Indiana University, Fall 2014 P309 Intermediate Physics Lab Lecture 1: Experimental UncertainBes •  Reading: •  Bevington & Robinson, Chapters 1-­‐3 •  Handouts from hMp://physics.indiana.edu/~courses/p309/f14/ •
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“Experimental UncertainBes” “Basic StaBsBcs” ScienBfic Method •  Essen%al goal of experimental physics: –  Precise & accurate measurement of quanBBes describing the natural universe –  Compare with models (“theories”) to idenBfy those that most accurately represent reality (or reject those that fail to do so) •  Cri%cal step in the process: –  Do measurement and theory agree? •  How do we answer this quesBon? à QuanBfy the uncertainty in the accuracy of the measurement (as well as that of the theory predicBon) Example: measuring “g” •  Theory: Do these agree? –  If uncertainty is 0.5 m/s2, then yes! •  Experiment: –  If uncertainty is 0.05 m/s2, then no! –  Experimenter has responsibility to provide a quanBtaBve esBmate of the experimental uncertainty!! Terminology •  SomeBmes (/o`en) you will hear the uncertainty in a measurement being referred to as the “error on the measurement”. This is an imprecise shorthand. •  Real errors (as in mistakes) in measurements should be corrected for, if they are known. •  Uncertainty reflects the noBon that not all errors can be determined & corrected for. •  We characterize the degree of our lack of knowledge of just how wrong our measurement is likely to be, and express this quanBtaBvely via something called the “uncertainty”, which describes a specific range of possible errors. •  SBll, people use “error” and “uncertainty” interchangeably, so don’t get too hung up on this shorthand. Two Categories of Measurement Uncertainty •  StaBsBcal uncertainBes: –  Due to random variaBons (noise) in the measurement process. –  If you repeat a measurement many Bmes, this noise will average to zero. •  SystemaBc uncertainBes: –  Due to limitaBons of the methods or apparatus being used to make the measurement. –  They affect all measurements the same way, and hence do not average out with repeated measurements. Example •  Suppose you use a stop watch that reads to 1/100th of a second to measure some Bme interval. •  From one trial to the next your reac1on may be different, resulBng in variaBon in measured Bme. This would be a source of random error; averaging over mulBple trials would reduce the size of the sta+s+cal uncertainty associated with this source of error. •  You’ll never be able to know the 1me to much be:er than 0.01 s, which is the limitaBon of the watch. This is a systema+c uncertainty. •  Your reac1on 1me may be systema1cally slow, meaning that averaging over mulBple trials will sBll leave a measurement error. If you knew what this was you would correct for it. But if not, you are le` with another systema+c uncertainty. •  By the way, the purpose of calibra+on is to control exactly this sort of systemaBc uncertainty. How to EsBmate UncertainBes •  Assessing systemaBc uncertainBes can be a bit of an art. –  It relies on the experimenter’s judgment and ability to calibrate or cross-­‐check the accuracy of the instrumentaBon or method. –  Assessing systemaBc uncertainBes (and designing experiments that are most immune to idenBfiable dominant sources or symptoms of error) o`en calls for creaBvity and ingenuity. •  On the other hand, staBsBcal uncertainBes can be treated according to precise mathemaBcal formulaBons. Treatment of Random Errors •  Suppose you are measuring a quanBty µ using a process that is subject to random noise: •  This distribuBon can be characterized by its central value (the mean) and its width (the standard deviaBon). You would report the mean as the best esBmate of the true value of µ; and the staBsBcal uncertainty would be related to the standard deviaBon. StaBsBcal Uncertainty CompuBng the “mean”: and “variance” The mean gives a good approximaBon to µ, and s (the r.m.s., or standard, deviaBon) approximates well the width σ, i.e., the single measurement uncertainty. Can compute the uncertainty on the mean: Thus report result as: How to answer: Do measurement and theory agree? •  With an esBmate of expt’l uncertainty in hand, one can answer: –  “What is the likelihood that my measurement is a fluctuaBon of the true value as predicted by theory?” •  In principle, need probability density funcBon for such fluctuaBons –  in most (but not all) cases, sufficient to assume “Gaussian staBsBcs” PropagaBon of Errors •  Suppose want to infer value for quanBty y that is a funcBon y(x1,x2,x3,…) of several independent (uncorrelated) measurements x1,x2,x3,… w/ uncertainBes σ1,σ2,σ3,… •  What is the overall uncertainty in y? (see Bevington, chap. 3) •  But use common sense: –  For complicated fcn’s, someBmes easier to shi` input values by their errors one-­‐by-­‐one ‘by hand’ to evaluate the above terms. •  Also useful to commit certain rules to memory: Sums & Differences: errors add in quadrature Products & QuoBents: per cent errors add in quadrature