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# EXPERIMENTAL PHYSICS PHYS 2150

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This 113 page Class Notes was uploaded by Mrs. Peter Toy on Friday October 30, 2015. The Class Notes belongs to PHYS 2150 at University of Colorado at Boulder taught by Eric Zimmerman in Fall. Since its upload, it has received 11 views. For similar materials see /class/232097/phys-2150-university-of-colorado-at-boulder in Physics 2 at University of Colorado at Boulder.

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PHYSICS 2150 I LABORATORY Prof Eric D Zimmerman Lecture 6 February 19 2008 ANNOUNCEMENT The second lab is due FRIDAY Problem set will be due WEDNESDAY next week THIS LECTURE The Poisson Distribution continued Goodnessoffit using x2 A POISSON PROCESS Assume a decay rate of 001 decayssec On average will get 1 decay100 seconds Sometimes you will get zero Sometimes you will get two Rarely you will get three or more What is the probability of getting exactly n decays in 100 seconds Answer Poisson distribution Applies to any discrete process where events occur randomly at a constant rate THE POISSON DISTRIBUTION 0 Characteristics the process has to have 0 Asymmetry Can t have fewer than zero counts so a symmetric function like a gaussian can t describe this process Also must vanish for nltO Discreteness Gaussian describes a variable that can take on a continuum of values There s no such thing as 048 events so Poisson must give probabilities not probability density of discrete outcomes 0 Summing we could view a process with a mean of 2p as a collection of two processes with a mean of 1 events from two halves of the same source for example So the following sum rule must be true P2u3 PMOPM3 I PM1PM2 I PM2PM1 I PM3PMO 2PMOPM3 I 2PM1PM2 Notation lf mean is p let probability of finding 3 events be Pp3 lf mean is 2p then probability of finding 3 events is P23 THE POISSON DISTRIBUTION u PM 6 ME 0 This is obvioust asymmetric doesn t allow negative events and is obviously discrete factorial is only defined for nonnegative integers Note that the mean p doesn t have to be an integer The sum rule works too try it out at home 0 Note that p factor keeps probability down for too few events n factor keeps it down for too many events MR GAUSS VS THE FISH Iu2 e 252 GAUSSIAN P 1 HUGE 0m Symmetric Continuous 1 is real Mean 2 u most probable point 2 LL standard deviation a Distribution describes results of measurements with accuracy 0 POISSON itquot PM 6 g Asymmetric PAH 2 0 Discrete n is integer Mean 2 u most probable point 3 u Standard deviation Distribution describes results from counting experiment POISSON DISTRIBUTION For a decay rate of 001 sec What is the probability of measuring in 100 seconds note that u 1 0 decays P10 1 100I 16 0368 1 decay P11 6 1 111 16 0368 2 decays P12 1 122I 126 0184 3 or more decays P13 P1z39 Easier way P13 1 P10 P11 P12 POISSON DISTRIBUTION 0394 Poisson pt 1 l Distribution for P101 6 1 Ego353 H 03 l Very asymmetric CH 025 I For u E II two most probable 02 points Pym PLu 1 015quot 01 I For u II most probable point 005 at n intu round down u G 0 l 2 I 5 6 Decays observed SMALL MEAN BEHAVIOR OF POISSON DISTRIBUTION 53 4 MI Poisson u 1 Gaussian u 10 l I For small u here u 10 Poisson and Gaussian different 7 l Standard deviation 2 a 7 0 25 68 probability discreteness 0239 l LL 2 1 implies 1 I 1 includes 015 0 1 2 92 of probability 01 l u 099 implies 099 I 0995 in 005 eludes 0 1 736 of probability 0 1 2 3 Probability o W UI 5 6 Decays observed LARGE MEAN POISSON BEHAVIOR GAUSSIAN LIMIT I For large u here u 2 75 014 I10isgosnz Gaussian Poisson and Gaussian become E061 Similar With a 17 008 l Standard deviation 2 a x 03906 68 probability 004 l Often use Gaussian approxima 002 tion for large M generally M gt 5 70 i i 3 4 5 6 8 9 101112131415 Decays observed HOW TO USE POISSON DISTRIBUTIONS IN COUNTING Uncertainty ou for large u can be interpreted similarly to gaussian 039 STATISTICAL uncertainty on the number of counts is thus the square root of the number of counts Background processes may be contributing to your rate uusignalubkg Often can measure ubkg by turning ohc the signal so when you then measure Htotal you can subtract the background to measure your signal rate How do you handle the error in this situation AN EXAMPLE FITTING RADIOISOTOPE LIFETIME Measure count rate background with no radioactive material present Introduce radioactive material Count the number of decays in a 1second period remeasure every 30 seconds Subtract the background rate Fit the rate vs time to an exponential lifetime MEASURING BACKGROUND Background rate should be constant so each trial is a remeasurement of the same thing Get the background rate by taking mean 40210402 note that this is exactly equivalent to simply measuring for 10 seconds and dividing by 10 to find the rate Uncertainty in the rate total counts is 402402 So in 10 seconds mean 40220 Divide by 10 to get rate in 1 sec 40220 Background rate is 40220 countssecond Error on the mean behaves the same way as with Gaussian errors a on each trial is 040 v40 Error on mean is a 040xN x40x 10 But that is exactly the same as 040010 x40010 Trial Counts in 1 sec 1 44 2 42 3 39 4 36 5 34 6 45 7 49 8 37 9 33 10 43 Total 402 SIGNAL DATA Now look at counts from an isotope with a short mean life SIGNAL DATA Seconds Counts Corrected Now look at counts from an isotope 0 2007 WIth a short mean life 30 1464 60 973 90 698 120 526 150 353 180 285 210 217 240 150 270 122 300 112 330 86 360 78 390 53 SIGNAL DATA Seconds Counts Corrected Now look at counts from an isotope 0 2007 With a short mean life 30 1464 60 973 0 Are all the trials measuring the same rate 90 698 0 What are the uncertainties in the number of counts 0 What is the measured signal rate 180 285 210 217 Subtract background 402 ctssec 240 150 2 O 122 0 Statistical error remains the square root of the total 300 112 number of counts 330 86 360 78 0 Also there Is a systematic error not shown due to 390 53 uncertainty in the background SIGNAL DATA Seconds Counts Corrected Now look at counts from an isotope 0 2007 lz45 With a short mean life 30 1464 l38 60 973 l31 0 Are all the trials measuring the same rate 90 698 i26 120 526 l23 o 7 What are the uncertainties In the number of counts 150 353 ilg 0 What is the measured signal rate 180 285 i 210 217 i15 Subtract background 402 ctssec 240 150 i12 2 O 122 ill 0 Statistical error remains the square root of the total 300 112 in number of counts 330 86 i9 360 78 i9 Also there is a systematic error not shown due to 390 53 i7 uncertainty in the background SIGNAL DATA Now look at counts from an isotope with a short mean life Are all the trials measuring the same rate What are the uncertainties in the number of counts What is the measured signal rate Subtract background 402 ctssec Statistical error remains the square root of the total number of counts Also there is a systematic error not shown due to uncertainty in the background Seconds Counts Corrected 0 2007 l45 1967 i45 30 1464 l38 1424 i38 60 973 l31 933 l31 90 698 l26 658 l26 120 526 l23 486 l23 150 353 i19 313 l19 180 285 i17 245 l17 210 217 i15 177 l15 240 150 i12 110 l12 270 122 ill 82 l11 300 112 ill 72 l11 330 86 i9 46 l9 360 78 i9 38 l9 390 53 i7 13 l8 420 51 i7 11 l7 FITTING FOR THE MEAN LIFE 0 Plot the rate vs seconds To fit to a line take the log 51500 012507 WU No 6Xp l ZI quot1nN 1nNo U7 10005 750 0 Linear fit y 1nN xt 500 250i 7 7 M H 0 Whats the uncertainty on y If uncertainty on N IS ON ObIIIiOO 200 300 400 seconds ay 1110N 1ncounts second a r I I I I I I I I O 100 200 300 400 seconds FITTING FOR THE MEAN LIFE quot5 52000 0 Plot the rate vs seconds 31750 0 To fit to a line take the log 315007 1250 Nt NO 6Xp l ZI quotquot1nN 1nNo U72 1000 1 U1 0 I 0 Linear fit y 1nN xt 500 Z 50 2 E 7 39 7 M H Whats the uncertainty on y If uncertainty on N IS ON 060100 200 300 400 seconds ay 1110N 1ncounts second a ir i quot N 7y 8N aN aNlnNaN NON N 3 2 7 I I I I O 100 200 300 400 seconds HOW GOOD IS THE FIT Often want to know how good a fit is Minimizing x2 told you what the best fit to your function was The value of that minimum x2 can tell you how well data actually fit your function THE CHISQUARED TEST l X2 test is a particular type of goodnessof t test I Generally for N measurements 01 02 ON N 0 v E 2 X2 Z where EZ is the expected value and al is the i1 0i error on measurement 0i I To use we need to know oi which could be our estimated measurement error xEZ or xOZ in the case of a counting experiment or if each measurement 02 is the result of many measurements the standard deviation of Oi I Also need degrees of freedom dof which is number of measurements N minus number of tted parameters 2 for a line slope and intercept l Reduced X2 is XZdof and should be N 1 in large def limit I Can convert to probabilities using Appendix D in Taylor if probability is high generally gt5 agreement is OK LEAST SQUARES AN D CH SQUARED If you do a straight line fit without using errors unweighted least squares fit Fit returns error on y assuming they are all the same Min XZdof is 1 by definition Can t use XZdof to determine fit quality If you use externally known uncertainties to do a weighted line fit See Taylor problems 89 819 Fit returns function parameters slopeoffset and X2 You can use the table to determine fit compatibility with data as a confidence probability LOOK AT THE DECAY DATA Examples I Fit with background subtraction has X2 106 with 13 degrees of freedom calculated con dence level is 64 good lFit without background subtraction has X2 843 with 13 degrees of freedom calculated con dence level is 17 gtlt10 10 very bad Uses of X2 I Can determine if t is good I If t is bad either data arebad or t function is wrong lncounts second lncounls second 3 background X L In6z 1 Fit with background subtraction fl 0 100 200 it seconds Fit without subtraction l t l t i 0 I 00 200 300 400 seconds WHAT YOU SHOULD KNOW FROM LECTURESNOTESBOOK I Error propagation for an artbtrary equation like what is If if f 055 and 0a 0b 0c ad are known I Understand difference between statistical and systematic uncertainties I Calculate mean standard deviation standard deviation of mean and know what they mean I Determine if measurements are compatible and how to calculate the weighted average I Perform least squares t to data and extract slope and intercept with uncertainties I How to determine if two variables are correlated l Understand what a counting experiment is and that the uncertainty on the counts is the square root of the number of counts PHYSICS 2150 I LABORATORY Prof Eric D Zimmerman Lecture 5 February 12 2008 ANNOUNCEMENT We will have a sixth lecture next week The problem set will be assigned then and due a week later THIS LECTURE More lab report feedback Covariance and Correlation Introduction to the Poisson Distribution MORE LAB REPORT FEEDBACK You should make your own schematic diagrams not copy the lab manual If this is not practical always cite the source of your figures 0 Data analysis must be described in text with readable equations not relegated to an appendix or your MathCAD printout Tell your readership that you squared x or solved form and make it clear Why 0 Again discrepancy is not error measuredexpected expected is not meaningful as an assessment of your result measuredexpected uncertainty is very meaningful Agreement confidence level from erf table is better 0 Your conclusion must include your result even if it has already appeared above in your report COVARIANCE AND CORRELATION Take a desired measurement qxy Error propagation says 6g Making two assumptions here errors are gaussian and x and y are uncorrelated If there is a correlation the error on 7 can be either larger or smaller than our estimate Sometimes epidemiology other complex systems the correlation itself is something interesting to learn AN EXAMPLE OF CORRELATED VARIABLES THE KO MASS Incoming K hits stationary neutron producing proton K and KO 0 The K0 travels a short distance p and decays to nn Need to measure angle GT between pions Also need to measure 9 angles between pions and KO 0 Measuring GT is easy but KO direction can be hard if its path is short AN EXAMPLE OF CORRELATED VARIABLES THE KO MASS 0 If direction of the blue line is K wrong then 9 and G will be wrong by equal amounts K I p but in opposite directions 9T 9 9 will still be OK 0 Thus we can say that our measurements of 9 and G will be correlated actually anticorrelated COVARIANCE AND ERROR PROPAGATION l Degree of correlation can be determined from covariance aw N 1 l Experimentally amy N E T g 39 1 z I Correct propagation of errors for qac7 y is now 9 2 a 2 9 a 039 6 6 0 COVARIANCE AND ERROR PROPAGATION 8g 2 3g 2 89 9 1 2 0 6m 6 2 0m 1 5 gt 8y y 8x 8y y I 7xy is small compared to 5x or 5y then the error reduces as expected to standard addition in quadrature I In case of maximum positive correlation worst case scenario addition is linear Thus aq 3 13 0x 2 1 83 ay l axy can be postive or negative in K 0 case would nd negative 0mg I If the covariance 7xy is negative then the error on the result is actually smaller than from addition in quadrature COVARIANCE VS CORRELATION The covariance oxy can be normalized to create a correlation coefficient roxy oxoy r can vary between 1 and 1 rO indicates that the variables are uncorrelated r 1 means the variables are completely correlated ie knowing the value of x completely determines the value of y Sign indicates direction of covariance positive means that XgtXmean indicates likely ygtymean negative means xgtxmean indicates likely yltymean CORRELATION AS A MEASUREMENT Sometimes the correlation itself is interesting In order to establish the significance of the correlation need to ask what r is as well as how many measurements were taken to establish it For given r and N look in table Taylor Appendix C to find out probability of randomly measuring a particular correlation value Random probabilities lt5 or lt1 are generally considered significant depending on the application Caution when you look at 20 correlations and their random probabilities you expect one to be lt5 just by chance CORRELATION A NON PHYSICS EXAMPLE From httphdrundporgstatisticsdata take nations with human development index in top 25 and at least 10 million people Country Physicians Smoking Life GDP per per 100K rate Expectancy capita people years US Australia 249 19 80 2 26 300 Canada 209 22 79 9 27 100 Belgium 418 24 788 29100 USA 549 23 77 3 37 600 Japan 201 29 81 9 33 700 Netherlands 329 28 78 3 31 500 166 27 783 30300 France 329 27 79 4 29 400 Italy 606 26 80 0 25 500 Germany 362 35 78 7 29 100 Spain 320 32 79 5 20 400 Greece 440 38 78 2 15 600 Average 3307 28 4 7885 24929 WITH WHAT DOES LIFESPAN CORRELATE DOCTORS SMOKING MONEY g 82 7 g 82 i o E 32 39 E 3 gt1 81 V 5 81 V 3 81 j a a g g 80 39 o 5g 80 39 5g 80 35 5 o 5 a 5 79 5 79 o 5 79 39 78 39 I u 78 7 u 78 39 o I D V I V 0 77 i w v v x w x 7 x t t t t w x 77 i v w t w w t 100 200 300 400 500 600 20 25 30 35 15 20 25 3O 35 Physicians 000 people Smoking rate 70 GDP per Capita US1000 I Calculate linear c01 relati0n coef cient i 532 55 y y 2 2 2031 56 E y y I For life expectancy versus physicians 7quot 040 I For life expectancy versus smoking T 0 17 I For life expectancy versus GDP per capita 7quot 004 IS IT SIGNIFICANT I The linear correlation coef cients of life expectancy versus physicians smoking and GDP per capita are 7040 017 and 004 respectively I Life expectancy is most closely correlated with number of physicians but negatively correlated I How signi cant are the correlations I Appendix C gives probability to nd 7 gt To for N measurements of two uncorrelated variables 7quot0 0 01 02 03 04 05 06 07 08 09 1 11 100 77 56 37 22 12 51 16 03 O 12 100 76 53 34 20 98 39 11 02 0 13 100 75 51 32 18 82 30 08 01 0 I 12 measurements are 20 likely to have Ir gt 04 I I Using linear extrapolation the probabilities for r 7017 and r 004 are 61 and 90 so are likely uncorrelated I I General rule lt 5 is signi cant lt 1 is highly signi cant COUNTING EXPERIMENTS THE POISSON DISTRIBUTION Start with a large amount of radioactive material with a very long halflife compared to the experiment Measure the decays with a detector at a fixed distance Measure the number of decays in five 100second intervals Should the five intervals all have the same number of decays Should there be a consistent trend COUNTING EXPERIMENTS Should there be a consistent trend NO The long halflife assures that over the time of the experiment the decay rate isn t changing significantly Should the five intervals all have the same number of decays NO The decay is a random process A POISSON PROCESS Assume a decay rate of 001 decayssec On average will get 1 decay100 seconds Sometimes you will get zero Sometimes you will get two Rarely you will get three or more What is the probability of getting exactly n decays in 100 seconds Answer Poisson distribution Applies to any discrete process where events occur randomly at a constant rate THE POISSON DISTRIBUTION 0 Characteristics the process has to have 0 Asymmetry Can t have fewer than zero counts so a symmetric function like a gaussian can t describe this process Also must vanish for nltO Discreteness Gaussian describes a variable that can take on a continuum of values There s no such thing as 048 events so Poisson must give probabilities not probability density of discrete outcomes 0 Summing we could view a process with a mean of 2p as a collection of two processes with a mean of 1 events from two halves of the same source for example So the following sum rule must be true P2u3 PMOPM3 I PM1PM2 I PM2PM1 I PM3PMO 2PMOPM3 I 2PM1PM2 Notation lf mean is p let probability of finding 3 events be Pp3 lf mean is 2p then probability of finding 3 events is P23 THE POISSON DISTRIBUTION u PM 6 ME 0 This is obvioust asymmetric doesn t allow negative events and is obviously discrete factorial is only defined for nonnegative integers Note that the mean p doesn t have to be an integer The sum rule works too try it out at home 0 Note that p factor keeps probability down for too few events n factor keeps it down for too many events PHYSICS 2150 I LABORATORY Prof Eric D Zimmerman TA Geruo A Lab Coordinator Jerry Leigh Lecture 3 January 29 2008 PHYSZ 150 Lecture 3 Rejection of data Weighted averages and compatibility of measurements ANNOUNCEMENT Your first lab report is due Thursday at 4 Graded reports will be returned in the blue file next to the turn in box Common issues in grading will be indicated by codes the key will be posted on the web and explained here Late lab reports If you turn in a report late write the date on your report If you can have Jerry or one of us initial the date If there s no date we can only assume the report was turned in the day we picked it up You may only have the same lab partner for two experiments 80 if you have the same partner for the first two be sure to work with someone different or by yourself for the third lab GRADING CODES RO round off appropriately lSF express uncertainty with one significant figure SFR state result in standard form x UNITS SPOT same power of ten for value and uncertainty PDAP plot data as points AXL label axes UNITS express units CAP no plot caption SPP Second person procedure procedure written like a lab manual ECO Excessiverepetitive calculation output REJ ECTING DATA We often find data points that are suspicious for one reason or another Is it ever legitimate to discard them If so when REJ ECTING DATA BE VERY CAREFUL HERE You are treading in the footsteps of a long line of practitioners of pathological science 0 Generally you should have external reasons for cutting data 0 In reality even that s not enough do you only go searching for problems when you get a result you didn t expect Your analysis is biased All prescriptions for deciding on the validity of data points are controversial l ll describe one that s common in textbooks but not in real life Chauvenet s criterion First a cautionary tale HOW TO LOOK FOR A PARTICLE Those of you doing the K meson experiment have already seen this Look in highenergy collisions for events with multiple output particles that could be decay products displaced from primary interaction if particle is longlived as with the KO Reconstruct a relativistic invariant mass from the momenta of the decay products Make a histogram of the masses from candidate events Look for a peak indicating a state of welldefined mass ONE PEAK OR TWO CERN experiment in late 1960s observed A2 mesons DOUBLE POLE x1 40 v Particle appeared to be a doublet E a a Statistical significance of split is very high There is really only one particle 1300 39 1350 mssmo MASS m HOW DID IT HAPPEN 0 In an early run a dip showed up It was a statistical fluctuation but people noticed it and suspected it might be real 0 Subsequent runs were looked at as they came in If no dip showed up the run was investigated for problems There s usually a minor problem somewhere in a complicated experiment so most of these runs were cut from the sample 0 When a dip appeared they didn t look as carefully for a problem 0 So an insignificant fluctuation was boosted into a completely wrong discovery 0 Lesson Don t let result influence which data sets you use CHAUVENET S CRITERION Assume data x i1N come from a gaussian distribution Determine mean ltxgt and standard deviation 0 0 Assuming gaussian distribution calculate how many data points fraction times N should be outside a particular number of standard deviations Pick a threshold Chauvenet says 05 in real life you would probably want to use a smaller number and reject data points farther from the mean 0 So if you use 05 as your threshold you would throw out data points beyond a number of sigma where you expect only 05 data points 0 Example Have 10 data points Know that 5 of data points should be outside 2 sigma so that s 05 data points So throw out an event if it s more than 20 from the mean 0 Problem Many experimental distributions have nongaussian tails Can t do this iteratively recaculate mean and sigma with remaining data then throw out some more points A FINAL COMMENT ON REJ ECTING DATA Rejecting data is a last resort If you are suspicious of a data point you should first If you can take more data Ask what could have gone wrong but be careful that you aren t going on a witch hunt Is there a problem with the way you recordedcalibrated that could cause this data point to be different Whatever you decide lay it all out in the report including the result including the rejected data NEVER reject data points because you don t like the answer WEIGHTED AVERAGES If you measure the same thing twice and the errors are different how do you combine the results A proper averaging gives more weight to measurements with smaller uncertainties The reported error on the average must reflect this Weighted averages are the most basic form of fitting WEIGHTED AVERAGES Showed that the mean of measurements gives the best measurement of the true value 0 This assumed that each measurement had the same uncertainty How to combine when they are different Assume two measurements of g g1980237cm82 g2977419Cm52 If unweighted mean9788 cms2 but this ignores the fact that measurement 2 has smaller uncertainty so the true value is likely to be closer to it than to measurement 1 JOINT PROBABILITIES Take two random discrete variables A and B works for more than two but this is simplest Let probability PAa E PA and PBbj 5 PB The joint probability PABij is the probability that Ai and 3 If the two variables are uncorrelated FABj PA PB JOINT PROBABILITY DENSITY FUNCTION Now consider the case where A and B are continuous variables fz y is the joint probability density for Ax and By Interpret this in the following way Probability that A is between X1 and X2 and B is between y1 and y2 is 12 div fyyf Gig at y Variables A and B are uncorrelated if 79 FA5EFB3 MAXIMUM LIKELIHOOD 0 Assume two measurements XAiO39A and XBiO39B where the true mean is an unknown u and errors gaussian uncorrelated A M2 0 Probability density form is PcA olt iexp 202 A 0A Similar fopr 0 Joint probability density for measuring both is PXAPXB 2 2 2 P93AaB OC 1 exp where X2 3 M O39AO39B 2 0A GB 0 Maximum likelihood is when X2 is minimized l Differentiate X2 With respect to u and set to 0 23 2333 0 0A CTB l Max likelihood u M where w wAlwB q Naequot TURNING MAXIMUM LIKELIHOOD ARGUMENT INTO A WEIGHTED AVERAGE I Take N measurements x1 x2 xN With uncertainties 0102 UN l Most likely true value given by welghted average an Z wz 1 I The we1ghts 101 are 101 2 0239 1 I The uncertainty on 13an 1s 0an V Z 10239 I If 01 02 2 UN weighted average equals straight average Weights go as 102 so more precise measurements count much more than less precise ones USING WEIGHTED AVERAGES Before averaging it s a good idea to check compatibility of the results I Same idea as discrepancy in Lecture 2 l Two measured values 1 A i 0A and 903 i O39B l Difference is xA xBl le 55339 m I Can convert signi cance of difference in a to probability using erf calculated or in tables I Signi cance of difference in 0 s is Integrate the gaussian l Probabihty of being w1th1n 6 of mean is 1 a a ex dxerf v 27r at p 202 g I Also in tables p 287 of Taylor I If probability is very low generally less than 1 measurements are incompatible and averaging might not make sense ARE OUR MEASUREMENTS CONSISTENT l Check our example of two 9 Ni measurements 9802 0037 ms2 E and 9774 0019 ms2 982 Signi cance of difference is 7 2 4 0028 m 2 2 0670 98 MW V00372 00192 0042 very reasonable 978 9 76 Meas 1 TAKE THE AVERAGE Straight average is 9788 ms2 Weighted average 97800017 98 Weighted average is much close to the more precise measurement Uncertainty is lower than on the 976 individual measurements we have more information in the weighted average 978 Meas 1 Weighted Average Meas 2 Average PHYSICS 2150 I LABORATORY Prof Eric D Zimmerman TA Geruo A Lab Coordinator Jerry Leigh Lecture 4 September 26 2006 PHYSZ 150 Lecture 4 Rejection of data Weighted averages and compatibility of measurements ANNOUNCEMENT The first labs have been returned except for late submissions Grades averaged about 7 Please read the comments If you have any questions see me or Geruo The second lab is due on October 4 next Wednesday Next lecture will be the final one The semester s only problem set will be assigned then and due a week later Make sure you are signed up for your third lab COMMON PROBLEMS IN LAB REPORTS Units and uncertainties Nearly all measurements should have units and uncertainties quoted If you believe that the uncertainty is negligible note that Idea section Discuss the relevant physics and explain the equations 0 Apparatus Describe don t just draw the equipment 0 Procedure This should NOT be a list of commands that s a lab manual 0 BAD Turn and tilt the coils until they are aligned with the earth s field Make sure that the coil doesn t fall off the table Take at least three measurements of the current lo 0 GOOD We aligned the coil with the earth s field This was done by testing different coil directions with a magnetic dip needle After the coil fell off the table we had to repeat the alignment process After finding the optimum angle we took five measurements of the offset current lo needed to cancel the earth s field COMMON PROBLEMS IN LAB REPORTS Notes on the analysis Include text equations that describe the analysis steps not just numbers and Mathcad output 0 Error propagation Often inadequate sometimes missing 0 Conclusion Several students didn t write scientific conclusions Must summarize results including important numerical values Example We measured em for the electron to be XX yy Dominant uncertainties were from Also this is where you should report your discrepancy with a known value if applicable Always quote your discrepancy in units of sigma and discuss whether there were likely other errors that you didn t account for 0 Page ordering Please start your report with the introduction Raw data sheets are supplemental and should be in an appendix STANDARD DEVIATION AND MEASUREMENT UNCERTAINTY Measure something N times take the average of the measurements to find best estimate X of the true value 0 Calculate standard deviation 0 What is the statistical error on your result 0 o ofV Something else Your measurement is the mean so your error has to be the error on the mean USING STANDARD DEVIATION EXAMPLE FRANCKHERTZ 0 Measure 10 ionization voltages in V 1132 1148 1145 1169 112711631161114011341145fora mean of 11464V 0 Meter has systematic error of 001 V 0 What is the uncertainty 0 Calculate standard deviation of the 10 measurements 0 0134 V Statistical error on mean is oflO 0042 V 0 This is way bigger than systematic error from meter error of course there may be other systematic errors 0 So we can write that Vion11464 i 0043 V WEIGHTED AVERAGE WHEN 39 NO if You don t know the intrinsic uncertainty or The intrinsic uncertainty is the same for all measurements You need to calculate the standard deviation to determine the uncertainty 39 YES if You know the intrinsic uncertainties AND They are not the same for all measurements FITTING A LINE Example from photoelectric effect calibration Measure voltage on capacitor by discharging through galvanometer converting charge to displacement so V D where V is calibration voltage D is galvanometer de ection and lkC converts D to V Here V is independent and D is dependent so use D 2 k0 V Voltage V De ection cm 000 O calibrate discharge 0 2 1 1 S 39 04 23 R 2 o 0 6 3 7 Operate 0 8 45 S o o U o O C NEW TOPICS TODAY Fitting a line to data points Fitting exponentials and other generic functions Covariance and correlation FITTING A LINE Use least squares formula from Taylor Sec 82 to fit the N measurements to the form yA BX where Xlt gtV and ylt gtD assume that errors on X are negligible compared to errors on y N Ii 1 I 1 2 E E 2y A NZ ZW 25c 00020420 110V 2x 00 022 042 202 154V2 Zyi 00 11 23 121 658cm chiyi 00 39 00 0211 04 23 20 121 9248V cm A N232 2 xi 11 x 154 1102 484V2 295221sz Iii A y E y 15 4x658 1410x9248 008 cm B NZziyrExIZyi A A ZifzyiAZMEww B 7 11gtlt92487111gtlt658 i 7 48A 606 CInV ICC FITTING A LINE From spread of measurements nd uncertainty on y A and B EIyt ABl iIQ N Z 014am op Calibration constant k0 k0 2 606i007 cmV 154 m ooacm 03 Mg 014 x 007cmV A is consistent with zero this is a good checksince in this case the intercept should be zero as D should be directly proportional to V FITTING TO AN EXPONENTIAL 0 Some functions can be easily converted to linear and then fitted to a line 0 Exponential decay shows up in many places especially radioactive decays 0 Decay rate drops with time Nt No exp tr where 1 is the mean lifetime halflife is related by factor of 1n2 0 However 1nN 1nNo tr which is linear 0 So can do a linear fit to 1nN and extract I FITTING TO AN EXPONENTIAL 800 700 g 600 7 r r t G 500 I Start w1th N t Noe 13400 7 300 e I We want the decay constant A 200 r 100 i ITakenatural logarithm to get 0 39 7 H10 lnN1nN0 t tune I Identify y as In N a as t A as In No and B as I Use previous formulas to nd A 1ndecays N DJ f i l t 0 l FITTING TO A MORE GENERAL FUNCTION Possible to fit a completely general function fx where f is dependent on parameters A B C Obtain N measurements y for N values of X N l foegt12 Construct the chisquared X2 Z i1 97 Scan over possible values of the parameters A B C and find the values that cause X2 to be minimized These values are the most likely values of the parameters Uncertainties on the parameters are determined from how far you have to scan away from the best values to cause X2 to increase by 1 CHISQUARED MINIMIZATION How do we know if we got a good fit ie does the function fdescribe the data adequately Can ask how small the minimum x2 value was Smaller values indicate a better fit fewer smaller deviations between the data points and the best fit function Actual numerical expectations for depend on the number N of data points and the number of parameters A B C in the function 7 More on this in Chapter 12 COVARIANCE AND CORRELATION Take a desired measurement qxy Error propagation says 6g Making two assumptions here errors are gaussian and x and y are uncorrelated If there is a correlation the error on 7 can be either larger or smaller than our estimate Sometimes epidemiology other complex systems the correlation itself is something interesting to learn AN EXAMPLE OF CORRELATED VARIABLES THE KO MASS Incoming K hits stationary neutron producing proton K and KO 0 The K0 travels a short distance p and decays to nn Need to measure angle GT between pions Also need to measure 9 angles between pions and KO 0 Measuring GT is easy but KO direction can be hard if its path is short AN EXAMPLE OF CORRELATED VARIABLES THE KO MASS 0 If direction of the blue line is K wrong then 9 and G will be wrong by equal amounts K I p but in opposite directions 9T 9 9 will still be OK 0 Thus we can say that our measurements of 9 and G will be correlated actually anticorrelated COVARIANCE AND ERROR PROPAGATION I Degree of correlation can be determined from covariance aw N 1 I Experimentally an N E T 3 11 I Correct propagation of errors for gQJ7 y is now 02 615 2l 634 22U 1 66 83 85v 8y my I If the covariance an is small compared to 3x or 3y then the error reduces as expected to standard addition in quadrature I In case of maximum positive correlation worst case scenario addition is linear Thus aq S g am 35 0y I am can be postive or negative in K 0 case would nd negative racy If the covariance an is negative then the error on the result is actually smaller than from addition in quadrature COVARIANCE VS CORRELATION The covariance oxy can be normalized to create a correlation coefficient roxy oxoy r can vary between 1 and 1 rO indicates that the variables are uncorrelated r 1 means the variables are completely correlated ie knowing the value of x completely determines the value of y Sign indicates direction of covariance positive means that large x indicates y is likely large negative means that large x indicates y is likely small CORRELATION AS A MEASUREMENT Sometimes the correlation itself is interesting In order to establish the significance of the correlation need to ask what r is as well as how many measurements were taken to establish it For given r and N look in table Taylor Appendix C to find out probability of randomly measuring a particular correlation value Random probabilities lt5 or lt1 are CORRELATION A NON PHYSICS EXAMPLE From httphdrundporgstatisticsdata take nations with human development index in top 25 and at least 10 million people Country Physicians Smoking Life GDP per per 100K rate Expectancy capita people years US Australia 249 19 80 2 26 300 Canada 209 22 79 9 27 100 Belgium 418 24 788 29100 USA 549 23 77 3 37 600 Japan 201 29 81 9 33 700 Netherlands 329 28 78 3 31 500 166 27 783 30300 France 329 27 79 4 29 400 Italy 606 26 80 0 25 500 Germany 362 35 78 7 29 100 Spain 320 32 79 5 20 400 Greece 440 38 78 2 15 600 Average 3307 28 4 7885 24929 WITH WHAT DOES LIFESPAN CORRELATE DOCTORS SMOKING MONEY g 82 7 g 82 i o E 32 39 E 3 gt1 81 V 5 81 V 3 81 j a a g g 80 39 o 5g 80 39 5g 80 35 5 o 5 a 5 79 5 79 o 5 79 39 78 39 I u 78 7 u 78 39 o I D V I V 0 77 i w v v x w x 7 x t t t t w x 77 i v w t w w t 100 200 300 400 500 600 20 25 30 35 15 20 25 3O 35 Physicians 000 people Smoking rate 70 GDP per Capita US1000 I Calculate linear c01 relati0n coef cient i 532 55 y y 2 2 2031 56 E y y I For life expectancy versus physicians 7quot 040 I For life expectancy versus smoking T 0 17 I For life expectancy versus GDP per capita 7quot 004 IS IT SIGNIFICANT I The linear correlation coef cients of life expectancy versus physicians smoking and GDP per capita are 7040 017 and 004 respectively I Life expectancy is most closely correlated with number of physicians but negatively correlated I How signi cant are the correlations I Appendix C gives probability to nd 7 gt To for N measurements of two uncorrelated variables 7quot0 0 01 02 03 04 05 06 07 08 09 1 11 100 77 56 37 22 12 51 16 03 O 12 100 76 53 34 20 98 39 11 02 0 13 100 75 51 32 18 82 30 08 01 0 I 12 measurements are 20 likely to have Ir gt 04 I I Using linear extrapolation the probabilities for r 7017 and r 004 are 61 and 90 so are likely uncorrelated I I General rule lt 5 is signi cant lt 1 is highly signi cant PHYSICS 2150 I LABORATORY Prof Eric D Zimmerman TA Geruo A Lab Coordinator Jerry Leigh Lecture 5 October 3 2006 ANNOUNCEMENT The second lab is due TOMORROW Problem set will be assigned tomorrow on the web page and due one week later Photoelectric effect Bonus opportunity THIS LECTURE The Poisson Distribution Goodnessoffit using x2 COUNTING EXPERIMENTS Start with a large amount of radioactive material with a very long halflife compared to the experiment Measure the decays with a detector at a fixed distance Measure the number of decays in five 100second intervals Should the five intervals all have the same number of decays Should there be a consistent trend COUNTING EXPERIMENTS Should there be a consistent trend NO The long halflife assures that over the time of the experiment the decay rate isn t changing significantly Should the five intervals all have the same number of decays NO The decay is a random process A POISSON PROCESS Assume a decay rate of 001 decayssec On average will get 1 decay100 seconds Sometimes you will get zero Sometimes you will get two Rarely you will get three or more What is the probability of getting exactly n decays in 100 seconds Answer Poisson distribution Applies to any discrete process where events occur randomly at a constant rate THE POISSON DISTRIBUTION 0 Characteristics the process has to have 0 Asymmetry Can t have fewer than zero counts so a symmetric function like a gaussian can t describe this process Also must vanish for nltO Discreteness Gaussian describes a variable that can take on a continuum of values There s no such thing as 048 events so Poisson must give probabilities not probability density of discrete outcomes 0 Summing we could view a process with a mean of 2p as a collection of two processes with a mean of 1 events from two halves of the same source for example So the following sum rule must be true P2u3 PMOPM3 I PM1PM2 I PM2PM1 I PM3PMO 2PMOPM3 I 2PM1PM2 Notation lf mean is p let probability of finding 3 events be Pp3 lf mean is 2p then probability of finding 3 events is P23 THE POISSON DISTRIBUTION u PM 6 ME 0 This is obvioust asymmetric doesn t allow negative events and is obviously discrete factorial is only defined for nonnegative integers Note that the mean p doesn t have to be an integer The sum rule works too try it out at home 0 Note that p factor keeps probability down for too few events n factor keeps it down for too many events MR GAUSS VS THE FISH GAUSSIAN 1 ltz 2 WW 2 6 2quot a 27T Symmetric Continuous a is real Mean 2 u most probable point 2 u standard deviation 2 a Distribution describes results of measurements With accuracy 0 POISSON PH 6 LL T Asymmetric Film 2 0 Discrete n is integer Mean 2 it most probable point 3 u Standard deviation 2 Distribution describes results from counting experiment POISSON DISTRIBUTION For a decay rate of 001 sec What is the probability of measuring in 100 seconds note that u 1 0 decays P10 1 100I 16 0368 1 decay P11 6 1 111 16 0368 2 decays P12 1 122I 126 0184 3 or more decays P13 P1z39 Easier way P13 1 P10 P11 P12 POISSON DISTRIBUTION 0394 Poisson pt 1 l Distribution for P101 6 1 Ego353 H 03 l Very asymmetric CH 025 I For u E II two most probable 02 points Pym PLu 1 015quot 01 I For u II most probable point 005 at n intu round down u G 0 l 2 I 5 6 Decays observed SMALL MEAN BEHAVIOR OF POISSON DISTRIBUTION 53 4 MI Poisson u 1 Gaussian u 10 l I For small u here u 10 Poisson and Gaussian different 7 l Standard deviation 2 a 7 0 25 68 probability discreteness 0239 l LL 2 1 implies 1 I 1 includes 015 0 1 2 92 of probability 01 l u 099 implies 099 I 0995 in 005 eludes 0 1 736 of probability 0 1 2 3 Probability o W UI 5 6 Decays observed LARGE MEAN POISSON BEHAVIOR GAUSSIAN LIMIT I For large u here u 2 75 014 I10isgosnz Gaussian Poisson and Gaussian become E061 Similar With a 17 008 l Standard deviation 2 a x 03906 68 probability 004 l Often use Gaussian approxima 002 tion for large M generally M gt 5 70 i i 3 4 5 6 8 9 101112131415 Decays observed HOW TO USE POISSON DISTRIBUTIONS IN COUNTING Uncertainty ou for large u can be interpreted similarly to gaussian 039 STATISTICAL uncertainty on the number of counts is thus the square root of the number of counts Background processes may be contributing to your rate uusignalubkg Often can measure ubkg by turning ohc the signal so when you then measure Htotal you can subtract the background to measure your signal rate How do you handle the error in this situation AN EXAMPLE FITTING RADIOISOTOPE LIFETIME Measure count rate background with no radioactive material present Introduce radioactive material Count the number of decays in a 1second period remeasure every 30 seconds Subtract the background rate Fit the rate vs time to an exponential lifetime MEASURING BACKGROUND Trial Counts in 1 sec 0 Background rate should be constant so each trial is a remeasurement of the same thing 1 44 2 42 0 Get the background rate by taking mean 3 39 40210402 note that this is exactly equivalent to simply measuring for 10 seconds and dividing 4 36 by 10 to find the rate 5 34 0 Uncertainty in the rate total counts is 402 x402 S 0 So in 10 seconds mean 402 20 8 37 0 Divide by 10 to get rate in 1 sec 40220 190 0 Background rate is 402 20 countssecond Total 402 SIGNAL DATA 0 Are all the trials measuring the same rate 0 What are the uncertainties in the number of counts 0 What is the measured signal rate 0 Subtract background 402 ctssec 0 Statistical error remains the square root of the total number of counts 0 Also there is a systematic error not shown due to uncertainty in the background Seconds Counts Corrected 0 2007 l45 1967 i45 30 1464 l38 1424 i38 60 973 l31 933 l31 90 698 l26 658 l26 120 526 l23 486 l23 150 353 i19 313 l19 180 285 i17 245 l17 210 217 i15 177 l15 240 150 i12 110 l12 270 122 ill 82 l11 300 112 ill 72 l11 330 86 i9 46 l9 360 78 i9 38 l9 390 53 i7 13 l8 420 51 i7 11 l7 FITTING FOR THE MEAN LIFE quot5 52000 0 Plot the rate vs seconds 31750 0 To fit to a line take the log 315007 1250 Nt NO 6Xp l ZI quotquot1nN 1nNo U72 1000 1 U1 0 I 0 Linear fit y 1nN xt 500 Z 50 2 E 7 39 7 M H Whats the uncertainty on y If uncertainty on N IS ON 060100 200 300 400 seconds ay 1110N 1ncounts second a ir i quot N 7y 8N aN aNlnNaN NON N 3 2 7 I I I I O 100 200 300 400 seconds HOW GOOD IS THE FIT Often want to know how good a fit is Minimizing x2 told you what the best fit to your function was The value of that minimum x2 can tell you how well data actually fit your function THE CHISQUARED TEST l X2 test is a particular type of goodnessof t test I Generally for N measurements 01 02 ON N 0 v E 2 X2 Z where EZ is the expected value and al is the i1 0i error on measurement 0i I To use we need to know oi which could be our estimated measurement error xEi or xOZ in the case of a counting experiment or if each measurement 02 is the result of many measurements the standard deviation of Oi I Also need degrees of freedom dof which is number of measurements N minus number of tted parameters 2 for a line slope and intercept l Reduced X2 is XZdof and should be N 1 I Can convert to probabilities using Appendix D in Taylor if probability is high generally gt5 agreement is OK LEAST SQUARES AN D CH SQUARED If you do a straight line fit without using errors unweighted least squares fit Fit returns error on y assuming they are all the same Min XZdof is 1 by definition Can t use XZdof to determine fit quality If you use externally known uncertainties to do a weighted line fit See Taylor problems 89 819 Fit returns function parameters slopeoffset and X2 You can use the table to determine fit compatibility with data as a confidence probability LOOK AT THE DECAY DATA Examples I Fit with background subtraction has X2 106 with 13 degrees of freedom calculated con dence level is 64 good lFit without background subtraction has X2 843 with 13 degrees of freedom calculated con dence level is 17 X 10 10 very bad Uses of X2 I Can determine if t is good I If t is bad either data is bad or t function is wrong lncounts second lncounts second L subtraction 2 t 62 l3 Fit with background subtraction i 100 200 300 400 seconds l 84 25 13 Fit without background l 0 00 l 200 300 400 seconds WHAT YOU SHOULD KNOW FROM LECTURESNOTESBOOK I Error propagation for an aribtrary equation like what is of if f 055 and 0a 05 ac ad are known I Understand difference between statistical and systematic uncertainties I Calculate mean standard deviation standard deviation of mean and know what they mean I Determine if measurements are compatible and how to calculate the weighted average I Perform least squares t to data and extract slope and intercept with uncertainties I How to determine if two variables are correlated I Understand what a counting experiment is and that the uncertainty on the counts is the square root of the number of counts

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