0: Systematic ErrorPhysical scientists. . . know that measurements are never perfect and thus wantto know how true a given measurement is. This is a good practice, for it keepseveryone honest and prevents research reports from degenerating into fi s h stories.Robert Laughlin (1998 Physics Nobel Laureate) p.10 A Different UniverseA hypothesis or theory is clear, decisive, and positive, but it is believed by noone but the man who created it. Experimental findings, on the other hand, aremessy, inexact things, which are believed by everyone except the man who didthe work.Harlow Shapley, Through Rugged Ways to the Stars. 1969Perhaps the dullest possible presentation of progress1in physics is displayed in Figure 1: themarch of improved experimental precision with time. The expected behavior is displayedin Figure 1(d): improved apparatus and better statistics (more measurements to average)results in steady uncertainty reduction with apparent convergence to a value consistent withany earlier measurement. However frequently (Figs. 1(a)–1(c)) the behavior shows a ‘final’value inconsistent w ith the early measurements. Setting aside the possibility of experimentalblunders, systematic error is almost certainly behind this ‘odd’ behavior. Uncertainties thatproduce different results on repeated measurement (sometimes called random errors) areeasy to detect (just repeat the measurement) and can perhaps be eliminated (the standarddeviation of the mean ∝ 1/N1/2which as N → ∞ , gets arbitrarily small). But systematicerrors do not telegraph their existence by producing varying results. Without any tell-tale signs, systematic errors can go undetected, much to the future embarrassment of theexperimenter. This semester you w ill be completing labs which display many of the problemsof non-random errors.Experiment: Measuring Resistance IConsider the case of the digital multimeter (DMM). Typically repeated measurement witha DMM produces exactly the same value—its random error is quite small. Of course,the absence of random error d oes not imply a perfect measurement; Calibration errors are1Great advancements is physics (Newton, Maxwell, Einstein) were not much influenced by the quest formore sigfigs. Nevertheless, the ability to precisely control experiments is a measure of science’s reach andhistory clearly shows that discrepant experiments are a goad for improved theory.910 Systematic Error(a) Neutron lifetime vs. Pub lication Date (b) B+lifetime vs. Publication Date(c) ω width vs. Publication Date (d) W mass vs. Publication DateFigure 1: Measured values of p article properties ‘improve’ with time, but ‘progress’ is oftenirregular. The error bars (δx) are intended to be ‘±1σ’: the actual value should be in therange x ± δx 68.3% of th e time (if the distribution were normal) and in the range x ± 2δx95.4% of the time. These figures are from the Particle Data Group, pdg.lbl.gov.Systematic Error 11–10 –5 0 5 10210–1–2 Voltage (V)Current (mA)AVRpowersupplyV δV I δI(V) (V) (mA) (mA)0.9964 .005 0.2002 .0012.984 .007 0.6005 .0034.973 .009 1.0007 .0056.963 .011 1.4010 .0078.953 .013 1.8009 .00910.942 .015 2.211 .012−0.9962 .005 −0.1996 .001−2.980 .007 −0.6000 .003−4.969 .009 −1.0002 .005−6.959 .011 −1.4004 .007−8.948 .013 −1.8001 .009−10.938 .015 −2.206 .012Figure 2: A pair DM-441B DMMs were used to measure the voltage across (V ) and thecurrent through (I) a 4.99 kΩ resistorexpected and reported in the device’s specifications. Using a pair of DM-441B multimeters,I measured the current through an d the voltage across a resistor. (The circuit and resu ltsare d isplayed in Figure 2.) Fitting the expected linear relationship (I = V/R), Linfitreported R = 4.9696±.0016 kΩ (i.e., a relative error of 0.03%) with a reduced χ2of .11. (Agraphical display showing all the following resistance measurements appears in Figure 3. Itlooks quite similar to the results reported in Figs. 1.)This r esult is wrong and/or misleading. Th e small reduced χ2correctly flags the fact thatthe observed deviation of the data from the fit is much less than what should have resultedfrom the supplied uncertainties in V and I (which were calculated from the manufacturer’sspecifications). Apparently the deviation between the actual voltage and the measuredvoltage does not fluctuate irregularly, r ather there is a high degree of consistency of theform:Vactual= a + bVmeasured(0.1)where a is small and b ≈ 1. This is exactly the sort of behavior expected w ith calibrationerrors. Using the manufacturer’s specifications (essentially δV /V ≈ .001 and δI/I ≈ .005)we would expect any resistance calculated by V/I to have a relative error of√.12+ .52=.51% (i.e., an absolute error of ±.025 kΩ for this resistor) whereas Linfit reported an error17 times smaller. (If the errors were unbiased and random, Linfit could properly reportsome err or reduction due to “averaging:” using all N = 12 data points—perhaps an errorreduction by a factor of N1/2≈ 3.5—but not by a factor of 17.) Linfit has ignored thesystematic error that was entered and is basing its error estimate just on the deviationbetween data and fit. (Do notice that Linfit warned of this problem when it noted the smallreduced χ2.)12 Systematic ErrorA B C4.9854.9804.9754.970 Resistance (k A B C4.994.984.974.964.95 Ω)Figure 3: Three different experiments are used to determine resistance: (A) a pair of DM-441B: V/I, (B) a pair of Keithley 6-digit DMM: V/I, (C) a Keithley 6-digit DMM directR. The left plot displays the results with error bars determined fr om Linfit; the right p lotdisplays errors calculated using each device’s specifications. Note that according to Linfiterrors the measurements are inconsistent whereas they are consistent using the error directlycalculated using each device’s specifications.When the experiment was repeated with 6-digit meters, the result was R = 4.9828 ±.0001 kΩ with a reduced χ2of .03. (So calibration errors were again a problem and the twomeasurements of R are inconsistent.) Direct application of the manufacturer’s specificationsto a V/I calculation produced a 30× larger error: ±.003 kΩA direct measurement of R with a third 6-digit DMM, resulted in R = 4.9845 ± .0006 kΩ.Notice that if Linfit errors are reported as accurate I will be embarrassed by future measure-ments which will point out the inconsistency. On the other hand direct use