{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 " " 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 } {CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 } {CSTYLE "" -1 295 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 51 "Experimental uncertaintie s: the normal distribution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1435 "We consider the statistical analysis of random \+ experimental uncertainties in this worksheet. This will allow us to un derstand how to analyze repeated measurements of the same quantity, as suming that the uncertainties are caused by random measurement errors. These can include fluctuations in currents in a circuit (depending on someone else's loads in the power grid), variable reaction time when \+ a stopwatch is used, parallax problems when reading a length. Some of \+ these can also give rise to systematic errors, whereby all data values taken are consistently under- (or over-) estimated by the experimente r. These are much harder to track. The most typical systematic error t hat I observed in our laboratory over the years was the uncalibrated o ut-of-tune timebase of a time chart recorder (used in a Cavendish expe riment to track the motion of two lead balls via a photodiode pick-up \+ of a laser beam). This error was the easily spotted with a stopwatch, \+ but no student was able to pick up on it (they simply believed the rea ding on the knob). A 20% systematic error in the timebase resulted in \+ a wrong value of Newton's constant. Most students had a more accurate \+ value from the manual experiment (where they tracked the motion by mar kings on graph paper pasted onto a wall), and were surprised that the \+ automated experiment gave inferior results (and often inflated their u ncertainties to avoid conflict with the literature value)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 386 "A detailed analy sis of experimental uncertainties is crucial in order to assess whethe r systematic errors are present in someone's experiment or not. We wil l not enter into the question here as to why the experimental uncertai nties can (usually) be assumed to follow a normal (Gaussian) distribut ion. The prerequisites for this situation are that there are many sour ces of small errors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 412 "We are interested in a single quantity that is measure d repeatedly. For the measurement of linear (or other) relationships a n extension of the arguments presented here leads to the linear least \+ squares (or quadratic) fit. In this worksheet we will simulate random \+ errors by using normally distributed random numbers to test the theore tical ideas which are proposed for the analysis of experimental uncert ainties." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 425 "Every experimenter will intuitively form averages from repeated m easurements in order to arrive at an improved experimental result for \+ the quantity to be measured. Understanding experimental uncertainties \+ will not only lead to a proper justification, but also to a measure of the expected accuracy of the averaged result. For this purpose we hav e to adopt a model which tells us how the repeated measurements are di stributed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "To illustrate the concept of aver age and standard deviation we follow the textbook approach (John R. Ta ylor: " }{TEXT 257 33 "An Introduction to Error Analysis" }{TEXT -1 418 ", chapters 4 and 5, University Science Books 1982, 2nd ed. 1998). Suppose that we have carried out a measurement for the spring constan t of a typical spring (out of a box containing a set of identically ma nufactured ones) repeatedly by timing the period of oscillation while \+ a well-known mass was attached. The series of 10 measurements is given as (in Newton/meter) the following list of values with 2-digit accura cy:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "kL:=[86,85,84,89,86, 88,88,85,83,85];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#kLG7,\"#')\"#&) \"#%)\"#*)F&\"#))F*F'\"#$)F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 707 " The choice of 2-digit accuracy was reasonable. Our timing device gave \+ more digits of readout, but due to the substantial fluctuations in the second digit we decided to round the numbers to two digits. (Common m istakes made by students is to: (a) carry all 10 digits that their cal culator produced while dividing two numbers that were measured only to 2 or 3 figures; (b) to trust digitial devices that display much more \+ than the stated accuracy in the manual (digital voltmeters and ammeter s); (c) ignoring the fact that timing motion with digital stopwatches \+ does not eliminate human error, and reaction time; (d) measurements wi th calipers can be tricky due to bending, improper holding of the inst rument." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 400 "Suppose that all numbers would have been equal in the set of meas urements. Then there would be no deviation from the average. In that c ase we might ask ourselves whether more digits can be read from the in strument, and whether it is worthwhile to make an attempt at more prec ision. At this point we probably could proceed and carry out an uncert ainty estimate with the data set with 3-digit accuracy." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The average value is presumably our best estimate:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "N:=nops(kL);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#5" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "k_a:=evalf(add(kL[i],i=1..N )/N,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k_aG$\"$f)!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "Before we proceed to calculate th e deviation we illustrate the data by forming a histogram (a frequency distribution or binning of the data)." }}{PARA 0 "" 0 "" {TEXT -1 189 "We choose a bin size and count how many measurements fall into ea ch bin. We could pick 10 bins around the calculated average value, or \+ simply cover the range between 80 and 90 with 20 bins." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "dk:=0.5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dkG$\"\"&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "k_i:=i->80+(i-0.5)*dk;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k_iGR6#%\"iG6\"6$%)operatorG%&arrowGF(,&\"#!)\"\"\"*&,&9$F.$! \"&!\"\"F.F.%#dkGF.F.F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "O ne way to proceed to bin the data is to subtract 80 from the value, mu ltiply the remainder by 20 and round it to an integer (by adding 0.5 a nd truncating):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "i_k:=k-> trunc(2*(k-80-0.5*dk)+0.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$i_kG R6#%\"kG6\"6$%)operatorG%&arrowGF(-%&truncG6#,(9$\"\"#$!%&f\"!\"\"\"\" \"%#dkG$!#5F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "j:=1 ; i_k(kL[j]),k_i(i_k(kL[j])),kL[j],k_i(i_k(kL[j])+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"jG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\" #7$\"%v&)!\"#\"#')$\"%D')F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "T he defined functions are consistent, i.e., the index used in the binni ng process works out such that it assigns data values to fall to the r ight of the bordering value." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "for i from 1 to 20 do: C[i]:=0: od:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 90 "for n from 1 to N do: ict:=i_k(kL[n]); if ict>0 and ict<=20 then C[ict]:=C[ict]+1; fi; od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "with(plots): P1:=plot([k_a,t,t=0..2.5],color=red):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "P2:=plot([seq([k_i(ict)+0. 5*dk,t*C[ict],t=0..1],ict=1..20)],color=blue): display(P1,P2);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "69-%'CURVESG6$7S7$ $\"1,+++++!f)!#9\"\"!7$F($\"1mmmT&)G\\a!#<7$F($\"1L$ek`o!>5!#;7$F($\"1 nm\"z>)G_:F37$F($\"1nmT&QU!*3#F37$F($\"1L$eRZXKi#F37$F($\"1n;z>,_=JF37 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"Is this data set following a Gaussian distribution? At f irst we might think that it does not (suspicious gap to the right of t he average). On second thought (and supported by testing done on the b inomial and Poisson distributions in a separate worksheet using a chi- squared test of a hypothesis) we remain open-minded: the data set is m uch to small to rule out that a much larger data set obtained with the same measurement procedure will reach a Gaussian as its limiting dist ribution (limit of infinitely many measurements!)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 467 "The standard deviation o f the data is a quantity that looks at the distances of the individual measurements from the calculated average. The distances are added qua dratically in order not to cancel positive against negative errors. We average the squared deviations, and then take the square root to obta in a width (the deviation). When averaging the sum of the squared devi ation we make a small adjustment to account for the fact that one degr ee of freedom from the " }{TEXT 258 1 "N" }{TEXT -1 97 " available fro m the data set is lost, as the average was obtained from the data set \+ itself. Thus:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "sigma:=eva lf(sqrt(add((kL[n]-k_a)^2,n=1..N)/(N-1)),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaG$\"$\">!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "This number represents the average " }{TEXT 262 35 "deviation o f individual data points" }{TEXT -1 276 " from the average value. It i s the typical uncertainty associated with an individual measurement. I f we make single measurements for other springs from the box of suppos edly identical springs, and find that the value is within sigma away f rom our calculated average (based on " }{TEXT 259 1 "N" }{TEXT -1 223 " measurements on the first spring), then we should not be concerned a t all about the question whether the springs are made identically to a sufficient degree, as they appear to be identical to within our measu ring accuracy." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 182 "We did promise, however, more. We promised to be able to assess from the distribution the accuracy of the determined average v alue, which is presumably the best value we have (after " }{TEXT 260 1 "N" }{TEXT -1 35 " measurements). The uncertainty or " }{TEXT 261 30 "standard deviation of the mean" }{TEXT -1 91 " - provided the data do have a Gaussian as its limiting distribution - can be estimated as :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sig_avg:='sigma/sqrt(N )';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(sig_avgG*&%&sigmaG\"\"\"-%%s qrtG6#%\"NG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(s ig_avg);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+J.&*Rg!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "This is smaller than the average uncertai nty of the individual data points by " }{TEXT 19 7 "sqrt(N)" }{TEXT -1 111 ", which is a good reason to take as many data as possible as t he reduction in this uncertainty with increasing " }{TEXT 263 1 "N" } {TEXT -1 9 " is slow." }}{PARA 0 "" 0 "" {TEXT -1 86 "Thus, the mean ( average) together with its deviation can be listed for our example as \+ " }}{PARA 0 "" 0 "" {TEXT -1 27 "(85.9 plusminus 0.6) [N/m]." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 264 19 "Normal distribution" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 445 "To compare the hi stogram to a normal distribution it has to be properly normalized. We \+ have just determined the two parameters that control the shape of the \+ limiting Gaussian distribution, namely the position of the average (a \+ Gaussian is symmetric), and the width (the deviation). The Gaussian (o r normal) distribution is normalized by an integral to unit area. The \+ frequency distribution at present is normalized to the number of measu rements:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "add(C[ict],ict= 1..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "It is expedient to normalize it to unity:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "for ict from 1 to 20 do: C[i ct]:=evalf(C[ict]/N,3); od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Th e average value of the spring constant can now be obtained by a probab ilistic sample average:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " evalf(add((k_i(ict)+0.5*dk)*C[ict],ict=1..20),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$f)!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "Ot her averages can be obtained similarly, e.g., the deviation squared of the individual data points:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "evalf(add((k_i(ict)+0.5*dk-k_a)^2*C[ict],ict=1..20),3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$I$!\"#" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "evalf(sqrt(%),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"$#=!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "This is slightly \+ less than the value obtained before, as it does not replace the 1/" } {TEXT 266 1 "N" }{TEXT -1 64 " factor in the formation of the average \+ squared deviation by 1/(" }{TEXT 265 1 "N" }{TEXT -1 4 "-1)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "To make compari son with the limiting distribution we plot:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "P1:=plot([k_a,t,t=0..0.25],color=red): P4:=plot([ [k_a-sig_avg,t,t=0..0.25],[k_a+sig_avg,t,t=0..0.25]],color=magenta): P 5:=plot([[k_a-sigma,t,t=0..0.15],[k_a+sigma,t,t=0..0.15]],color=violet ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "P3:=plot(exp(-(k-k_a) ^2/(2*sigma^2))/(sqrt(2*Pi)*sigma),k=80..90,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "P2:=plot([seq([k_i(ict)+0.5*dk,t*C[ ict],t=0..1],ict=1..20)],color=blue): display(P1,P2,P3,P4,P5);" }} {PARA 13 "" 1 "" {GLPLOT2D 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the comparison: the magenta-coloured lines indicate the standa rd deviation of the mean; the violet-grey coloured line indicate the s tandard deviation for the Gaussian distribution associated with the in dividual measurements." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 343 "It is the obligation of the experimenter to demonst rate that the error in the data are indeed normally distributed, i.e., that the agreement with a Gaussian distribution improves for larger d ata sets. Once this is demonstrated, the probability theory based on t he normal distribution can be used to further assess the confidence in our result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 189 "First we demonstrate that the continuous Gaussian distribution results in the expected values for the total probability (area under \+ the curve), the average value, and the deviation squared." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "int(exp(-(k-k_a)^2/(2*sigma^2))/(sq rt(2*Pi)*sigma),k=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'*********!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 " int(k*exp(-(k-k_a)^2/(2*sigma^2))/(sqrt(2*Pi)*sigma),k=-infinity..infi nity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'******e)!\")" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "int((k-k_a)^2*exp(-(k-k_a)^2 /(2*sigma^2))/(sqrt(2*Pi)*sigma),k=-infinity..infinity);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+(***4[O!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "One can ask how much probability is contained within the \+ standard deviation interval:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "int(exp(-(k-k_a)^2/(2*sigma^2))/(sqrt(2*Pi)*sigma),k=k_a-sigma.. k_a+sigma);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5\\*o#o!#5" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 334 "We say that we have 68% confidenc e that a measurement will fall within this interval. Why do we care ab out this statement (given that we have determined an accurate 'best' v alue by forming the average)? The answer is that what we learn about i ndividual measurements can be transferred later to assess the accuracy of the 'best' value." }}{PARA 0 "" 0 "" {TEXT -1 219 "So let us conti nue with considering the individual measurements. A 68% confidence sou nds not so terribly high. We can carry out area calculations, i.e., co nfidence estimates for larger than one-sigma intervals. For the " } {XPPEDIT 18 0 "2*sigma;" "6#*&\"\"#\"\"\"%&sigmaGF%" }{TEXT -1 6 "- an d " }{XPPEDIT 18 0 "3*sigma;" "6#*&\"\"$\"\"\"%&sigmaGF%" }{TEXT -1 19 "-intervals we have:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 " int(exp(-(k-k_a)^2/(2*sigma^2))/(sqrt(2*Pi)*sigma),k=k_a-2*sigma..k_a+ 2*sigma);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+_t*\\a*!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "int(exp(-(k-k_a)^2/(2*sigma^ 2))/(sqrt(2*Pi)*sigma),k=k_a-3*sigma..k_a+3*sigma);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+M?+t**!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 217 "Therefore, once we have determined sigma for our distribution, we can say with a definite confidence that subsequent measurements (by o urselves, or someone else provided our systematics are identical) will fall into (" }{XPPEDIT 18 0 "t*sigma;" "6#*&%\"tG\"\"\"%&sigmaGF%" } {TEXT -1 17 ")-intervals, for " }{TEXT 267 1 "t" }{TEXT -1 9 "=1,2,... ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 533 "Wha t relevance does this have for assessing our best value? One could per form a statistical analysis of the average value, i.e., run 10 dataset s with 10 measurements respectively, compute the averages, and then th e average of averages with known uncertainty. The previous assessment \+ of the standard deviation of the mean is a shortcut (but it is a guess as opposed to a measurement). It guesses that these measurements of t he averages ('best' values) will be distributed randomly according to \+ a Gaussian with a width estimated to be " }{TEXT 19 7 "sig_avg" } {TEXT -1 207 ". Therefore, we can consider the interval surrounding th e best estimate (red vertical line with magenta-coloured sidebars) to \+ be the 68% confidence interval for the true value. We can draw the int erval with " }{TEXT 19 9 "2*sig_avg" }{TEXT -1 218 " and be 95% confid ent that the true value lies within this interval. Taylor provides a p roof about the Gaussian distribution of the best values based on Gauss ian distributed individual data measurements in section 5.7." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The lessons to \+ be learned here are:" }}{PARA 0 "" 0 "" {TEXT -1 365 "a) the one-sigma interval is not a holy grail. it contains only a 68% confidence asses sment! One-sigma 'new' physics (propagated with big hoopla in Physical Review Letters or Physics Letters and suspected by the critical minds ) often goes away! For three-sigma events the chances are much less th at they are the result of hype rather than solid experimentation. By \+ " }{TEXT 270 1 "t" }{TEXT -1 72 "-sigma 'new' physics we mean unexpect ed results that lie outside of the " }{TEXT 271 1 "t" }{TEXT -1 51 "-s igma confidence interval of conventional physics." }}{PARA 0 "" 0 "" {TEXT -1 316 "b) the reader may wish to understand why the average rep resents the best estimate for the measured value. The argument involve s turning things around, i.e., assuming a limiting Gaussian distributi on with yet undetermined parameters to calculate the probability to ar rive at a certain measured data set (a product of " }{TEXT 268 1 "N" } {TEXT -1 298 " probabilities). A so-called principle of maximum likeli hood is invoked which results in a determination of the peak of the Ga ussian at the position given by the average of the measured data value s (the arguments are given in Taylor's book, and elaborated on in appl ied probability theory courses)." }}{PARA 0 "" 0 "" {TEXT -1 457 "c) t here are probably lessons to be learned also for error propagation. Gi ven that we are arriving at an understanding of uncertainties for one \+ variable, we can learn that normally distributed errors in two (or mor e) variables conspire to form errors in some final result (that may in volve some mathematical operation between the two basic variables). Ta ylor explains how addition of errors in quadrature follows from the no rmal distribution in section 5.6." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 269 11 "Simulations" }}{PARA 0 "" 0 "" {TEXT -1 223 "We can verify our claims by generating virtual measureme nts in Maple, i.e., computer simulations of normally distributed measu rements. Then we apply the formulae and check whether our findings are consistent with the input." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 225 "The first task is to find a distribution of ra ndom numbers which follows a Gaussian. Normally pseudo-random number g enerators produce randoms that fill the [0,1] interval uniformly. Norm ally distributed randoms fill the range" }{TEXT 19 19 "-infinity..infi nity" }{TEXT -1 63 " in such a way that a histogram produces a Gaussia n of width 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(stats ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "RNd:=[stats[random, n ormald](150)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(stat s[statplots]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "histogram (RNd,area=1);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6 %-%)POLYGONSG6-7&7$$!+k$y_e#!\"*\"\"!7$F($\"+LLLL8!#67$$!++)fh:#F*F-7$ F1F+7&F37$F1$\"+)*******RF/7$$!+O7/F7$FAF+7&FC7$FA$\"+jmmm')F/7$$!*4T!)o)F*FF7$FIF+7&FK7$FI$ \"+mmmm;!#57$$!*XboR%F*FN7$FRF+7&FTFQ7$$!)\")pc5F*FN7$FWF+7&FY7$FW$\"+ LLLL6FP7$$\"*$e^&=%F*Ffn7$FinF+7&F[o7$Fin$\"+pmmm9FP7$$\"*Y,nZ)F*F^o7$ FaoF+7&Fco7$Fao$\"+)*******fF/7$$\"+5()yw7F*Ffo7$FioF+7&F[p7$Fio$\"+IL LL`F/7$$\"+us!fq\"F*F^p7$FapF+7&Fcp7$Fap$\"+jmmmmF/7$$\"+Qe-N@F*Ffp7$F ipF+-%&COLORG6&%$RGBG$\"\"(!\"\"F`q$\"#5Fbq-%%VIEWG6$;$!+%y3t0$F*$\"+e i02EF*%(DEFAULTG" 1 2 0 1 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 } }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Without the option area=1 the \+ histogram would be arranged in such a way that each bar would contains the same area." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 314 "We can use the numbers produced with a deviation of 1 to generate errors by simply scaling the numbers with a constant factor. The average comes out to be zero, and thus we can add the scaled numb ers to some hypothetical measurement value. Suppose we have measured a n angle of val=65 degrees with some uncertainty." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "val:=38;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $valG\"#Q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "N:=nops(RNd); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"$]\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "MData:=[seq(val+0.5*RNd[i],i=1..N)]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "histogram(MData,area=1);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%)POLYGONSG6-7& 7$$\"+#3O2n$!\")\"\"!7$F($\"+XLLL8!#67$$\"+5?>#p$F*F-7$F1F+7&F37$F1$\" +M+++SF/7$$\"+Qzk8PF*F67$F9F+7&F;7$F9$\"+Pjmm')F/7$$\"+nQ5NPF*F>7$FAF+ 7&FC7$FA$\"+Tnmm')F/7$$\"+&zflv$F*FF7$FIF+7&FK7$FI$\"+\"ommm\"!#57$$\" +Bd,yPF*FN7$FRF+7&FTFQ7$$\"+^;Z*z$F*FN7$FWF+7&FY7$FW$\"+VLLL6FP7$$\"+z v#4#QF*Ffn7$FinF+7&F[o7$Fin$\"+zmmm9FP7$$\"+2NQUQF*F^o7$FaoF+7&Fco7$Fa o$\"+s(*****fF/7$$\"+O%RQ'QF*Ffo7$FioF+7&F[p7$Fio$\"+!QLLL&F/7$$\"+k`H &)QF*F^p7$FapF+7&Fcp7$Fap$\"+BnmmmF/7$$\"+#H^n!RF*Ffp7$FipF+-%&COLORG6 &%$RGBG$\"\"(!\"\"F`q$\"#5Fbq-%%VIEWG6$;$\"+hX8ZOF*$\"+8GNIRF*%(DEFAUL TG" 1 2 0 1 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "Let us apply our expressions to recover \+ the true value. In the example below it is important to carry more tha n 3 digits when carrying out the average!" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "v_a:=evalf(add(MData[i],i=1..N)/N,7);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$v_aG$\"(Ijz$!\"&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 57 "sigma:=evalf(sqrt(add((MData[n]-v_a)^2,n=1..N)/(N-1 )),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaG$\"&M:&!\"&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sig_avg:='sigma/sqrt(N)';" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(sig_avgG*&%&sigmaG\"\"\"-%%sqrtG6# %\"NG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sig_avg:=eval f(sig_avg);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(sig_avgG$\"+\"[Lx?%! #6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "We have a 68% confidence in terval which can exclude the true value." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "[evalf(v_a-sig_avg,4),evalf(v_a+sig_avg,4)];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"%#z$!\"#$\"%+QF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "For the 95% CI we almost certainly includ e the correct value for " }{TEXT 272 1 "N" }{TEXT -1 5 ">100:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "[evalf(v_a-2*sig_avg,4),eval f(v_a+2*sig_avg,4)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"%)y$!\"#$ \"%/QF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 211 "If we don't include t he correct value in the two-sigma interval, the cause for this discrep ancy comes from the fact that the average of the sample taken accordin g to a normal distribution has a systematic drift:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "evalf(add(RNd[i],i=1..N)/N);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$!++$ehM(!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 140 "Explore the quest ion whether the true value lies within the 1-sigma and within the 2-si gma confidence intervals around the 'best' answer by:" }}{PARA 0 "" 0 "" {TEXT -1 52 "a) changing the sample size (increase and decrease);" }}{PARA 0 "" 0 "" {TEXT -1 114 "b) changing the true value of the meas ured quantity (increase and decrease), while the error size is kept th e same" }}{PARA 0 "" 0 "" {TEXT -1 91 "c) changing the error size, whi le keeping the original true value of the measured quantity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 244 "Keep in mind t hat the simulation in Maple provides us with the luxury of many repeat ed measurements. In the undergraduate laboratory setting we often fore go this luxury (unless experiments are automated), and pay the price o f reduced precision." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 264 "We conclude the worksheet with an example of e rror propagation. Let us assume that the quantity measured above repre sents an angle of refraction in a simple experiment with a lightbeam a t an air/glass interface. Let us assume that we know the index of refr action (" }{TEXT 273 1 "n" }{TEXT -1 167 "=1.51 for glass) to a higher degree of precision than the angle measurement. What is the deviation in the incident angle. We first look at the transformed data sample: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "fun:=x->evalf(180/Pi*ar csin(sin(Pi/180*x)*1.51));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$funGR 6#%\"xG6\"6$%)operatorG%&arrowGF(-%&evalfG6#,$*&-%'arcsinG6#,$-%$sinG6 #,$*&%#PiG\"\"\"9$F;#F;\"$!=$\"$^\"!\"#\"\"\"F:!\"\"F>F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "IAngle:=map(fun,MData):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "IAngle[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/pwNq!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "histogram(IAngle,area=1);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%)POLYGONSG6-7&7$$\"+a**y\\k!\")\"\"!7$F($\"+L LLL8!#67$$\"+uT+>lF*F-7$F1F+7&F37$F1$\"+mmmmYF/7$$\"+%R=#)e'F*F67$F9F+ 7&F;7$F9$\"+++++5!#57$$\"+9EVdmF*F>7$FBF+7&FD7$FB$\"+mmmm')F/7$$\"+Mok EnF*FG7$FJF+7&FL7$FJ$\"+++++?F@7$$\"+a5'ez'F*FO7$FRF+7&FT7$FR$\"+mmmm; F@7$$\"+u_2loF*FW7$FZF+7&Ffn7$FZ$\"+++++7F@7$$\"+%\\*GMpF*Fin7$F\\oF+7 &F^o7$F\\o$\"+LLLL6F@7$$\"+9P].qF*Fao7$FdoF+7&Ffo7$FdoF67$$\"+MzrsqF*F 67$FjoF+7&F\\p7$Fjo$\"+KLLL`F/7$$\"+a@$>9(F*F_p7$FbpF+7&FdpFap7$$\"+uj 96sF*F_p7$FgpF+-%&COLORG6&%$RGBG$\"\"(!\"\"F^q$\"#5F`q-%%VIEWG6$;$\"+7 VltjF*$\"+;?G(G(F*%(DEFAULTG" 1 2 0 1 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "IA_a:=evalf( add(IAngle[i],i=1..N)/N,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%IA_a G$\"(8<$o!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sigmaIA:=e valf(sqrt(add((IAngle[n]-IA_a)^2,n=1..N)/(N-1)),5);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%(sigmaIAG$\"&^n\"!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sig_avgIA:='sigmaIA/sqrt(N)';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*sig_avgIAG*&%(sigmaIAG\"\"\"-%%sqrtG6#%\"NG!\"\"" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sig_avgIA:=evalf(sig_avgIA) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*sig_avgIAG$\"+BMrn8!#5" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Let us compare the true value with the confidence interval:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "IA_true:=evalf(fun(val),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%( IA_trueG$\"&#Qo!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "[eva lf(IA_a-sig_avgIA,4),evalf(IA_a+sig_avgIA,4)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"%=o!\"#$\"%YoF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "We notice that the deviation tripled. How can this be explained ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "We \+ made the claim that the uncertainty in the index of refraction is negl igible. Thus we use Snell's law " }{TEXT 19 17 "sin(IA)=n*sin(RA)" } {TEXT -1 52 " to relate the uncertainties in the reflected angle " } {TEXT 19 2 "RA" }{TEXT -1 42 " to the uncertainty in the incident angl e " }{TEXT 19 2 "IA" }{TEXT -1 57 ". Considering the differentials res ults in the statement:" }}{PARA 0 "" 0 "" {TEXT -1 5 "LHS: " }{TEXT 19 24 "d[sin(IA)]=cos(IA)*d[IA]" }}{PARA 0 "" 0 "" {TEXT -1 5 "RHS: " }{TEXT 19 28 "n*d[sin(RA)]=n*cos(RA)*d[RA]" }}{PARA 0 "" 0 "" {TEXT -1 30 "We can solve the equation for " }{TEXT 19 5 "d[IA]" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT 19 33 "d[IA] = n*(cos(RA)/cos(IA))*d[RA ]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "evalf(1.51*cos(val*Pi/ 180)/cos(Pi/180*IA_a)*sigma);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+A Cof;!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "This calculated value for " }{TEXT 19 7 "sigmaIA" }{TEXT -1 111 " is very close indeed to t he one measured above. Thus, it is important to consider error propaga tion carefully!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 282 "We can illustrate the error propagation for this nonline ar function by a graph. This also serves to justify why a linearized t reatment of the function in the neighbourhood of the mean argument val ue is jsutified (the linearization is implied by the use of first-oder differentials)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f_ex:=1 80/Pi*arcsin(1.51*sin(Pi/180*x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%f_exG,$*&-%'arcsinG6#,$-%$sinG6#,$*&%#PiG\"\"\"%\"xGF1#F1\"$!=$\"$^ \"!\"#\"\"\"F0!\"\"F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f_ lin:=convert(taylor(f_ex,x=38,2),polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&f_linG,&$!+!fdSV&!\")\"\"\"%\"xG$\"+;D\\HK!\"*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "sigma;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&M:&!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "P1:=plot([f_ex,f_lin],x=37..39,color=[red,blue]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "P2:=plot([[val-sigma,65],[val-sigm a,subs(x=val-sigma,f_lin)],[0,subs(x=val-sigma,f_lin)]],color=green): \+ P3:=plot([[val,65],[val,subs(x=val,f_lin)],[0,subs(x=val,f_lin)]],colo r=yellow):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "P4:=plot([[val+sigma ,65],[val+sigma,subs(x=val+sigma,f_lin)],[0,subs(x=val+sigma,f_lin)]], color=brown):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "display(P1 ,P2,P3,P4,scaling=constrained,title=\"incident vs refracted angle\"); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6+-%'CURVESG6$ 7S7$$\"#P\"\"!$\"1&y:H&**=Ll!#97$$\"1LL$3VfVq$F-$\"1%ej`$H\"ea'F-7$$\" 1n;H[D:3PF-$\"1WX%)[*Rob'F-7$$\"1LLe0$=Cr$F-$\"1:V.B/HplF-7$$\"1LL3RBr ;PF-$\"1:We$3x=e'F-7$$\"1n;zjf)4s$F-$\"1<>+6#eWf'F-7$$\"1L$e4;[\\s$F-$ \"109%ylrhg'F-7$$\"1+]i'y]!HPF-$\"1+m2N3N=mF-7$$\"1L$ezs$HLPF-$\"1;TZF :+JmF-7$$\"1+]7iI_PPF-$\"12!=@;oOk'F-7$$\"1mm;_M(=u$F-$\"1A**G)=dnl'F- 7$$\"1ML3y_qXPF-$\"1MK6yqLomF-7$$\"1++]1!>+v$F-$\"1$Q>%f9V\"o'F-7$$\"1 ++]Z/NaPF-$\"1%Rz!=Ek%p'F-7$$\"1++]$fC&ePF-$\"1^$\\3yMuq'F-7$$\"1L$ez6 :Bw$F-$\"1V.q:Q5>nF-7$$\"1nm;=C#ow$F-$\"1P()=-c/LnF-7$$\"1nmm#pS1x$F-$ \"1*=>yI8\\u'F-7$$\"1+]i`A3vPF-$\"1\"zm\\V¥F-7$$\"1nmm(y8!zPF-$\"1% R&*o\"f7rnF-7$$\"1+]i.tK$y$F-$\"1^8g+Gt%y'F-7$$\"1+](3zMuy$F-$\"1\"*pT .lv(z'F-7$$\"1nm\"H_?&oF-7$ $\"1+]PQ#\\\"3QF-$\"1Q:&[YrW'oF-7$$\"1LLe\"*[H7QF-$\"1A\\([**R!yoF-7$$ \"1+++dxd;QF-$\"1B(f]\"p8#*oF-7$$\"1++D0xw?QF-$\"1\")o8Rv+1pF-7$$\"1+] i&p@[#QF-$\"1=zHaW]>pF-7$$\"1++vgHKHQF-$\"1AhC!H!eMpF-7$$\"1nmmZvOLQF- $\"1?pwpF-7$$\"1++])Hxe%QF-$\"1Oi*['>(3*pF-7$$\"1n;H!o-*\\QF-$\"1y&=`!*pZ +(F-7$$\"1+]7k.6aQF-$\"16n')Q))Q>qF-7$$\"1nm;WTAeQF-$\"1V;`%*RxLqF-7$$ \"1+]i!*3`iQF-$\"1MM/1T$*[qF-7$$\"1MLL*zym'QF-$\"1K%)[%>MO1(F-7$$\"1LL 3N1#4(QF-$\"18'4:3q(yqF-7$$\"1m;HYt7vQF-$\"1t$o!\\e)Q4(F-7$$\"1+++xG** yQF-$\"1*)f)e?qy5(F-7$$\"1nmT6KU$)QF-$\"1U>z&*=,CrF-7$$\"1LLLbdQ()QF-$ \"1EY#\\)[bQrF-7$$\"1+]i`1h\"*QF-$\"1()fT7M3fLa&=(F--%'COLOURG6&%$RGBG$\"*++++\"!\")F*F*- F$6$7S7$F($\"1,+?>Z1:lF-7$F/$\"1p-XoM9HlF-7$F4$\"1mU*GJ$RTlF-7$F9$\"1x F9W&p^b'F-7$F>$\"1yW>$4P!plF-7$FC$\"1Ni?K(QGe'F-7$FH$\"1,HI=Yj&f'F-7$F M$\"1^$*=ZH ]mF-7$F[o$\"1E=rs&pEm'F-7$F`o$\"11&\\js+mn'F-7$Feo$\"1,Dk#3*e!p'F-7$Fj o$\"1&>5fXpSq'F-7$F_p$\"1'4!GE4J;nF-7$Fdp$\"1t-d=s'3t'F-7$Fip$\"1d?\\2 $)>VnF-7$F^q$\"1_jO&HUvv'F-7$Fcq$\"1:.8b\"R-x'F-7$Fhq$\"1(RNEipTy'F-7$ F]r$\"1Q)o>sMuz'F-7$Fbr$\"1#)H_haF6oF-7$Fgr$\"1l\\/\"e&)R#oF-7$F\\s$\" 1Kp-i^pPoF-7$Fas$\"1_'fOcN>&oF-7$Ffs$\"1lJ9v=LkoF-7$F[t$\"12M:J-sxoF-7 $F`t$\"1nN'Hr^:*oF-7$Fet$\"1s$z%>J30pF-7$Fjt$\"1n`0^a<=pF-7$F_u$\"1)3* 3`ArKpF-7$Fdu$\"1wnE;UxXpF-7$Fiu$\"1s\"z*\\1sfpF-7$F^v$\"1@-x\">eB(pF- 7$Fcv$\"1[+'HOuh)pF-7$Fhv$\"1y-L.V<**pF-7$F]w$\"1A(zn)Hw7qF-7$Fbw$\"1d nj3%[g-(F-7$Fgw$\"1_A49q&*RqF-7$F\\x$\"1pGAaEN`qF-7$Fax$\"1#fJ5j^q1(F- 7$Ffx$\"1xeRtrj!3(F-7$F[y$\"14)))e(37$4(F-7$F`y$\"153t2'Gu5(F-7$Fey$\" 11Sl8cA?rF-7$Fjy$\"11y1\"*)pQ8(F-7$F_z$\"1\"QBb?Hp9(F-7$Fdz$\"1****RAK '4;(F--Fiz6&F[[lF*F*F\\[l-F$6$7%7$$\"1++++gY[PF-$\"#lF*7$F[el$\"1++++` ermF-7$F*F`el-Fiz6&F[[lF*F\\[lF*-F$6$7%7$$\"#QF*F]el7$Fiel$\"1+++qR,Qo F-7$F*F\\fl-Fiz6&F[[lF\\[lF\\[lF*-F$6$7%7$$\"1++++S`^QF-F]el7$Fefl$\"1 ,++SEW/qF-7$F*Fhfl-Fiz6&F[[l$\")#)eqkF^[l$\"))eqk\"F^[lF_gl-%+AXESLABE LSG6$Q\"x6\"%!G-%(SCALINGG6#%,CONSTRAINEDG-%&TITLEG6#Q " 0 "" {MPLTEXT 1 0 13 "sigmaIA/IA_a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ks%>X#!#6" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "sigma/val;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+&*y:c8!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "The relative dev iation doubled when going from refracted to incident angle." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 276 11 "Exercise 3:" }} {PARA 0 "" 0 "" {TEXT -1 150 "Change the value of the exactly known re flected angle and repeat the simulation. Compare the prediction of the simulation on the relationship between " }{TEXT 19 5 "d[IA]" }{TEXT -1 5 " and " }{TEXT 19 5 "d[RA]" }{TEXT -1 148 ", and compare with the analytical result based on error propagation analysis. What happens i n the above example when the glass is replaced by water?" }}{PARA 0 " " 0 "" {TEXT -1 41 "The index of refraction for water equals " }{TEXT 277 1 "n" }{TEXT -1 6 "=1.33." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 275 11 "Exercise 4:" }}{PARA 0 "" 0 "" {TEXT -1 289 "Use some other physics law that involves a nonlinearity to relate the uncertainties in two variables (a measured variable with known un certainty is connected to a deduced variable). Perform a simulation, a nd confirm your results by an analytical derivation (as done above for Snell's law)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 287 17 "Systematic errors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 817 "So far we have assumed that the only s ource of errors were random, and that no systematic deviations were pr esent, i.e., that the experimental apparatus was optimized at the leve l of desired precision. One possibility to check whether the measuring apparatus is by comparison (e.g., identically built ammeters are comp ared in their readings of the same current, known objects are measured with different calipers or micrometer screws, etc.). In this way one \+ can establish the presence of systematic errors in the equipment (e.g. , one ammeter is measuring consistently high values, etc.). Often we d o not have the luxury of comparing equipment in this way and rely on t olerance specifications of the manufacturer who was able and obliged t o establish the systematic tolerances for the equipment as it left the factory." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 890 "An important question that arises in this context is how to deal \+ with this systematic error. Suppose we have established a standard dev iation interval for a measurement sequence. How should we add the syst ematic measurement error to the statistical uncertainty? John R. Taylo r points out (in chapter 4) that two situations can arise: either the \+ error is known for the equipment never to exceed a certain tolerance ( e.g., 1% of the reading; or 2% of the maximum reading on the scale). I n this case one would simply add the systematic error to the statistic al one. On the other hand, it might be true that based on a sequence o f comparisons the systematic error is known to fall within a normal di stribution, i.e., 70% of the instruments are better than 1%, but some \+ of them are outside this interval. In this latter case the systematic \+ error has to be added in quadrature to the uncertainty." }}{PARA 0 "" 0 "" {TEXT -1 50 "D[x]_tot = sqrt(D[x]_systematic^2 + D[x]_random^2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 488 "A just ification for adding independent errors (and uncertainties) in quadrat ure is given below. Usually it is the second method of adding systemat ic and random errors that we wish to use in a teaching laboratory. Eve n if equipment was certified to be within a certain tolerance range, w e cannot be assured that after years of use (and possibly abuse) it wa s still performing to norm. We have more confidence in the statement t hat with some likelihood it still performed to specifications." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 278 46 " Uncertainty in a function of several variables" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 188 "We have demonstrated in \+ the previous section how an error propagates for a non-linear function of one variable. Effectively, the function was linearized in the rang e around the mean value " }{TEXT 19 3 "x_a" }{TEXT -1 15 " and the ran ge " }{TEXT 19 22 "x_a-sigma .. x_a+sigma" }{TEXT -1 51 " was mapped i nto a correpsonding range surrounding " }{TEXT 19 6 "f(x_a)" }{TEXT -1 28 " using the first derivative " }{TEXT 280 1 "f" }{TEXT -1 2 "'( " }{TEXT 279 1 "x" }{TEXT -1 65 "). Depending on the magnitude of the \+ derivative the deviation in " }{TEXT 281 1 "f" }{TEXT -1 8 " (i.e., " }{TEXT 19 7 "sigma_f" }{TEXT -1 42 ") can increase or decrease as comp ared to " }{TEXT 19 5 "sigma" }{TEXT -1 79 ". Now we demonstrate for a function of two variables that the uncertainties in " }{TEXT 286 1 "x " }{TEXT -1 5 " and " }{TEXT 285 1 "y" }{TEXT -1 61 " have to be added in quadrature to obtain the uncertainty in " }{TEXT 284 1 "f" }{TEXT -1 1 "(" }{TEXT 283 1 "x" }{TEXT -1 2 ", " }{TEXT 282 1 "y" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 " First we generate a second sequence of normally distributed random num bers." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "RNd2:=[stats[rando m, normald](150)]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 287 "We pick Snell's law again, and consider the foll owing problem: suppose we measure corresponding incident and refracted angles in order to determine the index of refraction of the optically dense medium (we assume that for the purposes of the experiment air h as an index of refraction of " }{TEXT 290 1 "n" }{TEXT -1 28 "=1, i.e. , the vacuum value)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 216 "We measure the incident angle with the same accuracy a s the refracted angle. The data set for the refracted angle is kept fr om the previous section, and we produce a data set for the incident an gles (consistent with " }{TEXT 291 1 "n" }{TEXT -1 7 "=1.51)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "MDataIA:=[seq(IA_true+0.5*RN d2[i],i=1..N)]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 220 "For the purpo ses of our simulation we simply associate with each datapoint from the set of refracted angles a corresponding datapoint from the set of inc ident angles to obtain a set of values for the index of refraction:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "MDatan:=[seq(evalf(sin(Pi/ 180*MDataIA[i])/sin(Pi/180*MData[i])),i=1..N)]:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "histogram(MDatan,area=1);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%)POLYGONSG6-7&7$$\"+Al%zY\"!\"* \"\"!7$F($\"+'pmmm#!#67$$\"+OK3w9F*F-7$F1F+7&F37$F1$\"+rLLLLF/7$$\"+]* >U[\"F*F67$F9F+7&F;7$F9$\"+knmm')F/7$$\"+kmN#\\\"F*F>7$FAF+7&FC7$FA$\" +9+++7!#57$$\"+yL\\+:F*FF7$FJF+7&FL7$FJ$\"+^LLL:FH7$$\"+#4I'3:F*FO7$FR F+7&FT7$FR$\"+SJLLFH7$$\"+@N !\\_\"F*Fin7$F\\oF+7&F^o7$F\\o$\"+;MLLtF/7$$\"+N-/L:F*Fao7$FdoF+7&Ffo7 $Fdo$\"+YLLL6FH7$$\"+\\p " 0 "" {MPLTEXT 1 0 38 "n_a:=evalf(add(MDatan[i],i=1..N)/N,7);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$n_aG$\"(g;^\"!\"'" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 60 "sigma_n:=evalf(sqrt(add((MDatan[n]-n_a)^2,n=1..N)/( N-1)),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(sigma_nG$\"&Dw\"!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sig_avg_n:='sigma_n/sqrt( N)';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*sig_avg_nG*&%(sigma_nG\"\" \"-%%sqrtG6#%\"NG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "s ig_avg_n:=evalf(sig_avg_n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*sig_ avg_nG$\"+C_2R9!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "[n_a- sig_avg_n,n_a+sig_avg_n];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+D4A 5:!\"*$\"+v!*48:F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "The true va lue of " }{TEXT 292 1 "n" }{TEXT -1 55 "=1.51 falls barely outside the 70% confidence interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 35 "How can we predict the uncertainty " }{TEXT 19 7 " sigma_n" }{TEXT -1 67 " from the known uncertainties in the refracted \+ and incident angles?" }}{PARA 0 "" 0 "" {TEXT -1 233 "First we should \+ verify that the uncertainty in the incident angle is of the same magni tude as for the refracted angle (note that it independently given in t his section, and not determined via Snell's law as in the previous sec tion!):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "IA_a:=evalf(add( MDataIA[i],i=1..N)/N,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%IA_aG$ \"(S,%o!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sigma_IA:=ev alf(sqrt(add((MDataIA[n]-IA_a)^2,n=1..N)/(N-1)),5);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%)sigma_IAG$\"&h0&!\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "We need to apply the addition in quadrature rule given fo r a function " }{TEXT 19 18 "q(x1, x2, ..., xn)" }{TEXT -1 4 " as:" }} {PARA 0 "" 0 "" {TEXT 19 89 "D[q] = sqrt((diff(q, x1)*D[x1])^2 + (diff (q, x2)*D[x2])^2 + ... + (diff(q, xn)*D[xn] )^2)" }}{PARA 0 "" 0 "" {TEXT -1 60 "where the derivatives are evaluated at the average value \+ of " }{TEXT 19 17 "[x1, x2, ..., xn]" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 25 "Our function is given as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "RF_ind:=sin(x1)/sin(x2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'RF_indG*&-%$sinG6#%#x1G\"\"\"-F'6#%#x2G!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Dn:=sqrt((diff(RF_ind,x1)*Dx 1)^2+(diff(RF_ind,x2)*Dx2)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#D nG*$-%%sqrtG6#,&*&*&)-%$cosG6#%#x1G\"\"#\"\"\")%$Dx1GF1F2F2*$)-%$sinG6 #%#x2G\"\"#F2!\"\"\"\"\"*&*()-F8F/F1F2)-F.F9F1F2)%$Dx2GF1F2F2*$)F7\"\" %F2F \+ " 0 "" {MPLTEXT 1 0 82 "evalf(subs(x1=Pi/180*IA_a,x2=Pi/180*v_a,Dx1=Pi /180*sigma_IA,Dx2=Pi/180*sigma,Dn));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+2nf?=!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 204 "It is importan t in the above expression to provide the conversion factors for the an gles from degrees to radians not only in the arguments to the sine fun ctions, but also for the uncertainties themselves!" }}{PARA 0 "" 0 "" {TEXT -1 142 "The agreement of the predicted uncertainty with the one \+ observed in the simulation (which represents a statistical measurement !) is very good." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 294 11 "Exercise 5:" }}{PARA 0 "" 0 "" {TEXT -1 135 "Repe at the above simulation with a larger deviation in the incident and re fracted angles. Pick different incident and refracted angles." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 295 11 "Exercise 6:" }}{PARA 0 "" 0 "" {TEXT -1 264 "Repeat the simulation with a very much reduced statistical sample as corresponds to teaching laboratory measurements (or as happens sometimes in life science research when c ircumstances do not permit large sample sizes), e.g., N=5. Are the con clusions still valid?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{MARK "106 0 0" 35 }{VIEWOPTS 1 1 0 1 1 1803 }