PRIMER on ERRORS & Analysis – part II8.13 U.Becker Sept.,2006

Recall: ♦ Accura cy= closeness to “truth” (NIST)♦ Precision = quality o f data ~ 1/√N usually

♦ systemati c error = reproducible, but deviati on in system, apparatus, environmen t orcalibration♦ Statistical error = d uet o finite statist ~ 1ics /√N

Mean:

€

x=1N xii=1

N∑

Var ian ce :

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sx2= 1N−1 xi−x( )2

i=1

N∑For indepe nde nt me as urem entsnothing d epe nds on the pre vious opera tions or history.

Poisson Distr.

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P(x.μ)=μx

x!e−μ =(t/τ)r

r! e−t/τ

mean = μvariance = μ

The p roba bility dP to obse rve n o coun t in dt for rand om ev e ntssp a ced τ on ave rage is: dP(0;t,τ)=-P(0;t,τ) dt/τ. The obvious s olutionis: P (0;t,τ) = e -t/τ

Which means s hort interva ls are more like ly!!

Gau ssi an

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P(x;μ,σ)= 12πσe

(x−μ)2

2σ 2

mean = μvariance = σ = μ s tanda rd de viation

Applica tions are numero us ,folding two Gauss ians y ields aGau ssi an ag ain. The error s s imply add in quadra ture .

μ x

P

μ x

P

Γ = 2.354σ

PROPAGATION OF ERRORS Bevington ch.3

You measure u±σu a nd v±σv

bu t evaluate x= au+ b v or x = au/ v and wa nt theerro r forthose

Taylo r expansio :n

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xi−x=(ui−u)∂x∂u+(vi−v)∂x∂v+.........

then

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σx2=limN→∞1N(ui−u)∂x∂u+(vi−v)∂x∂v

⎛ ⎝ ⎜ ⎞

⎠ ⎟2

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=σu2∂x∂u ⎛ ⎝ ⎜ ⎞

⎠ ⎟2+σv2∂x∂v

⎛ ⎝ ⎜ ⎞

⎠ ⎟2+2σuv2 ∂x∂u

⎛ ⎝ ⎜ ⎞

⎠ ⎟∂x∂v ⎛ ⎝ ⎜ ⎞

⎠ ⎟

Sum&Diff: x = a u ± bv ->

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σx2=a2σu2+b2σv2±2abσuv2

Prod.&Div: x = auv or a u/v ->

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σx2x2=σu

2

u2+σv2

v2+2σuv2uv

Powers : x = a T4 ->

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σxx=± 4 σTT watch o ut!

= 0 if uncorrelated,otherwiseσμν covariance matrix -difficult

8.13

Fa mou s Confu sion – wa tch ou t

Best estimate= mean:

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x=1N xii=1

N∑ ±σxN error of the me a n

Bevington eq 4.1 2 ma ke m a ny me a sur e ments !

≠ S tan dard dev iat ion of the dist ribut ion

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σx2=sx2= 1N−1 xi−x( )2

i=1

N∑

In pract ice : Ofte n me a sure ments have differe nt e rrors σ I

Combine:

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x=xiσi2∑1σi2∑ ± 1

1σi2∑

They a re ca lled: we ighte d ave rag e ± co mbine d error

σx

Real life Part I

Measure resistors UB9/12/06 Manufacturer: 5% 2.20 ± 0.01 k

<R> = 2.178 ± 0.023 k

OK

Both histogram show the same data

Finer binning sometimes confusing -> Lower statistic/bin

Group1 9/10/04

Group1 9/10/04

Both 8.13 Groups ~ same

Systematics of DVM-> small difference, both:gap

Manufacturer 100 ± 5

<R> = 103 ± ??

Not an independent sample!! Best ones were selected out (gap)!!

Never a normal distribution.

Variance can still be quoted ± 7 !!!

Real Life Part II:

what

??

One more distibution:Lorentzian or

Breit-Wigner

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P(x:μ,Γ)=1π

Γ2m−μ( )2+Γ2( )

2

Line width Heisenberg Γ•τ ≥ h

Example: e+e- -> μ+μ- reconstruct the massevents yieldMμμ = 91.5, 92.1, 90.9, 93.0, 91.1…….n

Result:MZ = 91 187± 7 MeV/c2 better σM/√nΓZ = 2 490± 7 MeV/c2 σΓ/√n

μx

P

Γ

μ=Mz

x

P

Who cares ?? WE !!!! Particle data book http;//pdg.lbl.gov

Values and errors absolutely crucial.

1. EW Theory - γ,Z transmi t force, belon g to sa ‘me family’ Electro w. The : o Weinberg, Salam: bold! - revise s Maxwe ll a thigh E (Newton a t v->c)

2. MZ a s important a s fine structure constant α !!Bigge st systematic error: Mo on an d Sun.deformaccelerator LEP

3. Reca ll the uncertain tyrelation

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ΔE ⋅Δt ≥ h /2 every new de cay sh orten s Δ t hence increases Γ=ΔE

4. ΓZ = ΓZ-⇒ ee+ΓZ-⇒μμ+ΓZ->all: Each ne w kind adds 140 MeVFrom mea su red Γ -> There ar e thre e kinds of neu trinos . (Not DM)

Analysis –Measurements of Functional Dependences

Example Decay of cosmic muons

Log-lin good for powers&exp

Precise data Accurate databut inaccurate but not precisesomewhere a systematic error random errors too largerepeat differently (apparatus) measure more statistics

T=0 value seems wrong bad fit intervals too largedelete (must explain!) go finer

N(t) = (300± 20 stat±50syst) (N t) = (330± 60 stat±10syst) e {xp t/(2.05±.03μ }s e {xp t/(2.3±.6μ }s

good no t impressive

Theo

1 2 μsμs

Decays/s

Theo

1 2μs

log #/s

100

10

1

Theo

1 2μs

Theo

12μs

log #/s

100

10

1

The art of fitting Bevington,ch.4

Still one more: χ2- distribution:

Situation: We kno w th (e resul ) t function havin g value s μI it ha s in the -i th bin.We measu re n times with rand om errorσi in eac h bin an d evaluate

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χ2 = xi−μiσi

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2

i=1

n

∑

Clear ly this quan tity ha s a distribu tion itse lf,bec au se the next ra ndom samp le o f n measuremen ts will have different χ 2.

Why is this important? You u sually ha ve the rever se case ,n me as ureme nts and a “be st fit”, an d you wan t to kn ow: ”Is the fit accep table?”The χ 2 distribu tion gives y ou the proba bility tha t your me as ureme nt has this par ticular χ 2.Define χ 2=u and ν = n - #par ame ter s = “degree s of freedom”.(If we ha d 2 d a ta p oints a nd 2 para mete rs the curve must go th roug h the m, the re is n o “freed om to fit”.)The distribu tion is

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P(u)=u2( )ν2−1e

2Γν2( )

u2

where Γ = Gamma funct.

So, χ2 = 8 ha s still an acceptab le probabili tyfor ν = 5,bu t χ2 = 8 i s ver y unlikely for ν = 1. In thi s case the dat a orthe hypothesi s (= the theoretical fi t formula) may be wrong.Loo k for a better formul a or data!!

Note: for ν > 2 obtainin g χ2 = 0 ha s probabili ty= 0!!

8

Are the measurements shown in the histogram Gaussian?

Do not get confused; the question refers to the fit function.

Gaussian: χ2 = 3.0 for ν = 6-2 , or χ2/do f= 3/4, acceptable.

Bu t a Lorentzian yields χ2/do f= 1.2, also acceptable!-> Higher statisti cs i s needed to tell the difference.

Gaussian Shape?

Gaussian errors

Poisson error

1 2 s

Theo

Decays/s

Theo

1 2 s

log #/s

100

10

1

Theo

1 2 s

Theo

1 2 s

log #/s

100

10

1

Back to our measurements of the muon lifetime:

χ2/dof = 0.3 for ν =4-2, which is fine.

Result: (μ)= 2.30.6 μsnot accurate - nobody cares

χ2/dof = 18.1 for ν = 8 - 2, exponential (hypothesis) or data doubtful. Note: the first point is >4σ off the curve ~16 to χ2. Was started to late, delete. NOW (μ) = 2.050.03 μs

good!

Best Fits: Max. Likelihood or MIn. χ2 Bevingto nCh.6,11

Consider : f(x) = y = y(x;aset) with aset of parameters: aset = (a1, a2, a3,… an)

compared (fitted) to data with errors in x and y as shown.Step one is to convert those into common errors in y:

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σy' = σy2+σx2∂y∂x ⎛ ⎝ ⎜ ⎞

⎠ ⎟2

If the yi values are Gaussian distributed, the probability of finding

(xi,yi) is

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Pi =1

2πσie− yi−y(x,aset)( )2

2σi2

i=1

n

∏The probability of getting just the n observed values is the likelihood:

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L=P(aset,n)= Pii=1

n

∏ = 12πσi

e− yi−y(x,aset)( )2

2σi2

i=1

n

∏ = 12πσi

e−χ

2

2

i=1

n

∏

Fit now by maximizing L (or minimizing χ2) to find the best values of aset(=slope,intercept,……)

x

y

IF 68% OF THE ERROR BARS TOUCH THE CURVE. -⇒ OK !!

Less precise: if -χ2/Do F ~ 1, bu t see fig o f χ2 -distributions

HOW??♦Compute a “grid” o f χ2(a1,…an) interpolate for lowes t -⇒ slow, inaccurate♦ Foll owthe gradient, fas t -⇒ may ru n into a loca l ma /x min♦Use Markquar t combination - -⇒ fancy

Whe n is a fit good? LOOK at the grap !h Rule o f thumb:

In real life parameters often a re correlated,which one ca n show for two parameter s bytheir L- contour s (ellipse), because the error i s reac hed atΔχ2 = 1/2. The “normal “errors” correspo nd to the lines indicated.

a1

a2

More Bevington, Taylor…. Let’s go!