Scientific notation makes the correct use of significant figures extremely easy. Consider the following:

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1 Revised 08/1 Physics 100/10 INTRODUCTION MEASUREMENT AND UNCERTAINTY The physics laboratory is the testig groud of physics. Physicists desig experimets to test theories. Theories are usually expressed i the laguage of mathematics -- this expressio of a theory is referred to as a mathematical model. If the outcome of a experimet does ot agree with the predictio of the mathematical model, the, either the experimet was a failure, or else the model is wrog. If the experimet is desiged carefully, ad coducted carefully, ad repeated may times, by may researchers, ad the results are always differet from the predictio of the model, the the model is icorrect. If, o the other had, the outcome of the may experimets agrees with the predictio of the mathematical model, the it is highly likely that the model is correct. The outcomes of experimets, which are measuremets of physical quatities with appropriate uits, are compared with the predictios of mathematical models, which are always calculated umbers with appropriate uits. Clearly, the, the success of physics relies fudametally o the measuremet of physical quatities. SCIENTIFIC NOTATION AND SIGNIFICANT FIGURES You are required to use scietific otatio whe reportig measuremets ad calculated results. A umber writte i scietific otatio has three parts. I the umber writte as 3.45 * is called the matissa, 10 is called the base, ad is called the expoet, or sometimes the characteristic. Scietific otatio makes the correct use of sigificat figures extremely easy. Cosider the followig: How may sigificat figures are i the umber 300? Are the zeroes sigificat or are they there simply as place holders? If the zeroes are ot sigificat, you should write the umber as 3 * this has oe sigificat figure. If oe of the zeroes is sigificat, write it as 3.0 * 10. If both zeroes are sigificat, write it as 3.00 * 10. Rules for the proper use of sigificat figures i additio or subtractio are illustrated below: * * 10 I additio or subtractio, the sum or differece has sigificat figures oly i the decimal places where both of the origial umbers have sigificat figures. This does ot mea that the sum caot have more sigificat figures tha oe of the origial umbers. I the fourth example

2 Physics 100/10 - Itroductio Page above, ote that has oly oe sigificat figure, but the sum properly has four. It is the decimal place of the sigificat figure that is importat i additio ad subtractio. I the fial two examples there is aother case of the ambiguity of fial zeroes. If you estimate that there are 500 studets i a lecture, implyig a umber betwee 450 ad 550, your estimatio is ot chaged if 4 people leave. O the other had, if you draw out $500 from the bak ad sped $4, you have $496 left. Rules for the proper use of sigificat figures i multiplicatio or divisio are illustrated below: 5. * * / / * / I multiplicatio ad divisio the product or quotiet caot have more sigificat figures tha there are i the least accurately kow of the origial umbers. Cosider the first example: the product might be as large as (5.5)(3.15) or as small as (5.15)(3.05) The rule for sigificat figures i multiplicatio is evidetly justified i this case. Durig multiplicatio or divisio, carry a extra sigificat figure alog, the roud off the fial aswer. HOW TO REPORT UNCERTAINTIES Every measuremet of every quatity is made with referece to a scale (a ruler, for example). Every scale is defied with referece to a stadard. The stadards are defied arbitrarily, ad are usually based o a atural physical dimesio, or aturally occurrig time period. Every researcher does ot, of course, use the stadard measures, but istead relies o copies of the stadards, ad trusts that the copies are faithful to the origials. Researchers ca the use their ow scales to make measuremets, ad kow that the scales they use are cosistet with all others. Ultimately, the, all measuremets are made by people who judge, agaist a reliable scale, what the measuremet of a particular quatity is. This judgemet is based o the skill of the researchers, the precisio of the measurig devices, ad the physical coditios uder which the measuremet is made. Every measuremet really has two parts. The first is the measuremet itself, ad the other is the so-called readig error. This readig error is a estimate i the level of cofidece the experimeter has i the measuremet value beig reported. As a simple example, cosider the case of measurig the legth of this page usig a ruler. If the ruler has millimetres as its smallest divisio, this puts a limit o the precisio with which you ca measure the page. Certaily you ca measure to the earest millimetre, eve to the earest half. Ca you do better, though? What about to the earest fifth, or teth? What you quote as a readig error is really a attempt to aswer the questio: How good is your measuremet? If every researcher must judge what a measuremet is, the there ca be o absolutely correct value for ay measuremet (uless it is arbitrarily defied as a stadard). I other words, there is some ucertaity associated with every directly measured quatity. Some texts refer to these ucertaities as "errors". This is a ufortuate term sice it implies icorrect procedure or sloppiess. Although ucertaities are iheret to the process of measuremet, ad caot be

3 Physics 100/10 - Itroductio Page 3 elimiated, they ca be reduced by followig correct procedure, ad simply by takig careful measuremets. Absolute Ucertaity Whe a ucertaity i a quatity x is expressed i the same uits as x, it is called the absolute ucertaity ad is deoted δx, which is read as delta x or as the ucertaity i x. Whe reportig a absolute ucertaity, such as a readig error i some measured quatity x, for example, use the followig otatio: x ± δx For example: Legth of metal cylider L 5.00 ± 0.50 mm Relative Ucertaity ad Percet Ucertaity Sometimes it is importat to kow how large a ucertaity is ot i absolute terms, but i compariso to the quatity beig measured. If the ucertaity is expressed as a fractio of x, the it is called the relative ucertaity, ad is calculated as: δx (1) x For example: If L 5.00 mm ad δl 0.50 mm, the the relative ucertaity is: δl 0.50 mm 0.10 L 5.00 mm (Note: uits cacel) The relative ucertaity ca be expressed as a percetage by multiplyig by 100: Percet ucertaity Relativeucertaity*100 () So, i the previous example: Percet ucertaity 0.10 * % The origial measuremet would be expressed as L 5.00 mm ± 10% Percet Differece A fourth term sometimes used is a percet differece which is a compariso of a measured value to a stadard value, although ay two values ca be compared. For example, if two values A ad B are compared, the percet differece is: B A B A % diff 100 OR 100 A B The deomiator is usually the best kow value. This is ot a error; it is oly a compariso, ad is ot used i stadard practice. Do ot cofuse percet ucertaity with percet differece.

4 Physics 100/10 - Itroductio Page 4 Radom versus Systematic Ucertaity Ucertaities are categorized as beig either radom or systematic. A third type - the bluder - is ot a legitimate ucertaity, ad is ot discussed here. Radom ucertaities caot be elimiated from measuremets -- they are geerally iheret i the physical quatity beig measured, or i the measurig device itself. Systematic errors are ofte associated with improperly calibrated measurig devices or with the method of measuremet. For example, the ed of ruler might be wor dow so that the scale does ot start at zero. All measuremets reported will be systematically too log by the amout that the scale differs from zero at the wor ed. Systematic ucertaities ca be compesated for, although it is sometimes difficult to detect their source. More about Readig Errors Ucertaities associated with directly measured quatities (or Readig errors ) are assiged by the experimeter. These ucertaities ofte deped o the measurig device. For example, a measuremet made with a ruler may have a ucertaity o smaller tha ±0.0 cm, whereas the ucertaity associated with a verier caliper may be ± cm. Usually, readig errors are associated with the use of a measurig device uder ideal coditios. Ideal i this cotext meas the coditios for which the measurig istrumet was desiged to operate most effectively. Fortuately, the physical coditios of the first-year laboratory are coducive to the proper operatio of most of the measurig devices you will ecouter. The readig error of a measurig device is ot ecessarily the best estimate of a ucertaity associated with a directly measured quatity. For example, if you measure the legth of a metal cylider with a ruler, the readig error may well be ±0.0 cm. If the cylider has very rough edges, however, you may wat to icrease the ucertaity to, say, ±0.05 cm or more. The poit is, you are the oe makig the measuremet, therefore you must judge the size of the ucertaity, ad you will be accoutable for the ucertaity you assig. Whe reportig a value for a directly measured quatity, the umber of digits you report is limited by the precisio of the measurig device. For example, say you measure the legth, L, of a object with a ruler which has markigs i millimetres, ad fid L 5.5 ± 0.5 mm There are 3 sigificat figures i L, the least sigificat of which is the last digit. There is oly oe sigificat figure i δl. Ucertaities rarely have more tha oe sigificat figure -- remember, these are estimates. Note that the ucertaity occupies the same decimal place as the least sigificat figure does i the measured quatity L. It is usually a waste of time (ad just plai wrog) to quote more sigificat figures i a measured quatity tha justified by the size of the ucertaity i that measured quatity. For example, L 5.5 ± 5 mm suggests that the last digit i L is meaigless -- it would be better to report L 5 ± 5 mm.

5 Physics 100/10 - Itroductio Page 5 STATISTICAL UNCERTAINTY: Ucertaity i more tha oe measuremet It is usually the case that more tha oe measuremet of a quatity is made durig a experimet. With each measuremet, there will be a associated ucertaity. A questio aturally arises -- Does takig more measuremets somehow reduce the size of a ucertaity? (Ideed, if it does't, would't a sigle measuremet be eough?) Cosider the followig case: You use a ruler to measure the legth of a cylider, ad fid x 5.15 ±.05 cm. You put the cylider dow, ad pick it up agai to make aother measuremet, ad ow fid x 5.18 ±.05 cm. Maybe the cylider has a slated ed? You ask your lab parters to help you out. After 10 measuremets, you get the followig data table: x(cm) δx(cm) Table 1 You work out the average x to be rouded to 5.16 cm. (Why oly 3 sigificat figures here?) But what is the ucertaity? Is it still ±0.05 cm? By takig 10 measuremets, ad the workig out the average, have't you somehow gotte a better aswer, a more certai aswer, tha if you took just oe measuremet? The aswer to the last questio will be provided i the pages that follow. A few defiitios are required: x is the measured quatity. is the umber of measuremets take. x i is the value of the i th measuremet, where i 1,,...,. x i1 xi x 1 + x + x3 + + x is the average measuremet. d i x i - x is the deviatio of the i th measuremet from the average.

6 Physics 100/10 - Itroductio Page 6 If you plot the data i Table 1 o a bar graph with x as the abscissa ad the umber of times a measuremet of x occurs o the ordiate, you get the bar graph show i Figure 1. Frequecy vs. Legth Frequecy Legth (cm) Figure 1 5. If istead of 10 measuremets you spet a ifiite time at it ad took a ifiite umber of measuremets (i.e. ) you would ed up with a graph like that show i Figure. Gaussia Distributio Frequecy Measuremet Figure

7 Physics 100/10 - Itroductio Page 7 This is kow as a Gaussia distributio, ad has the form (x x) 1/ σx N N max e (3) where N max is the umber of measuremets that occurs the maximum umber of times. σ x is the stadard deviatio, defied as σ is the Greek letter sigma. Equatio (4) is read as follows: d i i 1 σ x lim (4) The stadard deviatio squared is defied as the sum of the squares of the deviatios divided by the umber of measuremets, i the limit as the umber of measuremets approaches ifiity. As you may have oticed, this defiitio is ot very practical -- o oe ever makes a ifiite umber of measuremets. It is possible to estimate the stadard deviatio of a umber of measuremets less tha ifiity. The estimate is σ i 1 d i 1 (5) This is the way you will calculate σ. But, what is σ? σ tells you that about 68% of the values of x that you measured will be withi σ of the average x value. Or, to put it aother way, if you choose oe of the measured values x at radom, there is a 68% chace that the value will be withi the rage: x - σ x x + σ I other words σ gives you a estimate of the ucertaity associated with each measuremet of x.

8 Physics 100/10 - Itroductio Page 8 Cosider the data i Table 1 agai. i x i d i x- x d (mm) (mm) (10-4 mm ) Table x 5.16 ; 10 Readig error ±0.05 mm σ di -4 i * *10 σ * 10 - rouded to 3. * 10 - mm Notice that i this example σ is expressed to oe sigificat figure oly (why?) From this sample of data, you ca say that there is a 68% chace that the most likely value for x, the legth of the cylider, is withi the rage x mm or 5.13 x 5.19 mm eve though the readig error you assiged to these data was ±0.05. But, the readig error is larger tha the stadard deviatio, so it would be far too optimistic to quote the fial ucertaity as the smaller of the two ucertaities. I this case, the readig error still represets the ucertaity i the measuremets. Why calculate a stadard deviatio the? Remember that the stadard deviatio gives you iformatio about the spread of your data.

9 Physics 100/10 - Itroductio Page 9 Cosider this example: Istead of a ruler, you use a more precise measurig istrumet such as a micrometer to measure the legth of the cylider. Suppose you get the followig data, ad perform the stadard deviatio calculatio that follows: σ i x i d i x- x d (mm) (10-3 mm) (10-6 mm ) Table 3 x mm; 10 Readig error ± mm di -6 i *10 9 σ 6.8 * 10-3 mm 46.98*10 Notice that i this case, σ is greater tha the readig error. I other words, the spread i the data is greater tha ca be accouted for by the precisio of the micrometer. I this case, the, you should quote the ucertaity associated with each measuremet as σ ±6.8 * 10-3 mm. This is a appropriate time to summarise the last few pages of iformatio. The importat poits are: 1. Every measuremet has associated with it a readig error.. Repeatig the same measuremet may times is useful because it ca provide iformatio about how the measuremets are distributed aroud the average value. 3. If you assume the distributio is ormal or Gaussia, the you ca assig a statistical ucertaity called the stadard deviatio to each of the measured values i the set. The stadard deviatio may be smaller or larger tha the readig error. What you eed to kow ow is what happes whe you have readig errors or stadard deviatios from differet measured values that must be combied. Assume that you have

10 Physics 100/10 - Itroductio Page 10 measured several quatities, say distaces, masses, ad so o. How do these ucertaities combie to give a overall ucertaity i some derived quatity? Surely they do t just add, or multiply, because uits would ot work out properly. This brigs the discussio to the ext topic: Propagatio of Error. PROPAGATION OF ERROR How do readig errors combie to give a ucertaity i a calculated value? The stadard approach is to assume that the readig errors are all idepedet of oe aother, ad the to quote the fial error as a probable error ot the maximum or the miimum, but a error somewhere i betwee, where the somewhere is determied by weightig the larger errors more heavily tha the smaller errors. The geeral method for calculatig the propagated error is to fid the square root of the sum of the squares of the cotributig errors. Souds simple eough, ad it is, provided that the cotributig errors are properly accouted for. Here s the geeral case: Let s say there is a quatity z for which you must calculate the error, ad you kow that z depeds o several other quatities, x, y, u, v, ad so o. The way that δz depeds o these is related to the rate of chage of each of x, y, ad the others. I other words it is the first derivative of z with respect to each of x, ad y ad the others that will determie the fial error i z. The calculus is used to fid these first derivatives i the way you have see (or will soo see) i your mathematics courses, except that you have probably see derivatives i oe variable, ot i several. So, what to do? Simply treat oe variable at a time! This is what the geeral expressio for fidig a ucertaity looks like: δz z x z y z u ( δx) + ( δy) + ( δu) + z This symbol is used to represet a partial derivative. Some people proouce this as die z x die x to distiguish it from a ordiary derivative. It meas the chage i the fuctio z with respect to the chage i the variable x, treatig all other variables (y, u, etc.) as costats. The symbol δ x (delta x) meas the ucertaity i x. This is simply a umber, usually a readig error. Here s a example: Let s say you are asked to compute the desity of a block of metal, ad that you have the followig measured values:

11 Physics 100/10 - Itroductio Page 11 Legth, l, of block i metres Width, w, of block i metres Height, h, of block i metres Mass, m, of block i kilograms Here are the ucertaities, or readig errors: I all the legths: ± metres I the mass: ± kilograms The Volume, V, is: V l w h This works out to be 5.0 x 10-6 m 3. mass The desity is: ρ Volume This works out to be 4.0 x 10 3 kg/m 3. What is the error i the volume due to errors i measurig legth, width, ad height? The first step is to compute the partial derivatives of V: V l wh V lh w V lw h The secod step is to substitute ito the geeral ucertaity equatio: δ V ( wh) ( δl) + ( lh) ( δw) + ( lw) ( δ h ) This still looks like it s goig to be a lot of work to put umbers ito, so divide both sides by the volume V: δv V δl l δw + w δh + h This has give us a useful shortcut. If the quatity you are tryig to fid a error for is a product of other terms, the relative error ca be foud by fidig the square root of the sum of the squares of the relative errors i those terms. Easy! (Well, maybe ot easy, but ot too bad). The terms i the umerators are simply the ucertaities i each of the legth l, width w, ad height h. These you kow are all ± metres each.

12 Physics 100/10 - Itroductio Page 1 So, substitute: δv V This gives: δv V (.01) + (.005) + (.00005) Note how small the last term is compared with the first term. You may as well drop the third term altogether to save time. Eve the secod term is four times smaller tha the first term, so you ca drop it too! The result is: δv V δv V (.01) 0.1 I other words, the ucertaity i the Volume is about 10% of 5.0 x 10-6 m 3. You would write the Volume 5.0 ±.5 x 10-6 m 3. The relative error i the desity is (usig the shortcut): δρ ρ δv V δm + m Substitutig gives: δρ ρ δρ ρ (.1) (.1) + (.05) Agai, oe term domiates. So the relative error i the desity is about 10 %. You would write the desity 4.0 ±.4 x 10 3 kg/m 3. This example was fairly straightforward sice it ivolved oly liear terms beig multiplied or divided.

13 Physics 100/10 - Itroductio Page 13 Below are the results of applyig the geeral rule to some commo fuctios. The variables are x ad y. Costats are a ad b. z axy z ax + by δ z δ z ( aδx) + ( bδy) δx δy + z x y z ax δz a x -1 δx z a l x aδx δ z x z δz a si( x) z a cos( x) a cos( x) δx δz a si( x) δx Whe dealig with trigoometric fuctios, the error term δx must be i radias. The ucertaity i the average of measuremets is estimated as σ δ x Ucertaity calculatios ca appear to be upleasatly complicated ad will be very timecosumig, at least i the begiig. Very quickly, though, you will lear to idetify the shortcuts. Most of the ucertaity calculatios you will ecouter i this laboratory will be domiated by oe error term. Keep that i mid. Remember that the purpose of doig the calculatio is simply to let others kow the cofidece you have i your result.

14 Physics 100/10 - Itroductio Page 14 GRAPHS Graphs are a useful way to represet data (usually two related parameters), but oly if you take the time ad make the effort to clearly idicate what you are plottig. Here are a few guidelies for makig a graph: Use a sharp pecil, ad ever a pe. Use as much of the graph page as possible, while usig reasoable scales o the axes. Use a sesible subdivisio such as, 4, or 5. Never subdivide by 3, 6, or 7. Break the scales if you have to so the graph fits icely o the page (i.e., you eed ot start each scale at the origi). Make the data poits small, but obvious. Label each axis clearly, ad iclude uits. Put a descriptive title o the graph (ot just y axis vs x axis ) If drawig a best-fit lie, draw it so the data poits are ot obscured. Idicate error bars (more o these below). Idicate where the slope calculatio poits are. Never use data poits to calculate slope. Error bars Just as readig errors are a way of idicatig your cofidece i measured values, error bars are a way of idicatig cofidece i your graphed data. I their simplest form, error bars ca simply be readig errors. I other cases, error bars are calculated from readig errors, or they ca be statistical errors based o stadard deviatios, for example. Drawig error bars o graphs ca serve two purposes: First, they idicate the rage of probable values for idividual data poits ad, secod, they provide a way to estimate error bouds for the slope of a fitted lie. Fittig Lies Data is preseted i a graph to show what sort of relatioship exists betwee two parameters, ad that relatioship is ofte a liear oe. I fact, the parameters are sometimes modified so as to be liearly related. If the sought for relatioship is liear, this is idicated by attemptig to fit a lie to the plotted poits. Oe way to do this is to draw a lie by eye. That is, you simply put a straight edge alog the set of poits, ad adjust it so that about half the poits are o oe side of the edge, ad half o the other side radomly distributed (i.e., do ot put the top few poits o oe side, ad the bottom few o the other, or vice versa). The draw a sigle lie usig the straight edge as a guide. Do ot force the lie through the origi, eve if you have a theoretical basis for claimig it ought to go there. A useful techique is to hold the graph page up to oe eye with the page horizotal ad sight alog the set of poits. You should see ay tred i the data very clearly, ad whether your draw lie is a good fit.

15 Physics 100/10 - Itroductio Page 15 Puttig it all together Figure 1 shows some data plotted o a graph. Error bars are draw. Notice that ot all error bars are ecessarily the same size. Also, i this example, the bars are vertical oly. The vertical lies exitig from the data poits i the example are the error bars. The short horizotal lies at the eds of the bars are ot error bars themselves: they are simply there to idicate where the vertical bars termiate. (There are cases, however, where error bars ca be o both axes.) Liear relatio betwee mass of course textbooks ad iverse frequecy of use durig the first week of classes Slopes calculated from poits o these verticals 1.0 Iverse frequecy of usage (days/hour) 0.5 Best fit lie Maximum lie Miimum lie 4 6 Mass (kg) Figure 1 Also draw o the graph are the best fit lie, a maximum lie, ad a miimum lie.

16 Physics 100/10 - Itroductio Page 16 The maximum lie is draw so that it itersects the ed of the lowest error bar of the lowest data poit, ad itersects the ed of the highest error bar of the highest data poit. Similarly, the miimum lie itersects the ed of the highest error bar of the lowest data poit, ad itersects the ed of the lowest error bar of the highest data poit. The slope of each lie is calculated i the usual way: slope y x y1 x 1 The poits (x,y ) ad (x 1,y 1 ) are chose so as to be: At coveiet grid itersectios Spaced far apart Not data poits Ofte it saves time to select the same x for all three lies, ad the same x 1 for all three lies. Oe way to estimate the ucertaity i the slope of the best fit lie is to calculate half the differece betwee the maximum ad miimum slopes: δ ( bestfitslope) max slope mi slope Uless specifically directed otherwise, draw error bars, draw the three lies, ad calculate the slopes ad error estimate for the best fit slope for every graph you make. Refereces Practical Physics, G. L. Squires, Cambridge Uiversity Press, Eglad, Data ad Error Aalysis i the Itroductory Physics Laboratory, W. Lichte, Ally ad Baco, Ic.,U.S.A., Data Reductio ad Error Aalysis for the Physical Scieces, P. R. Bevigto, McGraw-Hill, U.S.A., 1969.