Basic Error Analysis Physics 401 Spring 2015 Eugene V Colla
Errors and uncertainties The Reading Error Accuracy and precession Systematic and statistical errors Fitting errors Appendix. Working with oil drop data physics 401 2
T = 63 ±? Best guess T~0. 5 Wind speed 4mph±? Best guess ±0. 5mph physics 401 3
Clearance fit physics 401 4
Measurement of the speed of the light 1675 Ole Roemer: 220,000 Km/sec Does it make sense? What is missing? Ole Christensen Rømer 1644-1710 NIST Bolder Colorado c = 299,792,456.2±1.1 m/s. physics 401 5
L 0. 5mm L=53mm±ΔL(?) Acrylic rod L 0. 03mm How far we have to go in reducing the reading error? We do not care about accuracy better than 1mm 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). physics 401 6
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 physics 401 7
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 physics 401 8
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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 physics 401 10
Sources of systematic errors: poor calibration of the equipment, changes of environmental conditions, imperfect method of observation, drift and some offset in readings etc. Example #1: measuring of the DC voltage R in Current source I R U E off E off =f(time,temperature) expectation U=R*I U = actual result R I R R in 1 + R E off physics 401 R in 11
Example #3: poor calibration Measuring of the speed of the second sound in superfluid He4 20 Published data T l =2.17K 15 LHe Resonator U 2 (m/s) 10 P403 results T l =2.1K 10mA 5 HP34401A DMM 1.6 1.8 2.0 2.2 T (K) Temperature sensor physics 401 12
Result of measurement Systematic error X meas = X true + e s + e r Correct value Random error 0.35 B 0.35 e s B 0.30 0.25 e s =0 0.30 0.25 0.20 0.20 P 0.15 P 0.15 0.10 X true 0.10 0.05 0.05 0.00 0 20 40 60 80 100 120 140 X i 0.00 0 20 40 60 80 100 120 140 X X i true physics 401 13
) rt ) rt Pn t e n n! n 0,1,2,... r: decay rate [counts/s] t: time interval [s] P n (rt) : Probability to have n decays in time interval t Siméon Denis Poisson (1781-1840) P 0.3 0.2 0.1 0.0 rt=1 rt=4 rt=10 0 5 10 15 20 number of counts A statistical process is described through a Poisson Distribution if: o random process for a given nucleus probability for a decay to occur is the same in each time interval. o universal probability the probability to decay in a given time interval is same for all nuclei. o no correlation between two instances (the decay of on nucleus does not change the probability for a second nucleus to decay. physics 401 14
) rt ) rt Pn t e n n! n 0,1,2,... r: decay rate [counts/s] t: time interval [s] P n (rt) : Probability to have n decays in time interval t Properties of the Poisson distribution: 0.3 0.2 rt=10 < n >= rt σ = rt n n0 P ( rt) 1, probabilities sum to 1 P 0.1 n n P ( rt) rt, the mean n0 n 0.0 0 5 10 15 20 number of counts ( ) 2 n n P ( ), n0 n rt rt standard deviation physics 401 15
) rt ) rt Pn t e n n! n 0,1,2,... Poisson and Gaussian distributions 0.1 probability of occurence 0.08 0.06 0.04 0.02 0 0 10 20 30 40 "Poisson distribution" "Gaussian distribution" Carl Friedrich Gauss (1777 1855) number of counts 1 Pn ( x) e 2 ( xx) 2 2 2 Gaussian distribution: continuous physics 401 16
2 x 1 Pn ( x) e 2 ( xx) 2 2 2 Error in the mean is given as σ N physics 401 17
Source of noisy signal 4.89855 5.25111 2.93382 4.31753 4.67903 3.52626 4.12001 2.93411 Expected value 5V Actual measured values physics 401 18
10 100 10 4 10 6
Result U x c N - standard deviation N number of samples For N=10 6 U=4.999±0.001 0.02% accuracy
Ag b decay 40 20 800 600 Model 108 Ag t 1/2 =157s 110 Ag t 1/2 =24.6s ExpDec2 Equation y = A1*exp(-x/t1) + A2*exp(-x/t2) + y0 Residuals 0 Count 400 200 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 1.6525-20 0 50 100 150 time (s) 0 0 200 400 600 800 time (s) Count 20 Model Gauss Equation y=y0 + (A/(w*sq rt(pi/2)))*exp(-2 *((x-xc)/w)^2) Reduced Chi-S 4.77021 qr 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 t t y A1 exp A2 exp y t t 1 2 0 0-20 0 20 Residuals physics 401 21
40 Ag b decay Test 1. Fourier analysis 20 Residuals 0-20 0 50 100 150 time (s) Count 20 Model Gauss Equation y=y0 + (A/(w*sq rt(pi/2)))*exp(-2 *((x-xc)/w)^2) Reduced Chi-S 4.77021 qr 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 No pronounced frequencies found 0-20 0 20 Residuals physics 401 22
40 20 Ag b decay Test 1. Autocorrelation function Residuals 0-20 0 50 100 150 time (s) Count 20 Model Gauss Equation y=y0 + (A/(w*sq rt(pi/2)))*exp(-2 *((x-xc)/w)^2) Reduced Chi-S 4.77021 qr 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 Correlation function M 1 y( m) f ( n) g( n m) n0 0-20 0 20 Residuals autocorrelation function M 1 y( m) f ( n) f ( n m) n0 physics 401 23
Ag b decay Clear experiment Data + noise t 1 (s) 177.76 145.89 t 2 (s) 30.32 27.94 physics 401 24
Ag b decay Histogram does not follow the normal distribution and there is frequency of 0.333 is present in spectrum physics 401 25
Ag b decay Autocorrelation function Conclusion: fitting function should be modified by adding an additional term: t t y( t) y A exp A exp A3 sin( t ) t t 0 1 2 1 2 physics 401 26
FFT autocorrelation Clear experiment Data + noise Modified fitting t 1 (s) 177.76 145.89 172.79 t 2 (s) 30.32 27.94 30.17 physics 401 27
y = f(x1, x2... xn) 2 n f 2 i i i i1 xi f ( x, x ) x 1.15 f(x i ) f±fx 1.10 x i ± x i 1 1.5 x i physics 401 28
Derive resonance frequency f from measured inductance L± L and capacitance C± C L 1 1 f ( L, C) 2 LC 10 1mH, C 10 2μF 1 1 2 2 f f f ( L, C, L, C) L C L C 2 2 f L f C 1 C 4 1 3 2 2 L 1 L C 4 1 3 2 2 ; Results: f(l 1,C 1 )=503.29212104487Hz f=56.26977hz f(l 1,C 1 )=503.3±56.3Hz physics 401 29
time In general we could expect both components of errors Q meas = Q true + e s + e r e s - systematic error comes from uncertainties of plates separation distance, applied DC voltage, ambient temperature etc. V =V DC ±V, d=d 0 ±d e r - random errors are related to uncertainty of the knowledge of the actual t g and t rise. Uncertainty of time of crossing the marker line. It is random. physics 401 30
3 3 1 9d 2 x 1 1 1 Q F S T 32 fc V g t t g g t rise 2 2 2 2 2 2 ) ) ) Q S T F F T S F S T T 1 1 1 t t g g t rise 2 2 3 2 1 2 1 1 1 T t t t t t t t 2 2 5 2 3 2 g 1 2 2 rise g g rise g rise physics 401 31
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. Setup and environmental parameters physics 401 Raw data 32
Step 2. Working on personal Origin project Make a copy of the Millikan1 project to your personal folder and open it Prepare equations calculations of data in next columns (Set column values ). Switch Recalculate in Auto mode Paste these 5 parameters and raw data from Section L1-L4.opj projects Calculate manually the actual air viscosity physics 401 33
Millikan oil drop experiment Step 3. Histogram graph First use the data from the column with drop charges and plot the histogram physics 401 34
Step 4. Histogram. Bin size Millikan oil drop experiment Origin will automatically but not optimally adjust the bin size h. In tis page figure h=0.5. There are several theoretical approaches how to find the optimal bin size. 3. 5σ h = n 1/3 Is the sample standard deviation and n is total number of observation. For presented in Fig.1 results good value of h ~0.1 physics 401 35
Step 4. Histogram. Bin size Millikan oil drop experiment To change the bin size click on graph and unplug the Automatic Binning option Bin size in this histogram is 0.1 physics 401 36
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 physics 401 37
Step 4. Multipeak Gaussian fitting Millikan oil drop experiment This plot can be used for peak fitting. physics 401 38
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 Here w = 2 and error of the mean = σ N physics 401 39