lecture S12