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# Class Note for PHYS 3110 with Professor Forrest at UH

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This 25 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 18 views.

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Date Created: 02/06/15
Error Analysis PHYS 3110 Types of Error Instrumental Observational Environmental Theoretical Types of Error Instrumental Accuracy limits of instrument Poorly calibrated instrument Broken instrument Observational Environmental Theoretical Types of Error Instrumental Observational Parallax Misused instrument Environmental Theoretical Types of Error Instrumental Observational Environmental Electrical power brownout causing low current Local magnetic field not accounted for wind Theoretical Types of Error Instrumental Observational Environmental Theoretical Effects not accounted for or incorrectly ignored Friction Error in equations Types of Error Random n r Can be quantified by statistical l analysis True Value Systematic l I l Try to identify and get rid of l Hopefully found during True Value analysis may need to repeat experiment Statistical Analysis of Random Error For n measurements they should group x gt x around the true value 1 n For large n the average should tend x 2 xi to the true value 39 lfthe measurements are independent can 1 find the standard 5 Z a deviation 0 o is the 3921 Width Of the 4 distribution 39 39 3 Xi if Standard deviation of the mean o G Gn lZ m 1 n Gm szm X i1 For n gt 1 measured values report If om 0 use accuracy of measurement device for reported error If there is no systematic error there is a 23 probability that the true value is within ism Reporting Error Yicm Significant figures cm one sometimes two sig figs xave same accuracy as cm 6 1602176 5 X1039190000 OOO1X1O3919 C G and Gm 0 represents the error in one measurement cm represents the error in the mean of n measurements Gaussian Distribution Plot of measured value versus number N of times that value was measured IF error is random for large n this distribution tends to a 6 Gaussian distribution 24 PX NXn n X 2 0 1 2 3 4 5 6 7 8 9 10 x 262 MG 9 thm 25 1 2 3 4 5 6 7 8 9 10 Nx Gaussian Distribution The probability of a measurement being within YG 6 Ofxave39 Pwithin0 jPxdx Probability of being within 15 683 2c 955 35 997 Poisson Distribution X e PM 000 Q8382 v 005 Applies to processes a 000 003 described by an 000 001 exponential such as 0 radioactive decay 0 5 10 15 20 25 30 35 6 X Xave For large xave ie for long Gaussian 1 G jzo counting times the 447 Poisson distribution tends Gaussian 2 G 6 to the Gaussian distribution Reporting Error from a Poisson Process When measuring a physical process that you expect to follow a Poisson distribution the error in one measurement is xXioXiVX Example Measuring the intensity of radiation emitted by an or source and scattered by gold nuclei at an angle 9 over 30 seconds Propagation of Errors Determining the error in a quantity calculated from measured data Let x y 2 be measured values Let 8x 8y 82 be the corresponding estimated errors in the measurements x 1 8x etc If one measurement 8x precision of the instrument If n measurements of x then use x oX where ox is the standard deviation of the mean of x Propagation of Errors Let wxyz be a dz39 eremial function of measured aw aw aw values We want to dW dx dy dz 8x 8y 82 find 5w the error m w 39 If th Uncgrferlrgtfd raend 5 x 6502 5y z2 with dx z 5x 5W2 2 25 5 If the errors are i i correlated there are cov5x cross terms like Propagation of Errors M 6er 139 W ax by cz 5w Ja5x2 b5y2 cszY o Wzkxaybzc 2bwy5y2 2 awlax a k x6quot1 yb 2C 8 awax awx W a57quot2 W During lab for a quick error estimate just use the biggest term Density of a Cylinder p z 5p xS zamf Z 55d2 g 2 59 2 5m2 2 5 8dZ gshf your Density of a Cylinder 9 211 p J5m26d2 6h2 Assume after measuring d h and m three times each you get m 4920 05 g h 11001001 cm d 400 i 002 cm p 3559 gcm3 g 05s 2gtlt0020m 2 001m 2 50 339559 cm3 4920g2 400cm 1100cm 5023559 g 000120012000092 200356 g3 cm3 cm p 356 i 004 gCm3 Espp 1011 1 error Goal of Your 1St Experiment 002 error 5pp 0002 Consider the biggest term in 5pp only ESd 0002 8d 0002d2 0002400 cm2 0004 cm 6 G 8do 2 20004 before 002 o0035 d 5 2 8d203903520004 n 77 m J0 0004 Rule of thumb If more than 10 measurements of a single variable are needed use a better instrument or more care to reduce om Density of a cup d m 4 p 111 2D2H d2h D m D2H d2H x p 6p gar4f6D26H26d26x2 Weighted Average n n values each with Z xii2 their own errorxi oi Y 2 1 639 The error in this n 1 2 weighted mean can I1 be found from propagation of error a 2 2 1 i1 5 i1 2 aXi Weighted Mean Examples You measure 5 sets of E T data to try to find 0 in the StephanBoltzmann Law EGT4 You can t average E amp T and find one o from the average Find 5 as find the weighted mean using the 5 80 s calculated from propagation of error then find the uncertainty of the weighted mean Homework Find the density of your Melissinos textbook with estimated error

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