Basic Error Analysis - University Of Illinois · 2014-09-29 · data analysis. For data analysis...
Transcript of Basic Error Analysis - University Of Illinois · 2014-09-29 · data analysis. For data analysis...
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Basic Error AnalysisPhysics 401
Fall 2014
Eugene V Colla
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• Errors and uncertainties
• The Reading Error
• Accuracy and precession
• Systematic and statistical errors
• Fitting errors
• Appendix. Working with oil drop data
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𝑻 = 𝟔𝟑℉±? Best guess ∆𝑻~𝟎. 𝟓℉
Wind speed 4mph±? Best guess ±𝟎. 𝟓𝒎𝒑𝒉
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Clearance fit
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1675 Ole Roemer: 220,000 Km/sec
NIST Bolder Colorado c = 299,792,456.2±1.1 m/s.
Ole Christensen Rømer1644-1710
Does it make sense?What is missing?
Measurement of the speed of the light
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L=53mm±ΔL(?)∆𝑳 ≅ 𝟎. 𝟓𝒎𝒎∆𝑳 ≅ 𝟎. 𝟎𝟑𝒎𝒎
We do not care about accuracy better than 1mm
How far we have to go in reducing the reading error?
Acrylic rod
If ruler is not okay, we need to use digital caliper
Probably the natural limit of accuracy can be due to length uncertainty because of temperature expansion. For 53mm ∆𝑳 ≅ 𝟎. 𝟎𝟏𝟐𝒎𝒎/𝑲
Reading Error = ±𝟏
𝟐(least count or minimum gradation).
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Fluke 8845A multimeter
Example Vdc (reading)=0.85V
∆𝑽 = 𝟎. 𝟖𝟑 × 𝟏. 𝟖 × 𝟏𝟎−𝟓
+𝟏. 𝟎 × 𝟎. 𝟕 × 𝟏𝟎−𝟓 ≅ 𝟐. 𝟐 × 𝟏𝟎−𝟓
= 𝟐𝟐𝝁𝑽
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The accuracy of an experiment
is a measure of how close the
result of the experiment comes
to the true value
Precision refers to how closely
individual measurements agree
with each other
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• Systematic Error: reproducible inaccuracy
introduced by faulty equipment, calibration or
technique.
• Random errors: Indefiniteness of results due to
finite precision of experiment. Measure of
fluctuation in result after repeatable
experimentation.
Philip R. Bevington “Data Reduction and Error Analysis for the Physical sciences”, McGraw-Hill, 1969
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Sources of systematic errors: poor calibrationof the equipment, changes of environmentalconditions, imperfect method of observation,drift and some offset in readings etc.
Example #1: measuring of the DC voltage
RCurrent source
I
U
U=R*I
Rin
expectation
Eoff
𝐔 =𝐑 ∗ 𝐈 −
𝐑𝐑𝐢𝐧
𝐄𝐨𝐟𝐟
𝟏 +𝐑𝐑𝐢𝐧
actual result
Eoff=f(time,temperature)
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Example #2: measuring of the polarization current
Eoff
sample
Ip𝒆𝒐𝒖𝒕 = 𝒌𝟏 × 𝑰𝒑+𝒌𝟎 × 𝑬𝒐𝒇𝒇
200 250 300 350 400 450
-30
-20
-10
0
10
400 420 4400
2
4
6
8
10
296.4 296.6
-20
n0,2,6,8,12,14
Edc=0.76kV/cm(40V)
Field cooling
PMN-PT
T (K)
Ip (
nA
)
T (K)
𝒌𝟎×𝑬𝒐𝒇𝒇
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Example #3: poor calibration
LHe
HP34401ADMM
10mA
Resonator
Measuring of the speed of the second sound in superfluid He4
Temperature sensor
1.6 1.8 2.0 2.2
5
10
15
20
T (K)U
2 (
m/s)
Published data
P403 results
Tl=2.1K
Tl=2.17K
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Xmeas = Xtrue + es + er
Result of measurement
Correct value
Systematic error
Random error
0 20 40 60 80 100 120 1400.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
P
Xi
B
Xtrue
es=0
0 20 40 60 80 100 120 1400.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
P
Xi
B
Xtrue
es
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Siméon Denis Poisson (1781-1840)
) )
0,1,2,...!
n
rt
n
rtP t e n
n
r: decay rate [counts/s] t: time interval [s]
Pn(rt) : Probability to have n decays in time interval t
A statistical process is describedthrough a Poisson Distribution if:
o random process for a given
nucleus probability for a decay tooccur is the same in each timeinterval.
o universal probability the
probability to decay in a given timeinterval is same for all nuclei.
o no correlation between two instances(the decay of on nucleus does notchange the probability for a secondnucleus to decay.
0 5 10 15 20
0.0
0.1
0.2
0.3
P
number of counts
rt=1
rt=4
rt=10
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0 5 10 15 20
0.0
0.1
0.2
0.3
P
number of counts
) )
0,1,2,...!
n
rt
n
rtP t e n
n
r: decay rate [counts/s] t: time interval [s]
Pn(rt) : Probability to have n decays in
time interval t
rt=10
0
( ) 1 , probabilities sum to 1
n
n
P rt
2
0( ) ( ) ,
standard deviation
nn
n n P rt rt
0
( ) , the mean
n
n
n n P rt rt
Properties of the Poisson distribution:
𝝈 = 𝒓𝒕< 𝒏 >= 𝒓𝒕
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) )
0,1,2,...!
n
rt
n
rtP t e n
n
Poisson and Gaussian distributions
0
0.02
0.04
0.06
0.08
0.1
0 10 20 30 40
number of counts
pro
bab
ilit
y o
f
occu
ren
ce "Poisson
distribution"
"Gaussian
distribution"
2
2
( )
21
( )2
x x
nP x e
Gaussian distribution:
continuous
Carl Friedrich Gauss (1777–1855)
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2
2
( )
21
( )2
x x
nP x e
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x
2
Error in the mean is given as 𝝈
𝑵
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Source of noisy signal
4.898555.251112.933824.317534.679033.526264.120012.93411
Expected value 5V
Actual measured values
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10100
104 106
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Resultc
U xN
- standard deviationN – number of samples
For N=106 U=4.999±0.001 0.02% accuracy
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Ag b decay
0 200 400 600 8000
200
400
600
800
Co
un
t
time (s)
108Ag t
1/2=157s
110Ag t
1/2=24.6s
Model ExpDec2
Equation y = A1*exp(-x/t1) + A2*exp(-x/t2) + y0
Reduced Chi-Sqr 1.43698
Adj. R-Square 0.96716
Value Standard Error
C y0 0.02351 0.95435
C A1 104.87306 12.77612
C t1 177.75903 18.44979
C A2 710.01478 25.44606
C t2 30.32479 1.6525
0 50 100 150
-20
0
20
40
Re
sid
ua
ls
time (s)
-20 0 200
20
Co
un
t
Residuals
Model Gauss
Equation y=y0 + (A/(w*sqrt(PI/2)))*exp(-2*((x-xc)/w)^2)
Reduced Chi-Sqr
4.77021
Adj. R-Square 0.93464
Value Standard Error
Counts y0 1.44204 0.48702
Counts xc 1.49992 0.19171
Counts w 5.93398 0.40771
Counts A 219.24559 14.47587
Counts sigma 2.96699
Counts FWHM 6.98673
Counts Height 29.4798
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0
1 2
1 exp 2 expt t
y A A yt t
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0 50 100 150
-20
0
20
40
Re
sid
ua
ls
time (s)
Ag b decay
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-20 0 200
20
Co
un
t
Residuals
Model Gauss
Equation y=y0 + (A/(w*sqrt(PI/2)))*exp(-2*((x-xc)/w)^2)
Reduced Chi-Sqr
4.77021
Adj. R-Square 0.93464
Value Standard Error
Counts y0 1.44204 0.48702
Counts xc 1.49992 0.19171
Counts w 5.93398 0.40771
Counts A 219.24559 14.47587
Counts sigma 2.96699
Counts FWHM 6.98673
Counts Height 29.4798
Test 1. Fourier analysis
No pronounced frequencies found
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0 50 100 150
-20
0
20
40
Re
sid
ua
ls
time (s)
Ag b decay
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-20 0 200
20
Co
un
t
Residuals
Model Gauss
Equation y=y0 + (A/(w*sqrt(PI/2)))*exp(-2*((x-xc)/w)^2)
Reduced Chi-Sqr
4.77021
Adj. R-Square 0.93464
Value Standard Error
Counts y0 1.44204 0.48702
Counts xc 1.49992 0.19171
Counts w 5.93398 0.40771
Counts A 219.24559 14.47587
Counts sigma 2.96699
Counts FWHM 6.98673
Counts Height 29.4798
Test 1. Autocorrelation function
1
0
( ) ( ) ( )M
n
y m f n g n m
1
0
( ) ( ) ( )M
n
y m f n f n m
Correlation function
autocorrelation function
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Ag b decay
physics 401
Clear experiment Data + “noise”
t1(s) 177.76 145.89
t2(s) 30.32 27.94
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Ag b decay
physics 401
Histogram does not follow the normal distribution and there is frequency of 0.333 is present in spectrum
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Ag b decay
physics 401
Autocorrelation function
Conclusion: fitting function should be modified by adding an additional term:
0 1 2
1
3
2
( ) exp ex sin( )pt t
y t y At
tA At
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physics 401
Clear experiment Data + noise Modified fitting
t1(s) 177.76 145.89 172.79
t2(s) 30.32 27.94 30.17
FFTautocorrelation
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y = f(x1, x2 ... xn)2
2
1
( , )n
i i i
i i
ff x x x
x
1 1.5
1.10
1.15
xi
f(x
i)
xi±∆xi
f±fx
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Derive resonance frequency ffrom measured inductanceL±∆L and capacitance C±∆C
1
1
( )2
,
f L C
LC
1 110 1mH, 10 2μFL C
2 2
2 2( , , , )
f ff L C L C L C
L C
1 3
2 2
1 3
2 2
1;
4
1
4
fC L
L
fL C
C
Results: f(L1,C1)=503.29212104487Hz∆f=56.26977Hz
f(L1,C1)=503.3±56.3Hz
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Qmeas = Qtrue + es + er
In general we could expect both components of errors
es - systematic error comes from uncertainties
of plates separation distance, applied DCvoltage, ambient temperature etc.
er - random errors are
related to uncertainty of the knowledge of the actual tg and trise.
V =VDC±V, d=d0±d …
tim
e
Uncertainty of time of crossing the marker line. It is random.
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physics 401 32
3 3
3 2
1 9 2 1 1 1
c g riseg
d xQ F S T
f V g t tt
) ) )2 2 22 2 2
Q S T F F T S F S T
1 1 1
g riseg
Tt tt
2 2
2 2
5 2 3 2 1 2 2
3 2 1 2 1 1 1g rise
g g rise g rise
T t tt t t t t
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Step 1. Collect your data + parameters of the experiment in:\\Phyaplportal\PHYCS401\Common\Origin templates\Oil drop experiment\Section L1.opj
Use different columns for each
student or team. This Origin project is
for data collecting only but not for
data analysis. For data analysis you
have to copy these data and
experiment parameters obtained by
different students/team and paste it
in one in your personal Origin project.
physics 401
Setup and environmental parameters
Raw data
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Step 2. Working on personal Origin project
physics 401
Make a copy of the Millikan1 project to your personal folder and open it
Paste these 5 parameters and raw data from Section L1-L4.opj projects
Calculate manually the actual air viscosity
Prepare equations calculations of data in next columns (Set column values…). Switch Recalculate in Automode
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Step 3. Histogram graph
Millikan oil drop experiment
First use the data from
the column with drop
charges and plot the
histogram
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Step 4. Histogram. Bin size
Millikan oil drop experiment
Origin will automatically but notoptimally adjust the bin size h. In tispage figure h=0.5. There are severaltheoretical approaches how to findthe optimal bin size.
𝒉 =𝟑. 𝟓𝝈
𝒏𝟏/𝟑
Is the sample standard deviationand n is total number ofobservation. For presented in Fig.1results good value of h ~0.1
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Step 4. Histogram. Bin size
Millikan oil drop experiment
To change the bin size click ongraph and unplug the “AutomaticBinning” option
Bin size in this histogram is 0.1
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Step 4. Multipeak Gaussian fitting Millikan oil drop experiment
To do this you have to add
an extra plot to the graph
Counts vs. Bin Center
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Step 4. Multipeak Gaussian fitting Millikan oil drop experiment
This plot can be used for
peak fitting.
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Step 4. Multipeak Gaussian fitting Millikan oil drop experiment
This plot can be used
for peak fitting.
Final result for first
two peaks:
Q/e=0.93±0.01
Q/e=1.87±0.02
This pretty close to e
and 2e
physics 401
Here 𝒘 = 𝟐 𝒂𝒏𝒅 𝒆𝒓𝒓𝒐𝒓 𝒐𝒇 𝒕𝒉𝒆𝒎𝒆𝒂𝒏 =𝝈
𝑵
40