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by: Mrs. Peter Toy


Mrs. Peter Toy

GPA 3.96

Noel Clark

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Noel Clark
Class Notes
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This 24 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 Noel Clark in Fall. Since its upload, it has received 7 views. For similar materials see /class/232099/phys-2150-university-of-colorado-at-boulder in Physics 2 at University of Colorado at Boulder.




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Date Created: 10/30/15
PHYSICS 2150 I LABORATORY Instructors Noel Clark James G Smith Eric D Zimmerman Lab Coordinator Jerry Leigh Lecture 2 January 22 2008 PHY82150 Lecture 2 Announcementscomments The Gaussian distribution What it looks like Where and why it shows up Mean sigma and all that Statistical and systematic error A COMMENT YOUR FIRST LAB We hope you enjoyed the first experiment Your first lab reports are due next week Recall the advice in the syllabus and first lecture on your lab reports More info next week when you are deeper into writing GAUSSIAN DISTRIBUTION Shows up just about everywhere 06 Unit Gaussian Synonyms Normal Distribution Bell Curve Most basic form is unit Gaussian centered at zero unit integral unit I quotJr This is the lofli39ljy for a continUOUS normally distributed As with any probability density function random variable with mean zero and 00 standard deviation of 1 dxFc 1 OO GAUSSIAN DISTRIBUTION What can you do to the unit General normalized gaussian and still keep It a Gaussian gaussIan u 08 007 Change its mean from zero to u Change its width from 1 to I while increasing height by 10 Change its integral but then it s not a normalized probability distribution anymore So this isn t allowed here 1 027T As with any probability density exp I 202 function this still integrates to 1 WHERE IT SHOWS UP If you don t have a clue what the probability distribution of a random quantity is say the circumferences of cows at the EDZ Ranch it s highly likely to be approximately gaussian This is due to the Central Limit Theorem a sum of a large enough number of random numbers has a gaussian distribution no matter what the initial distribution shapes might have been Aside Distribution of counts of a process with a uniform rate in a finite amount of time is Poissondistributed see a later lecture but is approximately gaussian in the highnumber limit This is another example of the Central Limit Theorem If there is no systematic bias more on that later then the mean of measurements of a quantity is the best estimate of its true value MEAN SIGMA AND ALL THAT WARNING This is a histogram a plot showing 0 Say we measured 150 cows and how many times the result fell in each bin have the histogram of results NOT a normalized probability distribution Note that its integral is not 1 Calculate the mean circumference ltCgt 1 c Zc439m N Z Standard deviation is calculated by ac 311c W 2 00 m N 1 0 Read up in Taylor on 3 4 5 6 Bovme C1rcumferencem definitions and uses of variance standard deviation MEAN SIGMA AND ALL THAT 0 So we can now say that the mean circumference is 439 m and the standard deviation is 070 m What do we know 0 68 of cows have circumference between 439 070 and 439070 m This is because the integral of the gaussian from u o to u 0 is 068 0 How well do we know the mean circumference Need std dev on the mean Bovine Circumference m ac 0 006 C m 80 ltcgt439 006 m where quotiquot means 1 sigma or 68 probability that the true mean is within that interval 68 confidence level Note that we know the mean to much better than I of individual measurements USING THE GAUSSIAN Can use the same mathematics to describe the results of repeated measurements of the same quantity where there is random errorresolution in the instrument 0 The distribution w ll be centered on a mean assume for now that this is the correct value 0 The distribution will have a standard deviation 0 Can fit this to a gaussian Lecture 45 or just 239 15 Jifs o 05 1 15 2 Measured length cm calculate mean sigma directly l 0 Uncertainty on the mean IS now 0 Note More measurements gt smaller error on the mean Also means better determination of error WHAT DOES SIGMA MEAN First if the measurements are from a gaussian FuoX then the probability of measuring a value in the range ab is P f dema For a normalized gaussian dJJFM CE 068 The general integral can t be expressed analytically Use error function erfx tables for values other than 10 So saying J54O9 means one can say the true value of J is between 45 and 63 with 68 confidence level ANALYZING ERROR ON A QUANTITY 0 You are in a car on a bumpy road on a rainy day and are trying to measure the length of the moving windshield wiper with a shaky ruler 0 You measure it 37 times 8 The histogram of your results is at right It 7 doesn t look very gaussian But with only 37 e 13909quot measurements plotted in lots of bins E 5 one 0219quot distributions often look ratty E 4 0 You calculate the mean to be 562 cm and 73 the standard deviation to be 68 cm quot39 2 1 The 10 uncertainty on the mean is 11 cm 0 0 If we assume the underlying distribution is nevertheless gaussian and centered on the true Bar Length cm value we can turn this into a confidence level the wiper length is between 551 and 573 cm with 68 confidence ANALYZING ERROR ON A QUANTITY 0 Stop the car go outside and measure true the wiper properly it is 610 cm long 0 We said the wiper length was between 551 and 573 cm with 68 confidence 10 error OPT H198 0 We are off by over 4 sigma This is shockingly unlikely Frequency au 0 Clearly there is a systematic shift The distribution does not center on the true value 8 7 6 5 4 3 2 1 0 Bar Length cm 0 More data pornts won t get us any closer to the true value We need to make better measurements or find the source of the error and apply a correction to the data STATISTICAL RANDOM vs SYSTEMATIC UNCERTAINTIES STATISTICAL SYSTEMATIC NO PREFERRED DIRECTION BIAS ON THE MEASUREMENT ONLY ONE DIRECTION THOUGH OFTEN DON T KNOW WHICH CHANGES WITH EACH DATA POINT TAKNG MORE DATA REDUCES ERROR ON THE MEAN STAYS THE SAME FOR EACH MEASUREMENT MORE DATA WON T HELP YOU GAUSSIAN MODEL IS USUALLY GOOD EXCEPT COUNTING EXPERIMENTS WITH FEW EVENTS GAUSSIAN MODEL IS USUALLY TERRIBLE BUT WE USE IT ANYWAY IF DON T HAVE A BETTER MODEL Keep statistical systematic errors separate Report results as something like 0 g 965 30stat 12syst cms2 Add in quadrature note that this assumes gaussian distribution to compare with known values g 965 32total cms2 PROPAGATION OF ERRORS Often we aren t measuring directly the quantity our experiment is after we measure some lab quantities and our final physics result is a function of them Kaon experiment we measure curvature of tracks and from them calculate the momentum of the pions and then calculate the mass of the kaon from that We know the errors on the lab quantities How do we find the error on the final physics result This is a specific case of the general problem of finding the error on a quantity that is a function of random uncertain variables PROPAGATION OF ERRORS The general formula for error on a function q of random variables Xyz a 2 a 2 a 2 2 53596 87353 5352 Special case 1 addition with say multiples Let q X y 22 8g 2 8g 2 8g 2 5 5 5 5 2 ltaa 2quot ay 2 lt62 2 602 6202 25202 602 6202 4622 PROPAGATION OF ERRORS Special case 2 multiplication Let qxyz 8g 2 8g 2 8g 2 6 6 l 6 l 6z q 8 8y y 8z xyz52 Wait2 y6z2 6 670 2 6 2 sz 2 q llt gt lt ygt lt gt q 70 y z 0 We can add fractional errors in quadrature to get the fractional error on the final result This works for division too Derive it yourself COMPARING WITH KNOWN VALUE Measure g 965 i 32 cms2 Xi5X Often negligible Known value 980665 cms2 xoioxo Discrepancy is xxo6xxo where 6xxo2 6x2 6x02 add in quadrature Discrepancy in units of sigma often called 0 0 significance of discrepancy is a W Discrepancy here is 16 i 32 cms2 or 050 Use erf table to determine agreement confidence level 62 agreement good ANOTHER EXAMPLE em I Electrons accelerated to 40 60 or e 80 Volts l Acted on by magnetic eld 69 perpendicular to velocity B I 13 617 X g forces electrons into 39 electrons ejected CHCUlar paths K with voltage V 40 80 V l Measuring radius of circle gives centripetal force by F mfg l Energy of electrons given by accelerating voltage mv2 6V l Magnetic eld from Helmholtz coils is B 3 Va2 6 l Aft 39 i 3906 er m1x1ng m 1 qung ANOTHER EXAMPLE em 2 3 39062V a m MON 21 272 Variable De nition How determined V accelerating potential measured a Helmholtz coil radius measured MO permeability of free space artifact N number of turns in each coil given 7quot e beam radius given I IT 0 net current calculated I T total current measured IO cancellation current measured Want answer with statistical and systematic uncertainties ANOTHER EXAMPLE em l Measure IO three times and nd 017 020 023 giving a mean of 020 with uncertainty of 003 l Compute e m 14 times at different accelerating voltages and e radii Entry I A 7quot m V V Ckg 1 175 00572 400 208 X 1011 199 00509 400 208 x 1011 3 228 00447 400 200 x 1011 4 267 00384 400 198 X1011 5 320 00321 400 197 X1011 6 220 00572 600 197 x 1011 7 249 00509 600 194 x 10 8 285 00447 600 192 x 1011 9 334 00384 600 190 gtlt1011 10 1 1 12 13 403 00321 600 186 gtlt1011 258 00572 800 191 X1011 290 00509 800 191 x1011 330 00447 800 188 x 1011 14 390 00 8 185 gtlt1011 Entrv 8 used as tvDical for determining svstematic uncertaintv ANOTHER EXAMPLE em 55239 11 IMean a 1T 1946 x 10 Ckg W 11 I Standard deV1at10n am 2 2 0072 X 10 Ckg N 1 a 0072 I Uncertaint on mean a a 0019 X 1011 Ck y m m g Result with statistical uncertainty only 3 1946 i 0019 x 1011 Ckg m ANOTHER EXAMPLE em So 6 Va 3906 m M3N21272 lt6VVgt222lt267122 2 LEW V 60 5a 030m 7 i 3320m 6r 00002m 7 i m 610 from meter 0003x020001A 0011 A 610 from uncertainty setting IO 017 0207 023A gt IO 020 i003A 610 x00112 003 0032 FIT from meter 0003x285001A0019A 6139 61T26IO2 x0019200322AOO37A 61 0037A T 285A 039013 ANOTHER EXAMPLE em 62 6V 2 6a 2 SI 2 67 2 w E 3 WV 2agt 01 3 E 2 2 2 20037 2 200002 2 60 332 285 00447 000272 001812 002602 000892 00330 330 I gs 00330 x 1946 X 1011 0064 x 1011Ckg 6 0019 x 1011 Ckg I Answer 1946 3 0019stat j 0064sys X 1011 Ckg I Can round to 195 i 002stat i 006sys x 1011 Ckg ANOTHER EXAMPLE em I Compare measured value 1946 d 0019stat j 0064sys X 1011 Ckg to accepted value 175882 gtlt 1011Ckg I Absolute discrepancy 1946 1759 X 1011 Ckg 0187 x 1011 Ckg 0187 1011 Ck 0187 I Signi cance of discrepancy 28 V00192 00642 Ckg 0067 often stated as off by 280 I If uncertainties calculated correctly should get within 10 68 of time I What is probability of being off by 280 or more I Probability being inside 280 is 9949 from erf or table on p 287 of Taylor I Probability of being outside 280 is 1 09949 051 l Very unlikely mistake or underestimated systematic uncertainties


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