1B40 Practical Skills


 Bernadette Gray
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1 B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need to know wht is the error on the finl nswer in terms of the uncertinties on the individul quntities. We will discuss some specific cses; summry ppers t the end. Liner cses A simple cse is where the result Z is liner sum of two other quntities A nd B e.g.. If A hs the vlue ± δnd B the vlue ± δthen Z hs the vlue ± δ. How is δ relted to δ ndδ? We might expect tht δ δ δ. However this could led to the nonsense result thtδ 0! We could decide to consider the mximum possile error nd simply dd the mgnitudes of δ ndδ. This would e n overestimtion of the error. We know tht more sensile pproch is to consider stndrd devitions. Thus if we squre ( δ) ( δ) ( δ) δδ. Over lrge numer of mesurements we would expect tht if δ nd δ re oth eqully positive nd negtive nd not correlted to ech other then the verge vlue δδ is ero. Hence ( δ ) ( δ) ( δ) nd we dd the errors on nd in qudrture. Note the result would hve een the sme if. A forml derivtion of the result is s follows [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) cov.
2 The lst term involves the covrince of nd. This is mesure of whether their errors re correlted or not. It cn e positive or negtive or in the cse where they re uncorrelted ero. Its vlue is relted to the extent tht vlue of δ ffects tht of δ. Products nd quotients If then δ ( δ)( δ) δ δ δδ δ δ δ δδ. To firstorder in the errors δ δ δ. Since we don t know the signs of the uncertinties we squre nd verge over lrge numer of mesurements ( δ) ( δ ) ( δ) ( δ)( δ) ( δ ) ( δ) ( δ) ( δ)( δ). If the uncertinties in nd δ nd δ re uncorrelted ( δ)( δ ) 0 nd δ δ δ Thus in this cse we dd in qudrture the frctionl errors. If then ( δ / ) ( δ / ) δ δ δ δ δ δ δ δ δ δ δ. where in the second line we hve use the inomil pproximtion for δ δ when δ/. Just s in the liner cse we don t know the sign of the uncertinties so squring s efore.
3 δ δ δ δ δ δ δ δ if the errors re uncorrelted. Thus lso in this cse we dd in qudrture the frctionl errors.. Functions of single vrile Suppose we hve mesured quntity A nd determined its vlue s ± δ. The reson A ws mesured ws to provide n indirect mesurement of quntity Z which is known function of A i.e. f(). Wht is the est estimte for ± δ? The grph illustrtes such cse for. 0 8 (dz/da) DA/DZ DA ZA DZ A A The uncertinty δz in Z is relted to tht in A y 3
4 d δ d This is equivlent to ssuming tht f() is liner over the smll region we re considering nd we evlute the differentil t the men of the mesurements. This is generl result. Three exmples re d δ n n ) n giving n. ) d d δ ln δ d d exp exp d giving. δ 0 δ δ. 3) ( ) ( ) giving δ. Functions of Severl Vriles If f() nd the uncertinties in A nd B re given y δ nd δ respectively we mke the sme pproximtion s efore i.e. is liner function of nd in the relevnt region so tht we cn write The quntity δ. δ δ A 0 B 0 is clled prtil derivtive. You will lern more out these in your mthemtics courses (if you hven t lredy met them). To otin n expression for we consider s function of the vrile ll other quntities re considered s constnt nd differentite the expression for in the norml wy. As efore the signs of the uncertinties re not known so we squre the expression nd verge over mny mesurements ( δ) ( δ) ( δ) ( δ)( δ). ( δ ) ( δ) ( δ) ( δ)( δ) The lst term is ero if A nd B re independent vriles s there is equl proility of positive nd negtive comintions of δ nd δ whose verge is therefore ero. Thus ( δ ) ( δ) ( δ) This cn e extended to more vriles y dding further similr terms vi. 4
5 All the erlier exmples e.g. generl result given ove. c. c or / re just specil cses of the Correlted uncertinties Most of the mesurements of quntities you will meet in the lortory course re uncorrelted or t lest wekly correlted. So the sutle effects of correltions will usully e ignored. However it is worth considering n extreme cse to illustrte their effects. Consider the simple cse of. The error on is given y If then this result leds to.. Yet if we hd strted from we would immeditely get δ δnd hence in contrdiction to the previous result. The ltter is correct the former is wrong s it hs not llowed for the correltion etween nd ( ) which is of course 00%. We hve ( δ)( δ) 0 in fct cov( ). Exmple Consider tht quntity is dependent on quntities nd c in the following wy α β exp ( c) where α nd β re constnts. (This is contrived expression simply for the purposes of this exercise!) We wnt to find the error on in terms of those on nd c. We cn proceed either y pplying the formule for the vrious specil cses considered ove or the generl formul. Cse pply specil cse formule. Let / y β exp( c) nd p y then By differentition / α y α p. 5
6 nd so δ / p y δ y ( δ) ( αδ) ( δ p) 4. 4y ( δ) α ( δ) ( δy) But ( δ y) δ ( ) β δ ( c ) [ ] [ exp( )] ( δ y) ( δ) β c ( δc) Thus comining the results we get ( ) ( ) 4 exp( ). δ 4α δ 4 ( δ) β exp( c) ( δc). 4 exp( ) ( β c ) Cse ppliction of generl formul. By prtil differentition we hve The generl formul then gives α c exp( ) c c exp( ) ( β c ) c ( c) β exp. c 4α 4 β exp( c) c. 4 exp( ) 6
7 Summry of error propgtion formule If quntities nd hve errors nd respectively nd re uncorrelted nd k nd n re constnts then if quntity n error on quntity n ; ; ; ; / ; n n n ; n n ln( ) exp( ) e e k e k k k sin( k) kcos( k) Note nd re in rdins! ( ) f c f f f c c In complex expressions either pply the generl result from the lst row or rek the expression down into sums products nd quotients nd pply the seprte formule nd comine the results. 7
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