Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re performed on mesured untities. Suppose we hve simple eperiment where we wnt to mesure velocit, V, mesuring distnce, d, nd time, t. We tke mesurements nd come up with mesured untities d ± d nd t ± t. We cn esil estimte V dividing d t, ut we lso need to know how to find V. Below we investigte how error propgtes when mthemticl opertions re performed on two untities nd tht comprise the desired untit. Addition nd Sutrction If we re tring to find the uncertint,, ssocited with, we cn look t wht the highest nd lowest prole vlues would e. The highest vlue would e otined dding the estimtes for nd to the totl uncertint for oth vlues. Similrl, the lowest prole vlue would e otined dding the estimtes for nd nd sutrcting oth ssocited uncertinties: (highest prole vlue of ) ( ) () (lowest prole vlue of ) _ ( ) (3) Since, it is es to see is eul to nd cn e epressed s: ± ( ) (4) Similrl, for sutrction, we cn write tht: _ ± ( ) (5) A similr nlsis s shown ove cn e pplied to n numer of untities tht re dded or sutrcted, nd we cn stte generl rule: Rule : Uncertint in Sums nd Differences For z (u. w), z (u w) Or more simpl put, when dding or sutrcting untities, their uncertinties dd. Uncertint of Product We cn perform similr nlsis s ove for multipliction, ut first we must define frctionl uncertint of untit: M. Plmer
(frctionl uncertint in ). (6) The frctionl uncertint (or, s it is lso known, percentge uncertint) is normlized, dimensionless w of presenting uncertint, which is necessr when multipling or dividing. A mesurement nd its frctionl uncertint cn e epressed s: (vlue of ). (7) For simplicit, herefter the suscript will e omitted in the denomintor of the frctionl uncertint, ut it is ssumed. For, we hve mesured vlues for nd of the form: (mesured ) (mesured ) (8) (9) Once gin we look t the lrgest nd smllest prole vlues of : (lrgest prole vlue of ) (0) (smllest prole vlue of ) () We epnd (0) to get: (lrgest prole vlue of ) () We now must mke the ssumption tht the frctionl uncertinties re smll (tht is, the re less thn ). This is resonle ssumption since most well-designed eperiments will hve frctionl uncertinties tht re 0. or less. If we cn s tht the frctionl uncertinties re less thn one, then their product will e much less thn one nd their contriution to the uncertint negligile. From this rgument, we neglect the lst term in () nd simplif the eution to: (lrgest prole vlue of ) (3) M. Plmer
Similrl, we simplif the eution for the lowest prole vlue to: (smllest prole vlue of ) (4) This gives prole rnge of vlues for of: ± (5) The eution for is: (vlue of ) (6) nd since, we conclude from (5) nd (6) tht: (7) Therefore, to find the uncertint of two multiplied untities, we dd the frctionl uncertinties. Uncertint in Quotient To estimte the uncertint ssocited with the uotient /, we once gin look t the lrgest vlue of we could epect: (lrgest vlue of ) (8) We use little it of mthemticl mnipultion to simplif (8). The term on the right hs the form:, where nd re less thn (9) The inomil theorem llows us to epress the denomintor term s n infinite series:... (0) M. Plmer 3
M. Plmer 4 Since is ssumed less thn, nd ll of the higher order terms will ll e <<. These cn e neglected nd we cn s tht:. () Then, (9) ecomes ( )( ) Once gin we eliminte ecuse it is the product of two smll numers. We sustitute the frctionl uncertinties for nd nd simplif (8) to: (lrgest vlue of ) () A similr procedure pplied to the eution for the smllest vlue shows tht: (smllest vlue of ) (3) Knowing tht /, we find once gin tht: Now we etend the seprte results for multipliction nd division to ppl to n numer of untities tht re multiplied or divided: Quntities w re mesured with smll uncertinties. ω nd w u z....... Then the frctionl uncertint in the computed vlue of is: w w u u z z...... In summr, when n numer of untities re multiplied or divided, their frctionl uncertinties dd. Rule : Uncertint in Products nd Quotients
Summing errors in udrture The two rules we hve defined so fr stte tht for ddition or sutrction, we sum the uncertinties, nd for multipliction nd division, we sum the frctionl uncertinties. However, if the originl uncertinties ssocited with the mesured untities re independent nd rndom, these rules will produce vlues of uncertint tht re unnecessril lrge. Thus we consider nother method for computing propgted error. We previousl stted tht the highest vlue we would epect for the untit is ( ). For this to e the ctul vlue of, though, we would hve hd to underestimte oth nd their full mounts nd. If nd re independent errors nd rndom in nture, we hve 50% chnce tht n underestimte of is ccompnied n overestimte of. The proilit we would underestimte oth the full mount is ver smll nd n uncertint of would e too lrge. Without detiled justifiction, we present n lterntive w of clculting uncertint, ssuming errors in mesurement re governed the norml (or Gussin) distriution nd tht mesured untities re independent of ech other. This method, clled dding in udrture, provides the following rules for computing uncertint in. Rule 3: Uncertint in Sums nd Differences for Rndom nd Independent Uncertinties If,,w re mesured with independent nd rndom uncertinties,,w, nd re used to compute z (u w), then the uncertint in is the udrtic sum: ( )... ( z) ( u)... ( w) Rule 4: Uncertint in Products nd Quotients for Rndom nd Independent Uncertinties If,,w re mesured with independent nd rndom uncertinties... z,,w, nd re used to compute then the uncertint in u... w is the udrtic sum: z u w...... z u w M. Plmer 5
Notice tht the procedure of dding in udrture will lws produce vlues smller thn using ordinr ddition to sum uncertinties. Indeed, rules nd re upper ounds for the uncertint of mesured untit. The uncertint of will e no lrger thn the vlues produced rules nd. Rules 3 nd 4 re w uncertint cn e reduced under certin conditions. Another importnt conseuence of using rules 3 nd 4 is tht smll uncertinties re mde smller when the re sured, mking their contriution to the overll uncertint negligile. S, for emple, we re tring to mesure the densit of gs using the idel gs lw: ρ P/RT, where ρ is the densit of the gs, P is the pressure, R is the gs constnt, nd T is the temperture. R is constnt known with much precision, so we do not even consider its contriution to uncertint. For temperture, we re using high-precision thermometer nd we estimte tht T/ T is %. Our pressure guge, though, doesn t hve ver fine mrkings, nd we estimte P/ P is 0%. We then compute: ρ ρ P T P T ( 0.) ( 0.0) (4) nd find ρ/ ρ 0. or 0%. The smll error ssocited with temperture mde no contriution to the overll error. This tells us tht the w to reduce the overll uncertint in n eperiment is refine the mesurement procedures for the untities with the highest individul uncertinties. A note of cution on ssuming rndom nd independent uncertinties: If we use one instrument to mesure multiple untities, we cnnot e sure tht the errors in the untities re independent. If instrument clirtion is the cuse of the error, the errors re not independent nd the totl error should not e computed summing in udrture. Generl Formul for Error Propgtion Ver often, the untit eing investigted in n eperiment cnnot e computed through the simple mthemticl opertions discussed ove. In mn eperiments, we must understnd how error propgtes through more comple functions, such s those tht contin eponents, logrithms, nd trigonometric functions. Let us emine untit tht is some function of vrile (such s sin or 3 ). We lso ssume, s usul, tht our mesurement of hs the form ±. Then our estimte of is ( ). As we hve done efore, we estimte the lrgest prole vlue of. In this cse, the mimum vlue of is m ( ± ). To determine wht this vlue is, we must evlute the function t ±. We cn do this determining the slope of the function. From clculus, we know tht for continuous function () nd smll increment u, d ( u) ( ) u. (5) d Since is lws smll, we cn s tht: M. Plmer 6
d d m ( ) ( ) The uncertint in is simpl the difference m nd we conclude tht: d. (6) d More generll, we cn write formul for computing the propgted error in function of severl vriles. The onl mjor chnge from the single-vrile formul shown in (6) is tht we must compute prtil derivtives (i.e. / ) due to the fct tht cn e function of multiple vriles. Rule 5: Uncertint in Function of Severl Vriles If,,z re mesured with independent nd rndom uncertinties,,z nd re used to compute (,,z) then the uncertint in is... z z M. Plmer 7