Chapter 2 Error Analysis

Transcription

1 Chater Error Analsis Beore we discuss an eerients that we will eror in the laborator, ou will need to have soe tools to be able to design eerients and anale data. In articular, ou will need to know how to etract inoration ro grahs and how to eror an error analsis. This chater discusses the ethods b which we deterine the aount o uncertaint in an eerient, or an error analsis. Wh Anale Uncertainties? Suose two students eror an eerient. Each student uses his or her own ethodolog to achieve the objective o the eerient the class is conducting. For eale, the student could be tring to deterine the substance o an unknown sale b deterining its densit. One student obtains a value o 6000 kg/ 3 and the other student obtains a value o 900 kg/ 3. Iron has a known densit o 7960 kg/ 3 and coer has a densit o 8930 kg/ 3. Both students were given the sae sale. Figure - A grah o two students data. Student reorts the sale is iron. Student reorts the sale is coer. The irst student reorts that the data indicates that the sale is iron. The second student reorts that the data indicates the sale is coer. Who is right? Suose the students give soe additional inoration: The irst student states that 68% o the tie the data was within the range 3000 to 9000 kg/ 3. The second student states that 68% o the tie the data was in the range 800 to 0000 kg/ 3. These ranges are lotted in Fig. -. 5

2 Figure - A grah showing the error bars (range o high robabilit) or the two students. either data is conclusive. Given these ranges, ou can see that the irst student s data doesn t eclude coer. What the student should have reorted is that the data doesn t distinguish either coer or iron. It is ossible that the aterial was ade o either etal (and robabl soe others too). On the other hand, the second student s range o data is saller. He or she could reort that the etal was ost likel coer. However, the student couldn t sael eclude iron, since he or she still has a signiicant robabilit (about 3%) o reaching that answer. Both students, thereore, have ailed to show conclusivel that the aterial was either coer or iron, and thus both students should reine their eerient to show that the aterial is deinitivel one and not the other. How did the students deterine the robabilities? How do the students go about reining their eerient? How can the students coare their results in such a wa that all can agree uon their ethod o coarison? These and other questions can be answered b eroring an error analsis. Error analsis is the eaination o uncertaint in hsical easureents. Error analsis tells use how recisel we can give our reorted values, allows us to coare results with other eerients (or with theor), and hels us to deterine where an eerient can be iroved. It is essential that ou eror an error analsis on ever eerient. Deinition o Error What is an error anwa? Perhas it is unortunate that the words error and uncertaint are interchangeable ters in eeriental science. Most eole have the idea that an error is a istake or a isha (e.g. Darn! I blew u eerient! ). In the laborator, the word error is unortunatel interchangeable with the word uncertaint, which has an entirel dierent eaning. When we sa error we are 6

3 talking about the act that we cannot take easureents to ininite recision, nor can we ake easureents with ininite accurac. For eale, i I asked ou to easure the length o a en, ou ight get out our ruler and declare the en is 8. c long. You wouldn t sa that the en is c long. You know that ou can easure things with a ruler with onl so uch recision. You can onl easure to the nearest division on the ruler (), or erhas one hal o one division (0.5 ) i ou are reall careul. In order to reresent this recision, we give a range o values rather than just a single value. Another issue is accurac. To what etent can we sa that the anuacturer o the ruler is indeed reresenting according to our standard eter? The anuacturer o our ruler also has soe range o accurac which deines where the lines o the ruler will be drawn. These ranges (o recision and accurac) are called the uncertainties in the value. The error or uncertaint in a value is the range in which ou are reasonabl certain the value lies. In the case o the en, erhas ou ight sa to oursel: I can deinitel see that the en is longer than 8. c and shorter than 8.3 c, but I can t tell i it is longer than 8.5 c. It is this range, 8. c to 8.3 c, which trul reresents our eeriental value. That is what we ean b uncertaint in the value. Tes o Uncertaint How do ou deterine this range o values? Soeties it is obvious, like the length o a en, but other ties, it is not so obvious. What i ou had easured the value an ties and ou obtained an average? What i ou calculated the value ro two or three other easureents? What do ou do then? How ou deal with and calculate uncertainties deends uon the te o uncertaint which is resent in our data. In general, there are two tes o uncertainties: rando and ssteatic. Rando errors are arbitrar luctuations in our data. In other words, there is no reerred direction or such errors. Your data values can be greater than or less than the best value without reerence. For eale, i ou used a stowatch to easure an event, ou ight easure 3.45 s in the irst trial, 3.5 s in the second trial, 3.3 s in the third, and so on. Perhas at the end o the data collection ou deterine that the best nuber which reresents the tie is 3.35 s. Are all o our data oints greater than this value? Are all o our data oints less than this value? What ou robabl see is that ou are greater than this best value just as oten as ou are less than the best value. Another words, there is no reerence as to whether or not our net data oint will be overestiated or underestiated. Soe scientists even go so ar as to create rules about this. I have heard o several such rules, e.g. The sallest division equals the uncertaint. or Hal o the sallest division equals the uncertaint. However, I ask the reader to take such rules with a grain o salt. For ever rule, I can resent to ou an ecetion which will break the rule. It is better to understand uncertaint well enough such that ou can create our own rules. 7

4 Ssteatic errors are errors which skew our data in a reerred direction. For eale, suose the stowatch was slow. Then all o our data oints are consistentl underestiated. Your average with this watch will alwas be less than the actual tie elased. You can see then that alost ever data value which ou collect will generall have both tes o uncertaint resent in the sste. Even the easureent o the en would likel have both tes o uncertaint resent. The uncertaint ou reorted or the en reresented onl the rando art o the uncertaint. What ou are saing when ou reort that our rando uncertaint is is that the net tie ou tr to easure the value ou will likel be within o the reorted value, but the net value can be above or below it without reerence. There is a ssteatic uncertaint in this easureent also. All things stretch and coress with variations in teerature (that is how ost household theroeters work). Suose the ruler has stretched ro its original shae. This eans that when the ruler reads one illieter, that illieter is not the sae as the standard illieter. Ssteatic errors can onl be deterined quantitativel b coaring our easureent to a standard. For eale, ou could use a standard cesiu clock to calibrate our stowatch or use the standard eter (held in a bell jar in France) to calibrate our ruler. In ost cases, such calibration is iractical in a student laborator. evertheless, an anuacturers o scientiic instruents do rovide data on how well their device will eror under various circustances, so consult the equient anual i available. Another otion is to eror a calibrating eerient to deterine the aount o ssteatic error. I all else ails, attet to iniie the ssteatic uncertaint to the etent that ou can judge that the uncertaint is ver sall with resect to anthing which ou are easuring. Eale - Consider the uncertaint in a cannonball trajector. Draw a bull s ee target. Draw a dot on the target to indicate where an individual launch lands. Ater ultile launches what sort o attern on the target will the ollowing cases result in? Give an eale o wh each case ight arise in real lie. Cases: (a) Large rando uncertaint and large ssteatic uncertaint. (b) Large rando uncertaint and sall ssteatic uncertaint. (c) Sall rando uncertaint and large ssteatic uncertaint. (d) Sall rando uncertaint and sall ssteatic uncertaint. 8

5 Solution: Rando uncertaint could arise or several reasons: it is ver diicult to control an elosion (at least with gunowder) such that the aount, direction, and sread o the orce are accuratel controlled. Another actor is the soothness o the bore and the soothness o the cannonball. These actors were etreel aarent in earl cannon (and usket) designs. Ssteatic uncertaint could arise due to wind resistance (drag), cross winds, unaccounted orces (such as the Coriolis orce ), aong other things. (a) Large rando; large ssteatic (b) Large rando; sall ssteatic (c) Sall rando, large ssteatic (d) Sall rando, sall ssteatic Direct Measureents: Deterining the Rando Uncertaint b Estiation For certain easureents, it is silest to estiate the uncertaint using our coon sense. This can be done onl when ou alread have soe idea o the recision o our easureent (ro revious eerience). When ou are reading a scale or a eter o soe sort usuall ou can characterie the uncertaint b how well ou can read the scale. Measuring a en with a ruler is one such easureent. You are aware alread that ou can easure the en to at least a illieter, so it is accetable to estiate the uncertaint as the sallest unit on the scale ( ). This te o easureent is called a direct easureent, because no interediating sste is between ou and the easureent. However, use this ethod with etree rejudice! Do ou think that ou could easure the en as well i it were rolling about? Or sinning? To use this ethod, ou ust be sure that the uncertaint coes onl ro reading the scale and not ro an other actor! The Coriolis orce is actuall a seudoorce. A seudoorce is a orce which is added to account or accelerations observed in a non-inertial reerence rae, such as when the observer is on a sinning lanet! In World War I, the British shis ound this to their ebarrassent. Their engineers had assued that their cannons would be ired onl in the orthern heishere. Unortunatel, the oosite correction or the Coriolis orce should be alied in the Southern heishere. In a battle near the Falkland Islands, British shis ired hundreds o rounds which landed in the water (issing the target b about 00 ards) because o this. 9

6 Eale - How uch rando uncertaint would ou ascribe to the easureent o the length o this aercli? Solution: Figure -3 A centieter ruler held against a sall aercli Probabl the standard uncertaint which ost scientists would ascribe to this easureent is illieter. That is the sallest unit on the scale, and the scale is sall enough such that reading hal o the scale unit is airl diicult. Your range or this easureent would then be 3.0 ± 0. c. Or would it? I chose a aercli or a reason. In this case, the length o the ite is not so well deined. Is the length ro the ae o the end to the ae o the other end? What i the aercli is bent, as the norall are? Is the new length o the aercli the length? Or is the length when the aercli is unurled out o its cli shae? Figure -4 An iage o a aercli illustrating the roble o deinition While it sees like a trivial roble with a aercli, this question is robabl one o the ost diicult or students to gras. Here, soeone asking ou to easure the length o the aercli robabl eans ro ae to ae (3.0 ± 0. c), but that certainl deends on the alication. I the erson reall wanted to ind out how long the aercli is when it is being used (i.e. bent out o shae to accoodate a large stack o aers) then ou have not easured the length which is desired. The task o deterining eactl what easureent is desired is called the roble o deinition. 0

7 Multile Measureents: Deterining the Rando Uncertaint b Statistics In ost cases, it is not a good idea to estiate the uncertaint in a value. I ou have the slightest doubt about how well ou can read a direct easureent, then ou should sil take that easureent ultile ties and use statistics to deterine the uncertaint. You are robabl thinking to oursel: How can I go wrong assuing that the uncertaint is the sallest unit in easuring device? Isn t that alwas range o values? Suose I have a cannon which is at a ied angle. I let ou ire the cannon as an ties as ou wish. I ask ou to redict where the cannonball will land or an given launch. It would not be aroriate to sa that the uncertaint in where the cannonball alls is onl the uncertaint resent in our tae easure ( ). Where the cannonball will land deends uon an actors the elosive orce o the gunowder, the wind resistance, the aount o sin, the ath taken inside the bore o the cannon, aong other things. One illieter ight be the recision ou can easure an ied distance, but that is not what I asked ou to ind. I asked ou to ind the distance to the location where the cannonball will land or an given launch. I can guarantee ou that even the nicest cannon won t land consistentl within a illieter o the target; the sste itsel doesn t allow or that kind o recision. Another eale o how ou can go wrong with a sallest unit rule is the case o a digital stowatch. Man student stowatches can easure to a hundredth o a second. You, on the other hand, cannot easure events to a hundredth o a second. The huan reaction tie, our attention to the event in question, whether ou are anticiating the event all this coes into la when easuring an ties with a stowatch. These cases are all good candidates or using a statistical easure o uncertaint. We can use statistics to deterine the robabilit o inding the value again uon reetitions. With statistics, we can give an uncertaint which reresents the ost robable range o values or that easureent. In order to do statistics, ou need to reeat the easureent over and over. When reeating easureents, the best value, best, is the average o the easureents. The best value o a reeated easureent is the average. best (-) The average is given b equation -. i i 3..., (-) where is the total nuber o ties ou easured the value, is our irst easureent o the value, is our second easureent, and so on.

8 The uncertaint in the individual easureents is given b the standard deviation, or SD or short. The standard deviation reresents the uncertaint in individual easureents. The standard deviation is given b the equation: ( ) i i σ (-3) where is the average o the values, is the total nuber o ties ou easured the value, is our irst easureent o the value, is our second easureent, and so on. The standard deviation is the error in the values which ou have easured. However, it is not the uncertaint in the average value. The uncertaint in the average value is given b the standard deviation o the ean, or SDM or short. Logic suggests that ou can get the average o a bunch o values ore recisel than the values theselves. This is eactl what the standard deviation o the ean tells ou. The standard deviation o the ean reresents the uncertaint in the average easureent. The standard deviation o the ean is given b the equation: σ σ, (-4) where σ is the standard deviation and is the total nuber o ties ou easured the value. Eercise -3 Deterine the standard deviation and the standard deviation o the ean or the ollowing data. Trial Range () Solution: First, we need to ind the average. The average is given b Eq. -. i i ( )

9 To ind the standard deviation, we need to use Eq. -3. σ i. ( ) i ( 38 40) ( 4 40)... ( 40 40) 9 The standard deviation o the ean is given b Eq. -4 σ σ Thereore, the data can be characteried b the range 40. ± 0.3. Probabilit and Statistics: The Gaussian Distribution A histogra is a grah o the nuber o ties ou saw a articular value o a easureent versus the value o the easureent itsel. I ou were aking a histogra o the easureent o a length, would ou eect the histogra to show that ou easured all the values with the sae robabilit? O course not! You would eect that ou will get values close to the average ore oten than ou will get values ar ro the average. For eale, suose ou easured the range o a cannonball s trajector a total o 50 ties and then lotted the histogra. You ight get a grah like the one shown in Fig uber o ties Range () Figure -5 A histogra o a length easured 50 ties Soeties i we take ore easureents, the histogra begins to take on a deinite shae. 3

10 Figure -6 A histogra o a length easured 000 ties. The solid line drawn on to o the histogra is called the liiting distribution o the histogra. A liiting distribution is a sooth, continuous curve which reresents the likel distribution o data as the nuber o easureents aroaches ininit. One te o liiting distribution is the Gaussian distribution. The Gaussian distribution is widel used in an diverse ields, so it goes b an naes; it is also called the Gauss unction, the bell curve, the noral distribution, the noral densit unction, the noral error unction, or soeties sil the Gaussian. The Gaussian distribution is the ost iortant liiting distribution; other liiting distributions oten becoe aroiatel Gaussian under soe conditions. Soe other liiting distributions are the binoial distribution (used b biologists and sociologies to reresent oulation data) and the Poisson distribution (used b article hsicists to reresent article counting data). Deending on our ajor, ou a learn other liiting distributions later on in our college career. In general, a Gaussian unction is an unction o the or ( ) You will encounter the Gaussian again i ou take the last course in this series, quantu echanics, so this unction is hand to reeber. b Ae (-5) When a Gaussian distribution reresents the distribution o values o a rando easureent, it is given in ters o the standard deviation and the average. In this or, it gives the robabilit o easuring the value. The Gaussian distribution tells ou the robabilit o easuring the value. The robabilit, (), is: ( ) ( ) / σ e, (-6) σ π where σ is the standard deviation and is the average o the values. 4

11 The iortant thing that the Gaussian (or an other liiting distribution) tells ou is how robable our easured values are. In articular, the Gaussian relates the robabilit to the standard deviation. This allows us to sa nuericall how conident we are in the value. B integration o the Gaussian unction ro σ to σ, ou can ind what the robabilit is o easuring the value soewhere in this range. This total robabilit turns out to be 68%, which eans that we have a 68% robabilit o inding the value in this range the net tie we easure it, and onl a 3% chance (00% - 68% 3%) o inding soe other value. Table - A table o coonl used statistics o the Gaussian distribution. uber o standard deviations Probabilit o inding the net easured value in this range 3 Probabilit o inding the value outside this range 68.3% 3.7% 95.5% 4.6% %.% % 0.3% Figure -7 The robabilit ranges or a Gaussian distribution. The robabilit that the net easured value will be within standard deviation, _, o the average,, is 68.3%. The robabilit that the net easured value will be within _ o the average is 95.5% (68.3% 3.6% 3.6% 95.5%). This act allows us to deterine whether or not our data its a theor or known value, and allows us to ind the robabilit or which the are coatible. (See the section on Coarison o Values with Uncertaint or ore inoration.) 3 These values are ecerts ro the tables ound in Talor, John R. An Introduction to Error Analsis: The Stud o Uncertainties in Phsical Measureents, nd Ed. (Universit Science Books), Aendi A 5

12 Calculated Values: Deterining the Uncertaint b Proagation Suose ou are tring to ind the area o a table. You easure the length and ou easure the width. Since the length and width are relativel well deined or our regular rectangular table, and ou are airl sure that there are no other errors besides roerl reading our scale on the tae easure, ou estiate uncertaint in the length and width. How uch uncertaint does the area have? To ind the uncertaint in a value which is calculated ro other values which have uncertaint, we ust roagate. The general equation used to ind the uncertaint in a value which results ro the unction (a,b,c,..) is given b Eq. -7. a b c... (-7) a b c In Eq. -7, the sbol _a stands or the uncertaint in the value a, _b stands or the uncertaint in the value b, and so on. For those o ou unailiar with artial derivatives, several less general equations can be derived ro Eq. -7 which cover soe coon situations. Addition and subtraction: I the unction is (a, b, c, ) a ± b ± c ±, the uncertaint in the unction is given b: ( a) ( b) ( )... c (-8) Multilication and division: I the unction is (a, b, c, ) (ab)/c, the uncertaint in the unction is given b: a b c (-9) a b c Powers: I the unction is (a, b, c, ) (a n b )/c, the uncertaint in the unction is given b: a b c n (-0) a b c In all the given equations, -7 to -0, the araeters (a, b, c ) ust be indeendent o each other. In other words, b cannot be a unction o a and vice versa. 6

13 Eale -4 The easured length o a table was 85. ± 0. c, and the width was 59.4 ± 0. c. What is the uncertaint in the area? Solution: We have an area unction, Area length * width Using the ultilication and division orula ields an error in area given b Eq. -. length area area length (85.c)(59.4c) 8c width width 0.c 85.c 0.c 59.4c (-) Thus we would sa that the uncertaint in the area is 8 c. Ssteatic errors and rando errors can both be roagated in this ashion. However, ssteatic errors and rando errors cannot be ied in a roagation the ust be roagated searatel. Reorting Uncertainties It is clear ro our deinition o uncertaint that ever value we easure and ever value we calculate ro easureents will have soe associated uncertaint. Thereore, with ever easured value ou need to reort our uncertaint in that value. The standard wa to reort a value which has uncertaint is best ± _ where the sbol _ eans uncertaint in the value. The best value could be the idoint o an estiated range in the case o estiated errors, an average value or a statistical error, or a roagation or a calculated value. To get the uncertaint in the value, ou need to use one o the reviousl entioned ethods, such as inding the standard deviation o the ean or ultile easureents. In an cases ou can deterine the uncertaint to an decial laces. How an should ou kee? In the eale ro the revious section, should ou reort the area o the table as 358 ± 8 c or 3580 ± 0 c? In general, uncertainties should be rounded to one signiicant igure. 7

14 There are a ew ecetions to this rule. One ecetion is the case in which the leading nuber in the uncertaint is. In this case, it is reasonable to kee the uncertaint to two signiicant igures. Thereore, ou would robabl state the result as 358 ± 8 c. The reason or this is that we are oten interested in the ractional uncertaint. The ractional uncertaint is the uncertaint divided b the value. ractional uncertaint best (-) This ractional uncertaint can be changed into a ercentage b ultiling b 00%. The resulting ercentage is called the ercent uncertaint, the ercent error, or the tolerance o the easureent. For eale, the ercent uncertaint in 0.5 ± 0.5 c is.4% and the ercent uncertaint in 0.3 ± 0. c is.9%. Since the ractional uncertaint changes signiicantl between the choices 0.5 c and 0. c, we could choose to kee two signiicant igures to better reresent the uncertaint. However, the choice reains u to ou. The ercent uncertaint is oten a better easureent o how well the eerient was done than the absolute uncertaint. An absolute uncertaint o 0. c indicates a good easureent o a en o length 8. c (0.5% uncertaint) but it would indicate a oor easureent o the en s width o 0.8 c (3% uncertaint). Measureents can be reorted with their ercent uncertaint as ollows: best, ercentage The uncertaint in a easureent also dictates the nuber o signiicant igures that ou reort. Ater all, signiicant igures are a wa o indicating how an decial laces the value is good to. It would be sill to sa soething like: ± 0. c Even i ou can calculate our decial laces, our uncertaint is saing that ou are onl certain o the irst decial lace. Measureents should be rounded to the sae nuber o decial laces as the uncertaint. Coarison o Values with Uncertaint ow ou have deterined the uncertaint in our easureent. What do ou do with the nuber ou calculated? Recall the eale which started this entire discussion. Two students, who have data which we now know that we can reort as 6000 ± 3000 kg/ 3 and 900 ± 900 kg/ 3, are tring to deterine what kind o etal sale the have. ow that ou know about estiation, statistics, and roagation, ou know how the students were able to give these ranges and how the deterined that these were the 68% robabilit ranges. 8

15 What we want to do is have a standard wa to coare values with uncertaint. For eale, i we, (in Hawaii) sa that our value is coatible with soe theor, we want a scientist in Jaan to understand what we ean b coatible. Using the Gaussian distribution, we can ind the robabilit o inding a articular value o a easureent. Recall that there is a 68.3% robabilit o inding the value within one standard deviation, _, an additional 7.% robabilit o inding the value within two standard deviations, and so on. I we coare this easureent with a theoretical value (or a value easured b soe other eans), these robabilities will also tell us whether or not the values are coatible. I the coared value alls within one standard deviation o the easureent average, then there is a high robabilit that the are coatible, i.e. the easureent agrees with the coared value. I the coared value alls outside one standard deviation but within two standard deviations, there is still a reasonable robabilit that the are coatible (7% robabilit). I the coared value alls between and.5 standard deviations, we sa the result is inconclusive, i.e. that there is a low robabilit that the results are coatible and that urther tests should be done. Outside o.5 standard deviations, the values are considered signiicantl dierent. Table - Coarison results or two values Location o the coared value (in SD) Probabilit that the value is in this range Coarison result Less than 68.7% Coatible values Between to 7.8% Reasonable robabilit that the are coatible Between to.5 3.3% Low robabilit that the are coatible. Inconclusive. Above.5.4% Unlikel that the are coatible. The values are signiicantl dierent. ote that the divisions in Table - are not hard boundaries. There is still soe aount o gre area between the divisions. Phsicists don t debate uch about where eactl such boundaries all. Rather, what we would like is that our coarisons all well within standard deviations i we are to label soething coatible. I we would like to show that soething is incoatible, we would like our coared data to be well outside o.5 standard deviations. Does that ean that ou should tr to get the largest uncertaint ossible? The larger our uncertaint, ater all, the ore likel that the value ou are coaring will it in the high robabilit range. In general, we tr to iniie the uncertaint in our easureents. Let s look at the data ro the densit eerient again to ind out wh. Student (6000± 3000 kg/ 3 ) was in the high robabilit range to the true value o coer, so that student s data is certainl coatible with coer. However, it is also coatible with iron. In order to ind out whether or not it is deinitivel coer, the student needs to have and eerient which shows data which is coatible with coer but incoatible with an other etal. Thus, this student s data is not enough to satis the objective o the lab. Siilarl, the second student (900 ± 900 kg/ 3 ) has also not satisied the objective o the 9

16 lab. The second student cannot successull eclude iron ro the ossible etals, since iron is still within standard deviations o the result. How do ou go about iniiing the uncertaint? In general, ou want to locate the easureent which has the largest ractional (rando) error and reduce it either b increasing the recision o our easureent (e.g. easuring with a calier instead o a centieter ruler) or b increasing the value o the easureent (get a larger sale sie). You should also tr to ind and reduce ssteatic uncertaint. Is there soe resistance in our circuit? Is there air drag acting on the rojectile? Perhas ou can design a calibrating eerient to easure these eects alone so that ou can see how uch the will oset our answer. 0

17 Chater Review Error analsis is an essential ste in an eerient. It tells us how recisel we can give our reorted values, allows us to coare results ro other eerients, tells us the robabilit o getting the sae value uon reeating the eerient, and hels us to deterine where an eerient can be iroved. Error analsis is the eaination o uncertaint in hsical easureents. The error or uncertaint in a value is the range in which ou are reasonabl certain the value lies. There are two tes o uncertaint: rando uncertainties and ssteatic uncertainties Rando errors are arbitrar luctuations in our data. In other words, there is no reerred direction or such errors. Your data values can be greater than or less than the best value without reerence. Ssteatic errors are errors which skew our data in a reerred direction. Ssteatic errors can onl be deterined quantitativel b coaring our easureent to a standard. Rando errors can be estiated i easureents are ade directl (no other sources o error ecet that o using a scale) Rando errors can be calculated b statistics i ultile easureents are ade. The standard deviation reresents the uncertaint in individual easureents. The standard deviation is given b the equation: ( ) i i σ (-3) The standard deviation o the ean reresents the uncertaint in the average easureent. The standard deviation o the ean is given b the equation: σ σ, (-4) The Gaussian distribution tells how robabilit is related to the uncertaint. The robabilit, (), is: ( ) ( ) / σ e, σ π (-6) Both rando and ssteatic errors can be roagated i a calculated value is obtained ro easureents. The general roagation which deterines the error in a unction, _, is a b c... (-7) a b c Addition and subtraction: I the unction is (a, b, c, ) a ± b ± c ±, the uncertaint in the unction is given b:

18 ( a) ( b) ( )... c (-8) Multilication and division: I the unction is (a, b, c, ) (ab)/c, the uncertaint in the unction is given b: a b c (-9) a b c Powers: I the unction is (a, b, c, ) (a n b )/c, the uncertaint in the unction is given b: a b c n (-0) a b c The standard wa to reort a value which has uncertaint is best ± _ or best, ercentage In general, uncertainties should be rounded to one signiicant igure. The ractional uncertaint (-) best Measureents should be rounded to the sae nuber o decial laces as the uncertaint. When coaring results, results which all within standard deviation o another are coatible. Results which all within standard deviations are reasonabl coatible. Results which all between to.5 standard deviations are inconclusive. Results which all outside o.5 standard deviations are signiicantl dierent. Reerence Talor, John R. An Introduction to Error Analsis: The Stud o Uncertainties in Phsical Measureents, nd Ed. (Universit Science Books, CA, 98)

19 Further Eales. A student dros a ball ro a cli o known height and records the tie it takes or the ball to hit the ground. The student s data is given in Table -3. Deterine the uncertaint in the easured value o g, the gravitational acceleration, given that g H/t. Height o cli: 00.0 ± 0. Table -3 Student Data Trial Tie(s) Solution: First, we need to ind the uncertaint in the tie. We need to ind the average, the SD, and inall the SDM. The average tie t t i i ( ) s t t t3... t 0 s The SD The SDM σ t i ( t t ) i ( ) ( )... ( ) s t s σ t σ s 0 s Thus the average tie is 4.59 ± 0.03 s. To ind the uncertaint in the gravitational acceleration, ou need to roagate. Since the orula or g involves a squared ter, ou should use the general orula or the owers orula. 3

20 4 Using the owers orula Using the general orula The error in a unction o the or a n b /c is given b: c c b b a a n Since our orula is g H/t, a, b H, and c t, and all o the owers are ecet or t, our equation becoes: t t H H g g The error in a nuber (such as ) is ero. All the other errors we ound out reviousl, so we can lug in all o the values: s s s t H g So our error in g is 0.3 /s. The error in a unction with indeendent variables a, b, c is given b:... c c b b a a Our orula is g H/t, a H, and b t. The derivatives are 3 4 t H t g t H g I we lug this in, our orula becoes ( ) ( ) ( ) ( ) ( ) s s s s t t H H t g So our error in g is 0.3 /s. otice that our answer is 9.49 ± 0.3 /s. That is signiicantl dierent ro the acceted value (9.8 /s ). I ou were the investigator, what would ou sa about this result?. Mrs. Sith is teaching a 9 th grade hsical science class about calorietr. For an eerient, she asks her students to deterine the seciic heat o a sale in order to deterine what the sale is ade o. She gives her students three otions to choose ro: Material Silver Coer Aluinu Seciic heat, c (kj/kg K) either Mrs. Sith nor her students know how to eror an error analsis, so the coare b deterining which value their answer is closest to. Mrs. Sith knows the sale is ade out o coer. But being a good teacher, she does the eerient hersel, and she is leased to see that her data eactl atches the tetbook answer. Mrs. Sith s data: M HO 63 g, c HO 4.84 kj/kg K, _T HO 3 K, M sale 50 g, _T sale 57 K. So the seciic heat o her sale is:

21 c M M c O( ΔT ) H O ( ΔT ) kgk ( 63g) 4.84kJ ( 3K) H O H sale sale sale kj ( 50g)( 57K) kgk Assue that all o her students correctl ollowed the rocedure, and assue Gaussian statistics holds or her class. Mrs. Sith gives a grade based uon whether or not the student got the right answer. I asses (M) can be read to g, teerature dierences (_T) can be read to K, and c HO is a constant, what aroiate ercentage o her class has she graded unairl? (In other words, what is the robabilit that a student will ind that the aterial is soething other than coer?) Solution: The uncertaint in Mrs. Sith s seciic heat is deterined b the ultilication and division roagation orula: c sale c sale M M H O 0.386kJ kgk 0.9kJ kgk H O ΔT ΔT g 63g H O H O M M K 3K sale sale g 50g ΔT ΔT sale sale K 57K This eans that she should reall have stated her result as 0.4 ± 0. kj/kg K. This ilies that she has a 68% chance o getting within the range 0.3 to 0.5 kj/kg K net tie she does the eerient. Since we can onl get the irst decial lace, let s round all o the values to one decial to get our aroiate robabilities. Coer is aroiatel 0.4 kj/kg K, silver is aroiatel 0. kj/kg K, and 0.9 kj/kg K according to the tet. I the student gets below 0.3 kj/kg K, the student would sa that the value is closer to silver. The robabilit being between 0.3 to 0.5 kj/kg K is 68%. There is a reaining 3% chance to get values outside o this range; hal above 0.5 kj/kg K, and hal below 0.3 kj/kg K. Thus, the robabilit to get below 0.3 is still a signiicant 6%! I the student gets above 0.7 (again rounding), the student would sa that the value is closer to aluinu. That is 3 standard deviations awa ro 0.4 kj/kg K. The robabilit o obtaining a value outside o 3 standard deviations is 0.7%. So there is a negligible robabilit that this will occur. Thus 6% o her class has likel received an incorrect grade. 3. Derive equations (-9), (-0) and (-) ro the general orula (-8). Solution: The general orula or the error in a unction (a,b,c, ) is: a b c a b c... 5

22 6 An addition unction, characteried b (,), has an error roagation as derived below: The derivative b is ( ). The derivative b is ( ). otice that the derivative o ( ) would ield (- ). Thus the error in this unction is: ( ) ( ) ( ) ( ) This is Eq. (-9). otice that subtraction would ield the sae orula A ultilication/division, characteried b (,,) /, has an error roagation as derived below. The derivative b is. The derivative b is. The derivative b is. Thus, the error in this unction is: I ou recall that the unction itsel is (,,) /, the unction can be ulled out o each ter:

23 7 This is the sae as Eq. (-0). otice that the inus sign in the ter is canceled when the ter is squared. A unction o arbitrar owers, characteried b the equation (,,) n /, where, n, and are constants, has an error roagation as derived below. The derivative b is n n. The derivative b is n n n. The derivative b is n n. Thus, the error in this unction is: n n n n I ou recall that the unction itsel is (,,) n /, the unction can be ulled out o each ter: n n This is the sae as Eq. (-). otice that the inus sign in the ter is canceled when the ter is squared.

24 Eercises Tr these eercises on our own. Show our work.. You ind that the eriod o an oscillation, T, is 5. ± 0.05 s. What should ou reort or the requenc? ( /T). You want to easure the diaeter o a round ie. Discuss the uncertainties ou would eect to see. Elain how ou would go about easuring the diaeter. (Would ou estiate? Do ultile trials?) 3. A student is eroring an eerient in which the current in a coil, I, is varied. The student is told to reort the value o the current or which he sees a dot on his cathode ra tube screen (see Table -5). The student reorts that the current at which he sees a dot is.0 a and that the uncertaint in the current is 0. as because the sallest unit on the current eter is 0. as. Do ou agree with the student s assessent? Wh or wh not? Table -4 Student observations o the screen iage at dierent currents Screen iage Current (as) A student uses an inclined lane to accelerate an object. The student does 5 trials, starting at dierent ositions on the lane each tie. Given the student data in Table -5, what is the uncertaint in the acceleration o the object? (a /t ). (Hint: You can onl do the statistical ethod described in this chater when ou are reeating the sae easureent.) Table -5 Student Data Trial Distance (c) Tie (s)

25 9