MEASUREMENTS, CALCULATIONS AND UNCERTAINTIES

Transcription

1 MEASUREMETS, CALCULATIOS AD UCERTAITIES MEASUREMETS, CALCULATIOS AD UCERTAITIES (FALL 017) PROF. S.D. MAOLI PHYSICS & CHEMISTRY CHAMPLAI - ST. LAWRECE smaoli@slc.qc.ca WEPAGE: TALE OF COTETS I. ITRODUCTIO... 3 II. UMERS AD SIGIFICAT FIGURES... 3 III. MEA, AVERAGE AD UCERTAITY... 4 A. PARET POPULATIO SAMPLE POPULATIO... 5 C. UCERTAITY... 6 D. SUMMARY... 7 IV. DETERMIATIO OF UCERTAITIES... 7 A. TYPES OF UCERTAITIES SIMPLE MEASUREMETS... 8 V. PROPAGATIO OF UCERTAITIES... 9 A. THE DIFFERETIAL METHOD APPLICATIOS AD SIMPLIFICATIOS Additio ad Subtractio Expoetiatio Multiplicatio ad Divisio The Use of Fuctios Fuctios of the Form (pa + qb) Fuctios of the Form pa + qb m Polyomials C. SUMMARY AD COCLUSIOS VI. EXAMPLES VII. COCLUSIOS PAGE 1 OF 17

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3 MEASUREMETS, CALCULATIOS AD UCERTAITIES I. ITRODUCTIO Scietific experimets are characterized by their repeatability. Carryig out the same experimet may times must give the same result every time withi the practical limitatios of the experimet the precisio of the experimet. Experimets are repeated to test the validity of the hypotheses they are based upo ad also to icrease the precisio of the results. The accuracy of the results depeds o how well the experimet is desiged to test the hypothesis while the precisio of the result reflects how well the experimet was carried out. I theory, if the experimet is perfect, the results will be exact. However, i practice, this is rarely the case. Experimets are always carried out as precisely as possible, ad measuremets obtaied ad results calculated usig these measuremets collectively called experimetal quatities are reported i specific, uambiguous forms that reflect their precisio usig methods ad covetios that have bee developed for this purpose. II. UMERS AD SIGIFICAT FIGURES A perso wishes to kow how may jelly beas will fit i a jar. A straightforward method for carryig out this measuremet is to fill the jar with jelly beas while coutig them. The result obtaied is The sigificace of this result is clear: the jar cotais 1347 jelly beas, ot 1346 or This result is represeted by a iteger, ad every digit i this umber is meaigful. Hece, this umber has four sigificat figures. However, itegers should be expressed as real umbers quoted i scietific otatio. A real umber cotais a fractioal part, a base ad its expoet, ad the iteger 1347 is writte as This real umber also has four sigificat figures, the umber of digits i the fractioal part. If a similar result is quoted as 1500 jelly beas, uless further iformatio is give as to how the measuremet was take, this result ca be iterpreted i may ways; if it is assumed that the jelly beas were couted oe at a time, the the result could be quoted as , or, if the perso filled the jar by the hadful ad assumed that each hadful cotaied 10 jelly beas, the the result would be quoted as , or, if the perso filled the jar usig a scoop ad assumed that each scoop cotaied 100 jelly beas, the the result would be quoted as The poit is that there is o defiitive way of determiig whether trailig zeros i a iteger are sigificat uless more iformatio about the measuremet is kow (i.e. how did the perso fill the jar?) This meas that, if a result is quoted i iteger form ad has trailig zeros, the zeros are ot sigificat; a result such as 1500 jelly beas should be quoted as jelly beas uless it is kow how the jelly beas were couted. All experimetal quatities should be quoted i scietific otatio because their precisio ca be deduced straightforwardly. The umber of sigificat figures expresses its precisio directly. The umbers , , ad represet differet results ad expressig them i iteger form, 1500, ca lead to cofusio. A real umber which cotais leadig zeros should also be expressed i scietific otatio. A result such as should be expressed as , ad regardless of how it is writte it has 5 sigificat figures. Leadig zeros (the 0.0 ) are ot sigificat. Oce the proper umber of sigificat figures of a result has bee determied, the remaiig digits to the right of the least sigificat digit must be rouded off. If the first digit to the PAGE 3 OF 17

4 MEASUREMETS, CALCULATIOS AD UCERTAITIES right of the last sigificat digit is greater tha 5, the least sigificat digit is icreased by oe or rouded up. If the first digit to the right of the least sigificat digit is less tha 5 the this digit as well as all of the remaiig digits to the right are simply igored, they are trucated or rouded dow. The covetio for roudig off o-sigificat digits if the first digit to the right of the least sigificat figure is equal to 5 depeds o whether the least sigificat digit is odd or eve. If it is odd, the 5 is rouded up, while, if it is eve the 5 is trucated or rouded dow. There is aother commo techique for roudig off o-sigificat digits whe the first of these is equal to 5. Suppose is to be rouded off to five sigificat figures. The o-sigificat digits are 534 ad 534 is greater tha 500 which is halfway betwee ad Hece, the least sigificat digit 8 is rouded up to give eve though it is a eve umber. This is acceptable. The same logic applies to roudig off umbers such as to 3 sigificat figures. The o-sigificat digits 49 are less tha 50, the halfway poit, hece, they are trucated so that the result is The same result would have bee obtaied if oly the first o-sigificat digit 4 would have bee used to roud off the umber. However, o-sigificat digits should ot be rouded off i a cascadig maer, i.e. the 49 caot be rouded off to 50 which, i tur, is used to roud the last sigificat figure from 7 to 8 to give III. MEA, AVERAGE AD UCERTAITY Whe measuremets are repeated, the results of each oe will, i geeral, ot be exactly the same. However, they ca be used to fid the best result which ca the be compared to the exact result, assumig oe exists which may ot be evidet i some cases. I the jelly bea measuremet described previously, the aswer to the questio how may jelly beas will fit i the jar? seems to be exactly ut if the jar is emptied the refilled, it most probably will ot cotai 1347 jelly beas but maybe 1348, 1341 or eve The results will vary due to a variety of factors. Oe factor may be the size of the jelly beas, they are most probably ot all exactly the same size. Hece, it is possible that for a give measuremet there were more smaller jelly beas the larger oes, or vice versa, or eve equal umber of large ad small oes. A secod factor could be that the positioig of the jelly beas i the jar, did the perso pack the jelly beas as tightly as possible without distortig them? These are examples of variables that are difficult to cotrol. Most experimets ivolve such variables, however, if these variables vary radomly the their effects will be miimized the more ofte the experimet is repeated. Therefore, a defiitive, uique result should exist. A. PARET POPULATIO A measuremet is carried out times yieldig results. The differece betwee each result x i, ad the best result is called the deviatio ad their sum, xi, (1) i1 is expected to be small because variables which are difficult to cotrol vary radomly; just as may results will be greater tha tha less tha. I the limit as, the sum of the deviatios (1) approaches 0, PAGE 4 OF 17

5 MEASUREMETS, CALCULATIOS AD UCERTAITIES lim lim x i1 which ca be solved for i 0, () 1 lim xi, (3) i1 ad is called the mea of the results. Whe there are a ifiite umber of results,, the set of results is called the paret populatio, ad the spread of the results aroud the mea defies the paret distributio which ideally should be symmetric about the mea. Although the sum of the deviatios of the results from the mea is zero, see (), the sum of the squares of these deviatios will ot be zero, 1 lim i1 x i 0. (4) This quatity is called the variace of the results, ad the square root of the variace is called the stadard deviatio of the results, lim 1 i x. (5) i1 It is a measure of the spread of the results about the mea. Clearly, the stadard deviatio would oly be equal to zero if every experimetal result was equal to the mea ad, i this case, the mea would be the exact result. Therefore, the stadard deviatio is a estimate of how close the mea is to the exact result.. SAMPLE POPULATIO I practice, measuremets are repeated a fiite umber of times, say, ad the fiite set of results obtaied is called the sample populatio. Assumig a experimet ca yield a defiitive result, say, the deviatios of the idividual measuremets {x i} from are sample xi. (6) i1 As was the case for the paret populatio, the variace of the sample populatio s wrt is s 1 i x. (7) i 1 ad it is expected to be small. The best value of ca be obtaied by miimizig s wrt, ds d 1 x 0 d d i i1 xi xi i1 i1 1 x 0 x i i. (8) i1 i1 The best result is called the average or the mea of the sample populatio. Heceforth, the average will refer to the mea of the sample populatio, while the mea will refer to the mea of the paret populatio. Sice the mea represets the exact value of the required measured quatity while the average represets the best result obtaied from the fiite data set, a estimate of how close the average is to the mea is required. The use of (7) to calculate the variace of a fiite data set is problematic for two reasos: 1. If oly oe measuremet is made, the variace will be zero because the average will be equal to the measuremet. This would imply that the uique measuremet is the best oe.. If the mea is kow, the the variace of the sample populatio ca be expressed as PAGE 5 OF 17

6 MEASUREMETS, CALCULATIOS AD UCERTAITIES sample 1 x i (9) i 1 which should be close to the variace of the paret populatio. This form of the variace will be larger tha the oe give by (7) because is obtaied by miimizig the variace of the sample populatio. However, i practice, oly is kow, ad, hece, the variace calculated usig (7) will be uderestimated whe compared to the best oe (9). To compesate for these problems, the variace of the sample populatio is writte as 1 x i (10) 1 i1 If oly oe measuremet is made the variace of the sample populatio will be udefied due to divisio by zero. I additio, whe the average is used as a substitute for the mea, dividig by ( 1) icreases the variace, correctly implyig that the spread about the average is larger tha the spread about the mea. However, i the limit of a large umber of measuremets i.e. 1 ad (11) ad the variaces (9) ad (10) are close to oe aother. C. UCERTAITY From the sample populatio, the best result is the average of the measuremets (8), ad the stadard deviatio ad the variace are obtaied from (10). As the umber of measuremets icreases, the best result will approach the exact result ad the stadard deviatio will decrease. I practice, the icrease i precisio obtaied by icreasig the umber of measuremets is ot desirable because it is time-cosumig. As discussed previously, the average ad the mea should be close to oe aother but the stadard deviatio of the sample populatio will be larger tha the stadard deviatio of the paret populatio. However, if the paret populatio is sampled more tha oce, there will be a distributio of averages ad variaces, ad this distributio will give a best averagethe mea of the paret populatioad a best variace which will give the variatio about the mea. The variace of the mea will be give by a expressio similar to (9) as m 1 i (1) i 1 which ca be expressed i terms of the variace of the results as m (13) This meas that the stadard deviatio of the mea is m (14) ad it is smaller tha stadard deviatio of the results give by (5). This result is called the stadard error of the mea or ucertaity of the results ad it expressed explicitly as x m i1 x i 1 (15) where is the mea, or average, of the results. For coveiece, the ucertaity ca be approximated as x 1 x i (16) i1 PAGE 6 OF 17

7 MEASUREMETS, CALCULATIOS AD UCERTAITIES D. SUMMARY The same experimet carried out may times should give the same result every time. Repeatig a experimet times will give results each of which will be withi the rage of precisio of the experimet. The best result will be the average x x x x x 1 3 ave, (17) The stadard deviatio is a measure of the spread of the measuremets aroud the average. The differece betwee the average ad each of the measuremets is calculated, d x x, 1 ave 1 d x x, ave d x x, (18) ave where each of these quatities is called the deviatio from the mea, ad the stadard deviatio is calculated as s d1 d d3 d. (19) 1 The ucertaity of the average is the calculated as x s d1 d d3 d, (0) 1 ad it is called the stadard deviatio from the mea or the stadard error. It is sometimes approximated as d1 d d3 d x (1) if is large eough. The defiitive value of the average is quoted as x ave x where the ucertaity is give by (0) or (1), ad it will be smaller tha most, if ot all, of the idividual deviatios from the mea. Hece, the more ofte a experimet is repeated, the more precise the average will be. This is always the case because the variables which are ot uder the cotrol of the experimetalist that cause the deviatios of idividual measuremets from the mea vary radomly. Therefore, a experimet is repeated ot oly to test the validity of a hypothesis but also to icrease the precisio of the result. IV. DETERMIATIO OF UCERTAITIES Experimetal quatities must be writte as x x, where x is the mai value ad x is its ucertaity ad ot x oly. Quotig x oly is icomplete because it does ot idicate to what extet the mai value is ucertai. Sice the ucertaity associated with a measuremet reflects its precisio, it is oly the least sigificat figure that is ucertai, hece, it is the last digit that may prove to be iaccurate. It is for this reaso that ucertaities are oly quoted to oe sigificat figure. The magitude of the ucertaity of a measuremet depeds o the assumptios that may have made whe desigig ad/or carryig out the experimet. For example, i the jelly bea measuremets discussed previously, the jar was filled with jelly beas may times, the umber of jelly beas was couted each time, ad it was assumed that the jelly beas were of equal size ad the jar was filled to its optimum capacity every time. Therefore, the ucertaity associated with the measuremet of a experimetal quatity (the umber of jelly beas that the jar ca hold) is a measure of the precisio of the measuremet (how meticulous was the perso whe he/she couted the jelly beas) ad the quality of the experimet, i.e. the assumptios that were made. Huma errors such as ot readig istrumets properly should ot be reflected i PAGE 7 OF 17

8 MEASUREMETS, CALCULATIOS AD UCERTAITIES the estimate of the ucertaity sice these reflect poor experimetal techiques rather tha poor experimets. If such errors do occur, the measuremet should be take agai or the experimet repeated. The oly rigid rule for estimatig ucertaities is the use of commo sese. Cosider the assumptios that were made i the previous jelly bea experimet. The jar was filled several times ad the result was jelly beas. It was assumed that the jelly beas were roughly the same size ad that care was take to esure that the jelly beas were packed as tightly as possible without distortig them. The impact of these assumptios could have bee miimized thereby reducig the magitude of the ucertaity by first makig sure that all the jelly beas were approximately the same size by examiig each oe of them carefully before placig them i the jar, makig sure to reject jelly beas that were either too small, too big, broke or damaged i some way, ad secod by carefully placig each jelly bea i the jar as closely as possible to the other jelly beas. I this case, the time required for each of the experimets might icrease from, say, oe hour to twety hours. Furthermore, if the jar is to be filled several times, the fial, more accurate result will be obtaied i (say) oe week s time. Is the more accurate result worth the extra effort? The aswer depeds o the reaso for takig the measuremet; the more accurate result may ot be worth the effort required to obtai it. Therefore, the precisio of the ucertaity must be tempered by what the measuremet, or result, is meat to reflect. A. TYPES OF UCERTAITIES Ucertaities expressed i uits of the experimetal quatity are called absolute ucertaities, for example 5 jelly beas. However, ucertaities expressed as a percetage, or a fractio, are called relative ucertaities, ad they are calculated usig x xrel 100% () x where x/x is the fractioal ucertaity. Relative ucertaities ca be more useful tha absolute ucertaities because they give a very quick idea of the precisio of a experimet or a measuremet; for example, if the precisio of this experimet is 5% or if the precisio of that measuremet is 1% the a perso ca very quickly get a geeral idea of the precisio of the result ad/or the experimet.. SIMPLE MEASUREMETS The ucertaity of a measuremet take usig a istrumet is sometimes, but ot always, oe-half of the smallest divisio of that istrumet. If a ruler whose smallest divisio is oe millimeter is used to measure the legth of a object, a huma beig ca usually see that the edge of the object may fall betwee two millimeter marks, hece, the ucertaity ca be quoted to 0.5 mm (absolute ucertaity). If a perso caot see that the measuremet may fall betwee two of the smallest divisios, the ucertaity caot be quoted as half-thesmallest divisio but must be estimated some other way, usually a full divisio. Some digital istrumets, such as voltmeters, may be accurate to 1-% of the displayed value (relative ucertaity). I this case, all measuremets will have the same relative ucertaity but each oe will have a differet absolute ucertaity. I some cases, it may be ecessary to estimate the ucertaity some other way. For example, if a perso uses a stopwatch accurate to a hudredths of a secod to measure the time it takes for a evet to occur, the ucertaity of PAGE 8 OF 17

9 MEASUREMETS, CALCULATIOS AD UCERTAITIES the measuremet will ot be due to the precisio of the istrumet but to a estimate of a perso s reflexes. V. PROPAGATIO OF UCERTAITIES The result of a calculatio carried out usig experimetal quatities will have a ucertaity which is a combiatio of the ucertaities of the idividual quatities i.e. ucertaities propagate. Furthermore, the relative ucertaity of such a calculatio must be larger tha the largest of the idividual ucertaities because calculatios caot improve the relative precisio of a quatity. A. THE DIFFERETIAL METHOD Cosider a quatity f defied as a fuctio of a sigle experimetal quatity a such that f g a (3) where a has a ucertaity a. The variatio of f as a fuctio of a ca be obtaied by takig the derivative of (3) df dg (4) da da which ca be expressed i terms of differetials as dg df da. (5) da The differetial df is the deviatio of f from its actual value due to a ifiitesimal variatio da of a. However, sice the full variatio of a is a, a ivolves a large umber of differetials da which, i tur, give rise to a large umber of differetials df each of which represets a deviatio of f from its actual value; a deviatio from the mea. Therefore, the stadard deviatio (0) ca be used to calculate the average value of df df df df df 1 1 ave where is large. Usig (5) i (6) gives df ave dg da da 1 dg da1 da (6). (7) where each df correspods to a specific da. The derivatives i (7) are all equal, hece df da ave dg da ave. (8) The average df is the ucertaity f ad the average da is the ucertaity a, hece (8) becomes dg dg f a f a da da ad f is the ucertaity of f. (9) The derivatio of (9) from (5) may seem straightforward: da a hece df f, however, the rigorous techique used to do so is ecessary i order to calculate the ucertaity of a fuctio of more tha oe variable. Cosider f g a,b (30) where a ad b are quatities with ucertaities a ad b, respectively. The variatio of f is ow due to the variatio of g as a fuctio of a ad b. Usig the calculus of differetials o (30) gives g g df da db a b (31) where the derivatives i (31) are partial derivatives The partial derivative g/a meas differetiatig the fuctio g with respect to a while treatig all other variables (i.e. b) as costats ad evaluatig the derivative at a PAGE 9 OF 17

10 MEASUREMETS, CALCULATIOS AD UCERTAITIES ad b, ad similarly for the partial derivative g/b. Substitutig (31) i (6) ad usig (7) ad (8) gives (df) ave g g a b g g a b df da db ave ave ave da db ave ave. (3) Although it is beyod the scope of this work, it ca be show that the third term o the righthad-side of (3) is zero if the variables a ad b are idepedet of oe aother. This is usually the case experimetally because the ucertaity associated with oe measuremet does ot deped o the ucertaity associated with a secod measuremet. This meas that (3) reduces to g g a b df da db ave ave ave (33) ad usig the same derivatio that lead to (9) g g a b f a b. (34) Therefore, f cotais a cotributio from a ad b. The derivatio of (34) ca be geeralized to ay umber of variables to give g g a b f a b g g c z c z (35) where the variables a to z are idepedet of oe aother. This approach will be called the differetial method of determiig ucertaities, ad it is the most rigorous.. APPLICATIOS AD SIMPLIFICATIOS Ay calculated result ivolves additio, subtractio, multiplicatio, divisio, or the use of fuctios such as si, ta or exp or, more likely, a combiatio of these. I some cases, the applicatio of the differetial method (35) to calculate ucertaities ca lead to simplifyig expressios. 1. Additio ad Subtractio Cosider the fuctio f pa qb (36) where p ad q are ay real umbers, ad the variables a ad b have ucertaities a ad b, respectively, ad they are idepedet of oe aother. The partial derivatives of f ca be evaluated f p a ad f q b ad substituted ito (35) to give (37) f pa q b. (38) This applies to additio ad subtractio (q < 0). Therefore, whe a experimetal quatity is multiplied by a costat, its ucertaity is also multiplied by the same costat, ad the ucertaity of the result is the square root of the sum of the squares of the weighted absolute ucertaities. It is importat to ote that, for additio ad subtractio, oly absolute ucertaities ad ot relative ucertaities must be used to calculate the fial ucertaity This is remiiscet of Pythagoras theorem, ad it is expected because the quatities ivolved i the calculatio are idepedet of oe aother (recall the discussio ivolvig the simplificatio of (3)). If a ad b are cosidered to be idepedet vectors, they are vectors which are perpedicular to oe aother. Hece, a ad b are perpedicular to oe aother as well ad the vector a does ot have a compoet which lies alog the PAGE 10 OF 17

11 MEASUREMETS, CALCULATIOS AD UCERTAITIES directio of the vector b ad vice versa. Therefore, f is the hypoteuse of a rightagled triagle whose sides are a ad b. This esures that, if b is smaller tha a, f will at least be equal to a, as it should, or more geerally, that the miimum value of f will be either a or b whichever is the largest because, after a mathematical operatio, the result caot be more precise tha the least precise quatity. Furthermore, f caot be equal to the sum of the ucertaities a ad b because such a sum would imply that a ad b are colliear, hece, depedet o oe aother.. Expoetiatio Cosider the fuctio f a (39) where is ay real umber, ad the variable a has a ucertaity a. The partial derivative of f is f a a 1 such that the ucertaity of f (34) becomes 1 f a a f a (40) a a a f f frel arel (41) a 3. Multiplicatio ad Divisio Cosider the fuctio f m a b (4) where ad m are ay real umbers, ad the variables a ad b have ucertaities a ad b, respectively. The partial derivatives of f are f f a b ad ma b a b 1 m m1 (43) such that the ucertaity of f (34) becomes 1 m m1 f a b a ma b b a mb a b. (44) m m f a b a b Dividig both sides of (44) by the fuctio f gives f a mb f a b frel arel m brel (45) which shows that the relative ucertaity of f is the square root of the sum of the squares of the relative ucertaities of the variables used to calculate it after esurig that they are weighted by their respective expoets ad m. Subsequetly, the relative ucertaity is coverted to a absolute ucertaity usig ff f rel (46) 100% ad f must be rouded off to oe sigificat figure. Divisio is take ito accout by the fact that the expoet of either a or b (or both) ca be egative which will also give a ucertaity give by (45). Therefore, for multiplicatio ad divisio (whe m or < 0), the relative ucertaity of the fial result is the square root of the sum of the squares of the appropriately weighted idividual relative ucertaities. 4. The Use of Fuctios I some istaces, a experimetal quatity must be used as the argumet of a fuctio to calculate a result. For example, if a agle is measured experimetally, it will have a ucertaity associated with it but the formula used to calculate the fial result may require that the si, cos, ta ad/or some other trigoometric fuctio be evaluated. Cosider a fuctio f give by PAGE 11 OF 17

12 MEASUREMETS, CALCULATIOS AD UCERTAITIES f a g b (47) where a ± a ad b ± b are experimetal quatities, is ay real umber, ad the fuctio g may be a si, cos, ta or eve exp or log or a combiatio of these. Evaluatig the partial derivatives of f gives f a f b 1 a g b ad ag b (48) where g (b) deotes the derivative of g(b). Substitutig these partial derivatives i (34) gives 1 f a g b a a gb b a a f a g b a g b b (49) which is used to evaluate the ucertaity. However, (49) ca be simplified usig the defiitio of a derivative,. (50) g b b g b gb lim x0 b If the derivative of g is approximated by g b g b b g b, (51) b the ucertaity f (49) becomes a f a g b a g b b g b a g b gb.(5) Dividig both sides by the fuctio f gives frel arel grel (53) where g rel is relative ucertaity of the fuctio g, g g b b g b grel. (54) g g b The use of (53) ad (54) will give a small, but egligible, differece whe compared to rigorous use of (35) because of the approximatio (51) made i the evaluatio of the derivative of g. 5. Fuctios of the Form (pa + qb) Cosider a fuctio of the form f pa qb (55) where p, q ad are ay real umbers, ad a ± a ad b ± b are idepedet variables. The partial derivatives are f f p pa qb q pa qb a b ad 1 1 ad their substitutio i (35) gives 1 f p pa qb a 1 q pa qb b 1 f pa qb pa qb (56) (57) Dividig all the terms by f gives f pa qb f pa qb Alteratively, (55) ca be rewritte as (58) f g with g pa qb (59) This form suggests that the ucertaity of f ca be evaluated usig two steps: first, the ucertaity of g is calculated usig absolute ucertaities (additio or subtractio) as show i (38), the the ucertaity of f is calculated usig relative ucertaities to take ito accout the expoet (multiplicatio or divisio) ad the result is idetical to (58). PAGE 1 OF 17

13 MEASUREMETS, CALCULATIOS AD UCERTAITIES Therefore, the fial ucertaity of f ca be obtaied usig a combiatio of absolute ucertaities for additio ad relative ucertaities for expoetiatio. However, such shortcuts ca oly be used if the compoets of the fuctio a ad b are idepedet of oe aother. 6. Fuctios of the Form pa + qb m Cosider a fuctio of the form m f pa qb (60) where p, q, ad m are real umbers, ad a ± a ad b ± b are idepedet variables. The partial derivatives are f pa a f ad qmb b 1 m1 ad their substitutio i (35) gives 1 m1 f pa a qmb b pa m mqb f a b a b Alteratively, (60) ca be rewritte as (61) (6) m f pg qh g a ad h b. (63) This form suggests that the ucertaity of f ca be evaluated usig two steps: first, the ucertaities of the terms g ad h are calculated usig relative ucertaities g a h mb ad g a h b (64) which are the expressed as absolute ucertaities a m mb g a ad h b a b (65) ad the they ca be combied to calculate the absolute ucertaity of f which will be exactly the same as (6). As was the case i the previous sectio, the ucertaity of f ca be evaluated usig a combiatio of absolute ucertaities for additio ad relative ucertaities for multiplicatio. ut, agai, such shortcuts ca oly be used if the compoets g ad h of the fuctio are idepedet of oe aother. 7. Polyomials Cosider a polyomial fuctio of the form m p q f a b a b (66) where p, q, ad m are real umbers, ad a ± a ad b ± b are idepedet variables. The ucertaity of each of the terms a b m ad a p b q i (66) ca be calculated usig relative ucertaities for multiplicatio as i (4) but the results caot be combied to evaluate the ucertaity for additio as give by (36) because the two terms beig added are ot idepedet of oe aother. C. SUMMARY AD COCLUSIOS The most rigorous method for estimatig the ucertaity of a quatity is the differetial method which requires the evaluatio of partial derivatives to calculate the ucertaity. For additio ad/or subtractio, this gives f pa qb t z (67) where p, q ad so o are the coefficiets of the terms beig added/subtracted ad all ucertaities are absolute. For multiplicatio/divisio, this method gives frel arel mbrel q zrel, (68) where the relative ucertaities of the terms beig multiplied/divided are of the form a rel = a/a with beig the expoet of a ad so o for b rel to z rel. However, the applicatio of (35) for the use of fuctios gives PAGE 13 OF 17

14 MEASUREMETS, CALCULATIOS AD UCERTAITIES frel arel grel (69) where the relative ucertaity of g is give by g g b b g b grel (70) g g b which is a approximatio to the rigorous applicatio of (35), but it gives ucertaities that are very close to the best oes. The ucertaities of fuctios of more tha oe variable ca usually be expressed as a combiatio of additio ad multiplicatio so that (67) ad (68) ca be used i the appropriate sequece but this ca oly be doe after esurig the terms are idepedet of oe aother. VI. EXAMPLES As a first example, cosider a variatio of the jelly bea measuremet. Give that the volume of the jar used previously is cm 3, how may jelly beas will fit i a box of volume cm 3? Sice each jar cotais jelly beas ad if the umber of jars that fit i the box ca be determied, the the umber of jelly beas that the box ca hold ca be calculated as o. jelly beas o. jars o. jelly beas box box (71) jar The first ratio o the right side of this expressio is the umber of jars that will fit i the box which ca be calculated usig o. jars Volume Volume box box jar Equatios (71) ad (7) ca be expressed as v v (7) (73) where is the umber of jelly beas that will fit i the box, v ad v are the volumes of the box ad the jar, respectively, ad is the umber of jelly beas that fits i the jar. Suppose that these experimetal quatities are 3 j j jelly beas jar vb v b cm box 3 3 v j v j cm jar (74) These values alog with the relative ucertaities are give i Table 1. The umber of jelly beas that fit i the box depeds o three measuremets hece its ucertaity will iclude a cotributio from the ucertaities of, v ad v. The partial derivatives of (73) are evaluated v v 1 v v v v v v v v 1 v v Table 1. The quatities used to calculate the umber of jelly beas that fit i the box ad their ucertaities ad the fial result. Quatity Mai Value Absolute Ucertaity Relative Ucertaity v cm 3 /box cm 3 /box 3.% v cm 3 /jar 50 cm 3 /jar.9% jelly beas/jar 5 jelly beas/jar 0.37% jelly beas/box jelly beas/box 4% PAGE 14 OF 17

15 MEASUREMETS, CALCULATIOS AD UCERTAITIES v v 1 v v v v ad substituted i (35) v v v v v v v v v which reduces to rel rel rel rel (75) (76) v v. (77) Usig the relative ucertaities i Table 1,,rel is 3. %. 9% 0. 37% 4. 3% rel (78) The umber of jelly beas that fit i the box is v v cm box 3 jelly beas cm jar jar jelly beas box, (79) hece the absolute ucertaity is 100% rel jelly beas box4. 3% 100% jelly beas jelly beas (80) box because the ucertaity must be rouded off to oe sigificat figure. The mai result must also be rouded off to the same precisio.. (81) jelly beas box box As expected, the relative ucertaity of the fial result is greater tha the largest ucertaity of the quatities used to calculate it, see (78). The calculatio of the ucertaity of the fial result ca be simplified by examiig the relative ucertaities of each of the quatities used i the calculatio. I this case, the relative ucertaity of the umber of jelly beas that fits i the jar is approximately oe-teth of the other two relative ucertaities. Hece, it ca be eglected which meas that relative to v ad v, is exact ad the relative ucertaity of the result becomes v v rel rel rel rel 3. %. 9% 4. 3% (8) which shows that the impact of the ucertaity i is egligible. Therefore, a advatage to workig with relative ucertaities is that, before calculatig the ucertaity of the fial result, they ca be examied to see if ay of them ca be eglected relative to the others. I some cases, oe of the ucertaities is quite a bit larger tha the others, ad it is the oly oe that eeds to be take ito accout. As a secod example, cosider the calculatio of the kietic eergy of a object give the followig data: m g ad v m s (83) The mai value of the kietic eergy of the object is 1 1 K K mv. kg. m s (84) The derivatives of K with respect to m ad v are K m 1 v ad K mv v (85) PAGE 15 OF 17

16 MEASUREMETS, CALCULATIOS AD UCERTAITIES Usig (85) ad the ucertaities i (35) gives 1 K v m mv v 1. 5m s 0. 01kg 1 K 0. 35kg 1. 5m s 0. 1m s K (86) Hece, the kietic eergy is K 7 (87) Alteratively, the relative ucertaities of the measured quatities are m m rel m v 01. v v rel (88) Rememberig that the expoet of the velocity is, these relative ucertaities are substituted ito (68) K K K or % (89) rel ad the absolute ucertaity is KK rel K % K (90) Therefore, the kietic eergy is also give by (87). I this case, the relative ucertaity expressio (68) gives exactly the same ucertaity as (35) because these methods are exactly equivalet. A importat ote o the calculatio of the ucertaities is the cotributio from variables which have a expoet greater tha 1. I this example, the speed caot be cosidered as two separate variables v ad v such the total cotributio from these is (v rel) ad (v rel) or (v rel) because the idividual ucertaities are ot idepedet of oe aother, these separate speeds are the same ad their ucertaities are depedet o oe aother. Therefore, the proper cotributio is 4(v rel). As a third example, cosider the calculatio of the acceleratio of a object accordig to a g si with g ms ad The mai value of the acceleratio is (91) m s (9) a. m s si.. Usig (35), the ucertaity is a a g a g (93) The derivatives of the acceleratio with respect to each of the variables are a a si ad g cos g (94) Substitutig these ad the ucertaities of g ad ito (93) gives a sig g cos si m s a 9. 8 m scos a m s m s (95) where the ucertaity of has to be expressed i radias. Hece, the acceleratio ca be quoted as a m s (96) Alteratively, each absolute ucertaity ca be coverted ito a relative ucertaity, ad, for the fuctio si, the approximatio (54) is used PAGE 16 OF 17

17 MEASUREMETS, CALCULATIOS AD UCERTAITIES rel si si si si rel si si si si si 6. 5 si 65. si (97) rel while for g g g 98. g rel.. (98) The, usig (69), the relative ucertaity of a is a rel si a g a g si a rel or 7. 7% (99) ad the absolute ucertaity is aa a rel 100% m s a.. a a m s m s a m s (100) argue that ucertaities should be calculated as accurately as possible, a accurate ucertaity is somewhat of a oxymoro. It is importat to kow how precise a calculatio based o experimetal results is, but a icrease i precisio with the use of oe method rather tha aother may ot be worth the computatioal effort, as log as the precisio of the results is ot drastically over or uderestimated. I ay case, ay differeces i the values of the ucertaities will ot be of such magitude that they will alter the coclusios draw from the calculated values. which meas that the acceleratio is also give by (96). However, ote that, as metioed i sectio IV..3, the ucertaities (95) ad (100) are ot exactly the same due to the approximatio to the derivative of the fuctio (51) that was made whe derivig (54), but this differece is egligible. Ay such differece i ucertaities will ever be so differet so as to jeopardize the validity of a result. VII. COCLUSIOS Calculatio of ucertaities should be take with a grai of salt. Although some would PAGE 17 OF 17