ErrorPropagation.nb 1. Error Propagation


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1 ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then x s x where x = ÅÅÅÅÅÅ x, s x =  Hx  xl are the sample mean and varance. ext, suppose that we compute a derved quantty Our best estmate for ths quantty s f = D, but what s the uncertanty n ths quantty? Ths queston falls under the headng of error propagaton. Let us assume that s x s small enough to use a lnear approxmaton to near x, such that f  f º x Hx  x L where the dervatve s evaluated at x. Thus, f we expect the true value of x to le n the range x dx, the true value for should le n the correspondng range f df where d f = ƒ ƒ ƒ x ƒ dx Under the present crcumstances we would nterpret the uncertanty dx as the standard devaton s x, but often we must estmate ths quantty by other means. As llustrated n the fgure below, the steeper s near x the larger s d f for gven dx; ths follows smply from the defnton of dervatve as rate of change. ote that we use the absolute value because uncertantes are expressed as postve numbers gvng the wdth of an nterval. f df x dx x ow suppose that we measure two varables x and y and wsh to compute f = yd. How accurately do we know f f the uncertantes n the measured quanttes are dx and d y? A crude estmate mght smply add two contrbutons analogous to the onedmensonal result above. However, ths s lkely to be an overestmate because sometmes the fluctuatons n the two varables wll add and sometmes they wll subtract. A more realstc estmate requres statstcal analyss. The smplest stuaton occurs when x and y are normally dstrbuted random varables. Our best estmates of the true values for these parameters are then ther mean values, x and y, wth uncertantes gven by ther standard devatons,
2 ErrorPropagaton.nb s x and s y. Furthermore, we agan assume that the uncertantes are small enough to approxmate varatons n yd as lnear wth respect to varaton of these varables, such that f  f º x Hx  x L + y Hy  y L where the partal dervatves are evaluated at Hx, y L. If we perform many measurements, the varance of f becomes s f =  H f  f L =  j k x Hx  xl + y Hy  yl y { Expandng ths expresson s f =  and dentfyng ts terms, we fnd where k j k j y x { s f = k j y s x + x { k j y y { s x = Hx  xl + k j y y { s y + x  Hx  xl, s y = are the sample varances for each varable and s x,y = s ther covarance.  Hx  xl Hy  yl y s x,y  Hy  yl Hy  y L + x y Hx  xl Hy  yl y { Ths analyss can readly be generaled to an arbtrary number of varables. Let x = 8x m, m =, m< represent the set of varables, such that where m s f = s m,n = m,n= ÅÅÅ x m s m,n ÅÅÅ x n  Hx m,  x ml Hx n,  xnl s the covarance matrx and x m, s the th measurement of x m. ote that the dagonal elements of the covarance matrx, s m,m =s m, are smply varances for each varable.
3 ErrorPropagaton.nb 3 The covarance measures the tendency for fluctuatons of one varable to be related to fluctuatons of another. A closely related quantty s the correlaton C x,y = s x,y Å s x s y ï  C x,y whch s normaled to the range  C x,y. If a postve devaton n x (such that x  x > 0) s more lkely to be accompaned by a postve devaton n y, then C x,y wll be postve, whereas C x,y would be negatve f a postve devaton n one varable s lkely to be accompaned by a negatve devaton n the other. If the devatons n one varable are equally lkely to be accompaned by devatons of ether sgn n the other varable, the sum of products of fluctuatons wll tend to average to ero and C x,y wll be small. Thus, when C x,y s neglgbly small, the varables x and y are descrbed as statstcally ndependent or as uncorrelated. It s often mpractcal to repeat measurements many tmes. We must then estmate the uncertantes n varous quanttes by other means. For example, f we are usng a ruler, the uncertanty n length wll be about half the smallest dvson. In the absence of contrary nformaton, we usually assume that random fluctuatons n dfferent quanttes are ndependent and omt the covarance. The error propagaton formula then reduces to Hd f L = k j x dxy { + j k y d yy { + where we use the notaton dx to represent an uncertanty nstead of s x because we use an estmated uncertanty nstead of an observed varance. Ths formula can be extended to an arbtrary number of statstcally ndependent varables whose contrbutons to the net uncertanty are sad to add n quadrature because fluctuatons sometmes add and sometmes subtract. Each term s a partal uncertanty determned by the uncertanty n one varable and the rate of change wth respect to that varable. otce that f the partal uncertantes vary sgnfcantly n se, only the largest contrbutons matter because squarng before addng strongly emphases the larger terms. For example, suppose that d f conssts of sx contrbutons where one term s fve unts and the other fve terms are each one unt, such that Hd f L = H5L + HL + HL + HL + HL + HL ï df = 5.5 The total contrbuton of the fve smaller terms s only one tenth the contrbuton of the sngle largest term. Thus, the net uncertanty d f ƒ ƒ x dxƒ ƒ + ƒ ƒ y d yƒ ƒ + s less than the lnear sum of partal uncertantes. When desgnng an experment, dentfy the domnant partal uncertantes and attempt to mnme them; the smaller terms do not requre as much attenton f ther contrbuton to the quadrature sum s neglgble. A couple of smple examples are lsted below. Here a, b, m, n, l are consdered exact numbers whle x, y are expermental quanttes measured wth fnte precson. f = ax + by ï Hd f L = Ha dxl + Hb d yl f = ax m y n ï k j d f y f { = Jm ÅÅÅÅÅÅÅ dx x + k jn d y y y {
4 ErrorPropagaton.nb 4 f = a xd ï df = l f dx ü example: measurng g wth a pendulum The perod, T, of a smple pendulum s related to ts length, L, by L T = p $%%%%%% ÅÅÅÅÅ g Therefore, f we measure L and T, we can deduce the gravtatonal acceleraton, g, usng g = 4 p L ï dg T g = $%%%%%%%%%%%%%%%% J dl ÅÅÅÅÅÅ Å L + %%%%%%%%%%%%%%%% J ÅÅÅÅÅ dt T %%%% otce that the relatve uncertanty n the perod carres a greater weght than the precson of the length because t enters the formula for g wth a larger power. ü example: mean of measurements Suppose that we make observatons of the random varable x and compute ts mean value x = ÅÅÅÅÅÅ x What s the uncertanty n the mean? If we assume that the uncertanty, dx, s the same for each observaton, then Hd x L = j x y ÅÅ dx k x { = J dx = HdxL ÅÅÅÅÅÅ ï d x = dx ÅÅÅÅÅ è!!!! Therefore, the uncertanty n the mean s smaller than the uncertanty n a sngle observaton by a factor of ë è!!!!. ü example: weghted mean ext consder the weghted mean x = x ÅÅ s ì ÅÅ s where the uncertantes n each observaton may be dfferent. otce that the terms wth the smallest uncertanty carry the most weght. The uncertanty n the weghted mean can be evaluated usng standard error propagaton to be HsL x = Hs  s L ÅÅ ï s H s  L x = $%%%%%%%%%%%%%%%%% ÅÅÅÅ s 
5 ErrorPropagaton.nb 5 otce that f uncertantes are unform, such that s =dx s the same for each observaton, then ths result s the same as the precedng result for the unweghted average.
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