ENGI 441 Propagato of Error Page 8-01 Propagato of Error [Navd Chapter 3; ot Devore] Ay realstc measuremet procedure cotas error. Ay calculatos based o that measuremet wll therefore also cota a error. The errors are propagated from measuremets to the calculated value. From Chapter 1, for ay costats a, b ay rom quatty X we kow that [ ] E[ ] E ax + b = a X + b V[ ax + b] = a V[ X] = a X From Chapter 7, for ay costats { c1, c,, c } { X1, X,, X }, we kow that ax+ b ay set of depedet rom quattes [ ] E cx 1 1 + cx + + cx = c1e X 1 + ce X + + ce X V cx 1 1 + cx + + cx = c1 V X 1 + c V[ X] + + c V X A rom sample of observatos draw from a populato of mea µ varace a sample mea X for whch has E X = µ X V X = = X Example 8.01 The dameter of a crcle s measured to be 6.0 cm, wth the ucertaty represeted by the stard devato of 0.1 cm. Fd the ucertaty the crcumferece of the crcle. Let D = measured dameter of the crcle C = resultg calculated crcumferece of the crcle the
ENGI 441 Propagato of Error Page 8-0 Example 8.0 A wegh scale s used to estmate masses of objects up to 100 kg wth a ucertaty of 0.5 kg. The error each measuremet s depedet of the errors all other measuremets. Four compoets of a mache are measured masses of (8.7 ± 0.5) kg, (10. ± 0.5) kg, (1.7 ± 0.5) kg (15.1 ± 0.5) kg are reported. What s the total mass of the four compoets? Let M = total mass M = mass of the th compoet, the Note o quoted ucertaty: The ucertaty b the expresso (a ± b) s, by default, the approprate stard devato. However, b s sometmes take to be the maxmum possble error. O other occasos t s some multple of the stard devato (ofte or 3). I chapters 11 1 we wll express cofdece tervals the form (a ± b). Be sure whch of these terpretatos s teded whe you ecouter (a ± b).
ENGI 441 Propagato of Error Page 8-03 Example 8.03 The mass of a compoet s measured sx tmes o a wegh scale whose ucertaty s ukow. The sx measuremets ( kg) are 5.09, 5.16, 5.08, 5.10, 5.14 5.1. Estmate the mass of the compoet fd the ucertaty ths estmate. Let M = sample mea of the measured masses S = sample stard devato. The true ucertaty s ukow, but t may be estmated by the observed s. m = 30.69 6 = 5.115 Repeated Measuremets wth Dfferet certates Example 8.04 Two depedet estmates of the durato of a chemcal process, usg dfferet clocks, are (4.0 ± 0.) s (4.1 ± 0.1) s. Fd the ucertaty the average tme of 4.05 s. Let T = average of the tmes o the two clocks. 4.0 + 4.1 t = = 4.05
ENGI 441 Propagato of Error Page 8-04 Example 8.04 (cotued) However, oe clock s more precse tha the other. It makes sese that greater weght should be placed o the value provded by the more precse clock. Let us use a weghted average T W place of the smple average T. W 1 V V 1 V T ct c T T c T c T 1 W 1 where c s a weght the terval [0, 1]. We eed the value of c that mmses ( therefore geerates the best possble precso). I geeral, whe two depedet estmates for the same value are reported as x x, the the most precse average value wll be x, where 1 1 x c x c x c c c c 1 1 1, 1 1, 1 1 1
ENGI 441 Propagato of Error Page 8-05 Lear Combato of Depedet Measuremets The geeral expresso for the varace of a lear combato of rom quattes Y a X s 1 VY V a X a a Cov X, X Ths ca be re-wrtte as j j 1 1 j 1 V Y a V X a a Cov X, X where j j 1 1 j 1 a aa jj j 1 1 j 1 j s the correlato coeffcet betwee X X j V X The maxmum possble value of V[Y] occurs whe sg a a, j We the obta V. j j. Y a a a a Y 1 1 1 I all other cases, a coservatve estmate of the ucertaty Y a a a Y 1 1 a X s 1 Example 8.05 Suppose that we are ot sure whether or ot the two estmates, usg dfferet clocks, of the durato of a chemcal process Example 8.04 are depedet. The estmates are (4.0 ± 0.) s (4.1 ± 0.1) s. Fd a coservatve overall estmate. T T 1 4.0 4.1 T T1 T t 1 As before, 4.05
ENGI 441 Propagato of Error Page 8-06 No-Lear Fuctos of Oe Measuremet Suppose that (X) s a o-lear fucto of a rom quatty X whose true mea s μ. If X s a ubased estmator of μ, the, geeral, (X) wll be a based estmator of ( μ ). If X s close to μ, the the Taylor seres approxmato for (X), d 1 d ( X) = ( μ) + ( X μ) + ( X μ ) + dx! dx X = μ X = μ may be trucated to just the frst order, d d d ( X) ( μ) + ( X μ) = ( μ) μ + X dx X = μ dx X = μ dx X = μ Ths s a lear fucto of X, whose stard devato follows from the geeral formula ax+ b = a X : d X dx X = μ Ths s the formula for propagato of error. I practce we do ot kow the true value of μ, so we evaluate the dervatve at the observed value of X stead. Example 8.06 I Example 8.01, the dameter D of the crcle was measured to be (6.0 ± 0.1) cm. Estmate the area A of the crcle.
ENGI 441 Propagato of Error Page 8-07 Relatve certaty [for bous questos oly] If s a ubased measuremet whose true value s µ whose absolute ucertaty s, the the relatve ucertaty s the dmesoless quatty µ (also kow as the coeffcet of varato). I practce, the relatve ucertaty s calculated as d Note that l ( l ) =. d Therefore the absolute ucertaty (l ) s also the relatve ucertaty. Example 8.06 (aga) [for bous questos oly] Fd the relatve ucertaty the area of the crcle of dameter (6.0 ± 0.1) cm hece fd the absolute ucertaty the area of the crcle D A= π = π D l A= l π D = l π + l D = l π + l D ( ) ( ) 4 4 4 4 d 1 ( l A) 0 dd = + D = 6 = 3 A 1 1 1 D = 0.1 = = 3.3% A 3 3 30 1 1 6 9π 3π A A = π = = 0.94 cm 30 30 30 10
ENGI 441 Propagato of Error Page 8-08 Example 8.07 (Navd textbook, exercses 3.3, page 185, questo 8) [examable] 1 The refractve dex η of a pece of glass s related to the crtcal agle θ by η =. sθ The crtcal agle s reported to be (0.70 ± 0.0) rad. Estmate the refractve dex fd the ucertaty the estmate. No-Lear Fuctos of Several Measuremets [for bous questos oly] If { X1, X,, X } are depedet measuremets whose ucertates { 1,,, } small f = ( X1, X,, X ) s a o-lear fucto of { X1, X,, X }, the are + + + X X X 1 1 where the partal dervatves are evaluated at the observed value of ( X1, X,, X ) Ths s the multvarate propagato of error formula..
ENGI 441 Propagato of Error Page 8-09 If the { X1, X,, X } are ot depedet, the a coservatve estmate s + + + X X X 1 1 Example 8.08 (Navd textbook, exercses 3.4, page 193, questo 8(b)) [for bous questos oly] The pressure P ( kpa), temperature T ( K) volume V ( ltres) of oe mole of a deal gas are related by the equato PV = 8.31T Gve that P = (4.5 ± 0.03) kpa T = (90.11 ± 0.0) K, estmate V. V 8.31T 8.31 90.11 = = = 9.94068 P 4.5 1 V 1 8.31 V = 8.31TP = 8.31P = = 0.034 65 T 4.5 V 8.31 90.11 = 8.31TP = = 0.0409891 P ( 4.5) Assumg that the measuremets of pressure temperature are depedet, V V V T P ( ) ( ) 0.034 0.0 0.040 0.03 0.001408 T + P = + Therefore V = ( 9.9407 ± 0.0014 ) ltre If we caot assume depedece, the a coservatve estmate s gve by V V V T P 0.034 0.0 0.040 0.03 0.001915 T + P = + Therefore V = ( 9.9407 ± 0.0019 ) ltre [The aswer s gve to the ffth sgfcat fgure V, because P T were both quoted to that level of precso. However, the costat s quoted to oly 3 s.f., so the aswer may be urelable beyod the thrd sgfcat fgure.] [Ed of Chapter 8]
ENGI 441 Propagato of Error Page 8-10 [Space for Addtoal Notes]