Introduction to Measurements & Error Analysis

Introduction to Measurements & Error Analsis The Uncertaint o Measurements Some numerical statements are eact: Mar has 3 brothers, and + 4. However, all measurements have some degree o uncertaint that ma come rom a variet o sources. The process o evaluating this uncertaint associated with a measurement result is oten called uncertaint analsis or error analsis. The complete statement o a measured value should include an estimate o the level o conidence associated with the value. Properl reporting an eperimental result along with its uncertaint allows other people to make judgments about the qualit o the eperiment, and it acilitates meaningul comparisons with other similar values or a theoretical prediction. Without an uncertaint estimate, it is impossible to answer the basic scientiic question: Does m result agree with a theoretical prediction or results rom other eperiments? This question is undamental or deciding i a scientiic hpothesis is conirmed or reuted. When we make a measurement, we generall assume that some eact or true value eists based on how we deine what is being measured. While we ma never know this true value eactl, we attempt to ind this ideal quantit to the best o our abilit with the time and resources available. As we make measurements b dierent methods, or even when making multiple measurements using the same method, we ma obtain slightl dierent results. So how do we report our indings or our best estimate o this elusive true value? The most common wa to show the range o values that we believe includes the true value is: measurement best estimate ± uncertaint (units) Let s take an eample. Suppose ou want to ind the mass o a gold ring that ou would like to sell to a riend. You do not want to jeopardize our riendship, so ou want to get an accurate mass o the ring in order to charge a air market price. B simpl eamining the ring in our hand, ou estimate the mass to be between 10 and 0 grams, but this is not a ver precise estimate. Ater some searching, ou ind an electronic balance that gives a mass reading o 17.43 grams. While this measurement is much more precise than the original estimate, how do ou know that it is accurate, and how conident are ou that this measurement represents the true value o the ring s mass? Since the digital displa o the balance is limited to decimal places, ou could report the mass as m 17.43 ± 0.01 g. Suppose ou use the same electronic balance and obtain several more readings: 17.46 g, 17.4 g, 17.44 g, so that the average mass appears to be in the range o 17.44 ± 0.0 g. B now ou ma eel conident that ou know the mass o this ring to the nearest hundredth o a gram, but how do ou know that the true value deinitel lies between 17.43 g and 17.45 g? Since ou want to be honest, ou decide to use another balance that gives a reading o 17. g. This value is clearl below the range o values ound on the irst balance, and under normal circumstances, ou might not care, but ou want to be air to our riend. So what do ou do now? 1

Introduction to Measurements & Error Analsis To help answer these questions, we should irst deine the terms accurac and precision: Accurac is the closeness o agreement between a measured value and a true or accepted value. Measurement error is the amount o inaccurac. Precision is a measure o how well a result can be determined (without reerence to a theoretical or true value). It is the degree o consistenc and agreement among independent measurements o the same quantit; also the reliabilit or reproducibilit o the result. The statement o uncertaint associated with a measurement should include actors that aect both the accurac and precision o the measurement. Caution: Unortunatel the terms error and uncertaint are oten used interchangeabl to describe both imprecision and inaccurac. This usage is so common that it is impossible to avoid entirel. Whenever ou encounter these terms, make sure ou understand whether the reer to accurac or precision, or both. Notice that in order to determine the accurac o a particular measurement, we have to know the ideal, true value, which we reall never do. Sometimes we have a tetbook measured value, which is known precisel, and we assume that this is our ideal value, and use it to estimate the accurac o our result. Other times we know a theoretical value, which is calculated rom basic principles, and this also ma be taken as an ideal value. But phsics is an empirical science, which means that the theor must be validated b eperiment, and not the other wa around. We can escape these diiculties and retain a useul deinition o accurac b assuming that, even when we do not know the true value, we can rel on the best available accepted value with which to compare our eperimental value. For our eample with the gold ring, there is no accepted value with which to compare, and both measured values have the same precision, so we have no reason to believe one more than the other. The onl wa to assess the accurac o the measurement is to compare with a known standard. For this situation, it ma be possible to calibrate the balances with a standard mass that is accurate within a narrow tolerance and is traceable to a primar mass standard at the National Institute o Standards and Technolog (NIST). Calibrating the balances should eliminate the discrepanc between the readings and provide a more accurate mass measurement. Precision is oten reported quantitativel b using relative or ractional uncertaint: Relative Uncertaint uncertaint measured quantit (1) For eample, m 75.5 ± 0.5 g has a ractional uncertaint o: 0.5g 75.5g 0.0066 0.66% Universit o North Carolina

Introduction to Measurements & Error Analsis Accurac is oten reported quantitativel b using relative error: measured value - epected value Relative Error epected value () I the epected value or m is 80.0 g, then the relative error is: 75.5 80.0 0.056 5.6% 80.0 Note: The minus sign indicates that the measured value is less than the epected value. When analzing eperimental data, it is important that ou understand the dierence between precision and accurac. Precision indicates the qualit o the measurement, without an guarantee that the measurement is correct. Accurac, on the other hand, assumes that there is an ideal value, and tells how ar our answer is rom that ideal, right answer. These concepts are directl related to random and sstematic measurement errors. Tpes o Errors Measurement errors ma be classiied as either random or sstematic, depending on how the measurement was obtained (an instrument could cause a random error in one situation and a sstematic error in another). Random errors are statistical luctuations (in either direction) in the measured data due to the precision limitations o the measurement device. Random errors can be evaluated through statistical analsis and can be reduced b averaging over a large number o observations (see standard error). Sstematic errors are reproducible inaccuracies that are consistentl in the same direction. These errors are diicult to detect and cannot be analzed statisticall. I a sstematic error is identiied when calibrating against a standard, appling a correction or correction actor to compensate or the eect can reduce the bias. Unlike random errors, sstematic errors cannot be detected or reduced b increasing the number o observations. Our strateg is to reduce as man sources o error as we can, and then to keep track o those errors that we can t eliminate. It is useul to stud the tpes o errors that ma occur, so that we ma recognize them when the arise. Department o Phsics and Astronom 3

Introduction to Measurements & Error Analsis Common sources o error in phsics laborator eperiments: Incomplete deinition (ma be sstematic or random) - One reason that it is impossible to make eact measurements is that the measurement is not alwas clearl deined. For eample, i two dierent people measure the length o the same rope, the would probabl get dierent results because each person ma stretch the rope with a dierent tension. The best wa to minimize deinition errors is to careull consider and speci the conditions that could aect the measurement. Failure to account or a actor (usuall sstematic) The most challenging part o designing an eperiment is tring to control or account or all possible actors ecept the one independent variable that is being analzed. For instance, ou ma inadvertentl ignore air resistance when measuring ree-all acceleration, or ou ma ail to account or the eect o the Earth s magnetic ield when measuring the ield o a small magnet. The best wa to account or these sources o error is to brainstorm with our peers about all the actors that could possibl aect our result. This brainstorm should be done beore beginning the eperiment so that arrangements can be made to account or the conounding actors beore taking data. Sometimes a correction can be applied to a result ater taking data to account or an error that was not detected. Environmental actors (sstematic or random) - Be aware o errors introduced b our immediate working environment. You ma need to take account or or protect our eperiment rom vibrations, drats, changes in temperature, and electronic noise or other eects rom nearb apparatus. Instrument resolution (random) - All instruments have inite precision that limits the abilit to resolve small measurement dierences. For instance, a meter stick cannot distinguish distances to a precision much better than about hal o its smallest scale division (0.5 mm in this case). One o the best was to obtain more precise measurements is to use a null dierence method instead o measuring a quantit directl. Null or balance methods involve using instrumentation to measure the dierence between two similar quantities, one o which is known ver accuratel and is adjustable. The adjustable reerence quantit is varied until the dierence is reduced to zero. The two quantities are then balanced and the magnitude o the unknown quantit can be ound b comparison with the reerence sample. With this method, problems o source instabilit are eliminated, and the measuring instrument can be ver sensitive and does not even need a scale. Failure to calibrate or check zero o instrument (sstematic) - The calibration o an instrument should be checked beore taking data whenever possible. I a calibration standard is not available, the accurac o the instrument should be checked b comparing with another instrument that is at least as precise, or b consulting the technical data provided b the manuacturer. When making a measurement with a micrometer, electronic balance, or an electrical meter, alwas check the zero reading irst. Re-zero the instrument i possible, or measure the displacement o the zero reading rom the true zero and correct an measurements accordingl. It is a good idea to check the zero reading throughout the eperiment. Phsical variations (random) - It is alwas wise to obtain multiple measurements over the entire range being investigated. Doing so oten reveals variations that might otherwise go undetected. These variations ma call or closer eamination, or the ma be combined to ind an average value. Paralla (sstematic or random) - This error can occur whenever there is some distance between the measuring scale and the indicator used to obtain a measurement. I the observer s ee is not squarel 4 Universit o North Carolina

Introduction to Measurements & Error Analsis aligned with the pointer and scale, the reading ma be too high or low (some analog meters have mirrors to help with this alignment). Instrument drit (sstematic) - Most electronic instruments have readings that drit over time. The amount o drit is generall not a concern, but occasionall this source o error can be signiicant and should be considered. Lag time and hsteresis (sstematic) - Some measuring devices require time to reach equilibrium, and taking a measurement beore the instrument is stable will result in a measurement that is generall too low. The most common eample is taking temperature readings with a thermometer that has not reached thermal equilibrium with its environment. A similar eect is hsteresis where the instrument readings lag behind and appear to have a memor eect, as data are taken sequentiall moving up or down through a range o values. Hsteresis is most commonl associated with materials that become magnetized when a changing magnetic ield is applied. Personal errors come rom carelessness, poor technique, or bias on the part o the eperimenter. The eperimenter ma measure incorrectl, or ma use poor technique in taking a measurement, or ma introduce a bias into measurements b epecting (and inadvertentl orcing) the results to agree with the epected outcome. Gross personal errors, sometimes called mistakes or blunders, should be avoided and corrected i discovered. As a rule, gross personal errors are ecluded rom the error analsis discussion because it is generall assumed that the eperimental result was obtained b ollowing correct procedures. The term human error should also be avoided in error analsis discussions because it is too general to be useul. Estimating Eperimental Uncertaint or a Single Measurement An measurement ou make will have some uncertaint associated with it, no matter how precise our measuring tool. How do ou actuall determine the uncertaint, and once ou know it, how do ou report it? The uncertaint o a single measurement is limited b the precision and accurac o the measuring instrument, along with an other actors that might aect the abilit o the eperimenter to make the measurement. For eample, i ou are tring to use a meter stick to measure the diameter o a tennis ball, the uncertaint might be ± 5 mm, but i ou used a Vernier caliper, the uncertaint could be reduced to mabe ± mm. The limiting actor with the meter stick is paralla, while the second case is limited b ambiguit in the deinition o the tennis ball s diameter (it s uzz!). In both o these cases, the uncertaint is greater than the smallest divisions marked on the measuring tool (likel 1 mm and 0.1 mm respectivel). Unortunatel, there is no general rule or determining the uncertaint in all measurements. The eperimenter is the one who can best evaluate and quanti the uncertaint o a measurement based on all the possible actors that aect the result. Thereore, the person making the Department o Phsics and Astronom 5

Introduction to Measurements & Error Analsis measurement has the obligation to make the best judgment possible and report the uncertaint in a wa that clearl eplains what the uncertaint represents: Measurement (measured value ± standard uncertaint) unit o measurement where the ± standard uncertaint indicates approimatel a 68% conidence interval (see sections on Standard Deviation and Reporting Uncertainties). Eample: Diameter o tennis ball 6.7 ± 0. cm Estimating Uncertaint in Repeated Measurements Suppose ou time the period o oscillation o a pendulum using a digital instrument (that ou assume is measuring accuratel) and ind: T 0.44 seconds. This single measurement o the period suggests a precision o ±0.005 s, but this instrument precision ma not give a complete sense o the uncertaint. I ou repeat the measurement several times and eamine the variation among the measured values, ou can get a better idea o the uncertaint in the period. For eample, here are the results o 5 measurements, in seconds: 0.46, 0.44, 0.45, 0.44, 0.41. The best estimate o the period is the average, or mean, o these N independent measurements: 1 + +... + Average(mean) N Whenever possible, repeat a measurement several times and average the results. This average is the best estimate o the true value. The more repetitions ou make o a measurement, the better this estimate will be. Consider, as another eample, the measurement o the width o a piece o paper using a meter stick. Being careul to keep the meter stick parallel to the edge o the paper (to avoid a sstematic error which would cause the measured value to be consistentl higher than the correct value), the width o the paper is measured at a number o points on the sheet, and the values obtained are entered in a data table. Note that the last digit is onl a rough estimate, since it is diicult to read a meter stick to the nearest tenth o a millimeter (0.01 cm). Observation Width (cm) #1 31.33 # 31.15 #3 31.6 #4 31.0 #5 31.0 N Average sum o observed widths no. o obs ervations 155.96 cm 5 31.19 cm 6 Universit o North Carolina

Introduction to Measurements & Error Analsis This average is the best available estimate o the width o the piece o paper, but it is certainl not eact. We would have to average an ininite number o measurements to approach the true mean value, and even then, we are not guaranteed that the mean value is accurate because there is still some sstematic error rom the measuring tool, which can never be calibrated perectl. So how do we epress the uncertaint in our average value? One wa to epress the variation among the measurements is to use the average deviation. This statistic tells us on average (with 50% conidence) how much the individual measurements var rom the mean. 1 + +... + N AverageDeviation, d N However, the standard deviation is the most common wa to characterize the spread o a data set. The standard deviation is alwas slightl greater than the average deviation, and is used because o its association with the normal distribution that is requentl encountered in statistical analses. Standard Deviation To calculate the standard deviation or a sample o 5 (or more generall N) measurements: 1. Sum all the measurements and divide b 5 to get the average, or mean.. Now, subtract this average rom each o the 5 measurements to obtain 5 deviations. 3. Square each o these 5 deviations and add them all up. 4. Divide this result b (N-1) and take the square root. We can write out the ormula or the standard deviation as ollows. Let the N measurements be called 1,,..., N. Let the average o the N values be called. Then each deviation is given b δ i i, or i 1,,..., N. The standard deviation is: s ( δ + δ +... + δ ) 1 N δ i ( N 1) ( N 1) In our previous eample, the average width is 31.19 cm. The deviations are: Observation Width (cm) Deviation (cm) #1 31.33 +0.14 31.33-31.19 # 31.15-0.04 31.15-31.19 #3 31.6 +0.07 31.6-31.19 #4 31.0-0.17 31.0-31.19 #5 31.0 +0.01 31.0-31.19 The average deviation is: d 0.086 cm Department o Phsics and Astronom 7

Introduction to Measurements & Error Analsis The standard deviation is: s (0.14) + (0.04) + (0.07) 5 1 + (0.01) 0.1 cm The signiicance o the standard deviation is this: i ou now make one more measurement using the same meter stick, ou can reasonabl epect (with about 68% conidence) that the new measurement will be within 0.1 cm o the estimated average o 31.19 cm. In act, it is reasonable to use the standard deviation as the uncertaint associated with this single new measurement. However, the uncertaint o the average value is the standard deviation o the mean, which is alwas less than the standard deviation. Consider an eample where 100 measurements o a quantit were made. The average or mean value was 10.5 and the standard deviation was s 1.83. The igure below is a histogram o the 100 measurements, which shows how oten a certain range o values was measured. For eample, in 0 o the measurements, the value was in the range 9.5 to 10.5, and most o the readings were close to the mean value o 10.5. The standard deviation s or this set o measurements is roughl how ar rom the average value most o the readings ell. For a large enough sample, approimatel 68% o the readings will be within one standard deviation o the mean value, 95% o the readings will be in the interval ± s, and nearl all (99.7%) o readings will lie within 3 standard deviations rom the mean. The smooth curve superimposed on the histogram is the gaussian or normal distribution predicted b theor or measurements involving random errors. As more and more measurements are made, the histogram will more closel ollow the bell-shaped gaussian curve, but the standard deviation o the distribution will remain approimatel the same. s ± 1s ± s ± 3s 8 Universit o North Carolina

Introduction to Measurements & Error Analsis Standard Deviation o the Mean (Standard Error) When we report the average value o N measurements, the uncertaint we should associate with this average value is the standard deviation o the mean, oten called the standard error (SE). Standard Deviation o the Mean, or Standard Error (SE), s N (3) The standard error is smaller than the standard deviation b a actor o1 N. This relects the act that we epect the uncertaint o the average value to get smaller when we use a larger number o measurements, N. In the previous eample, we ind the standard error is 0.05 cm, where we have divided the standard deviation o 0.1 b 5. The inal result should then be reported as: Average paper width 31.19 ± 0.05 cm Anomalous Data The irst step ou should take in analzing data (and even while taking data) is to eamine the data set as a whole to look or patterns and outliers. Anomalous data points that lie outside the general trend o the data ma suggest an interesting phenomenon that could lead to a new discover, or the ma simpl be the result o a mistake or random luctuations. In an case, an outlier requires closer eamination to determine the cause o the unepected result. Etreme data should never be thrown out without clear justiication and eplanation, because ou ma be discarding the most signiicant part o the investigation! However, i ou can clearl justi omitting an inconsistent data point, then ou should eclude the outlier rom our analsis so that the average value is not skewed rom the true mean. Fractional Uncertaint Revisited When a reported value is determined b taking the average o a set o independent readings, the ractional uncertaint is given b the ratio o the uncertaint divided b the average value. For this eample, Fractional uncertaint uncertaint average 0.05 cm 0.0016 0.00 31.19 cm Note that the ractional uncertaint is dimensionless (the uncertaint in cm was divided b the average in cm). An eperimental phsicist might make the statement that this measurement is good to about 1 part in 500 or precise to about 0.%. The ractional uncertaint is also important because it is used in propagating uncertaint in calculations using the result o a measurement, as discussed in the net section. Department o Phsics and Astronom 9

Introduction to Measurements & Error Analsis Propagation o Uncertaint Suppose we want to determine a quantit, which depends on and mabe several other variables, z,etc. We want to know the error in i we measure,,... with errors,, Eamples: (Area o a rectangle) p cosθ (-component o momentum) / t (velocit) For a single-variable unction (), the deviation in can be related to the deviation in using calculus: Thus, taking the square and the average: d δ δ d and using the deinition o, we get: δ d d δ Eamples: a) b) cosθ d d 1, or d sinθ dθ d d sinθ θ, or tanθ θ Note: in this situation, θ must be in radians. 10 Universit o North Carolina

Introduction to Measurements & Error Analsis Department o Phsics and Astronom 11 In the case where depends on two or more variables, the derivation above can be repeated with minor modiication. For two variables, (, ), we have: δ δ δ + The partial derivative means dierentiating with respect to holding the other variables ied. Taking the square and the average, we get the law o propagation o uncertaint: δ δ δ δ δ + + ) ( ) ( ) ( (4) I the measurements o and are uncorrelated, then 0 δ δ, and using the deinition o, we get: + Eamples: a) + 1, 1 + When adding (or subtracting), the absolute uncertaint o the sum (or dierence) is the rss o the individual absolute uncertainties. Adding or subtracting a constant does not change the absolute uncertaint o the value. b), + Dividing the above equation b, we get: +

Introduction to Measurements & Error Analsis c) / 1, 1 + Dividing the above equation b /, we get: + When multipling (or dividing), the relative uncertaint o the product (quotient) is the rss o the individual relative uncertainties. Multipling or dividing b a constant does not change the relative uncertaint o the value. Note that the relative uncertaint in, as shown in (b) and (c) above, has the same orm or multiplication and division: the relative uncertaint in a product or quotient is the square root o the sum o the squares o the relative uncertaint o each individual term, as long as the terms are not correlated. Eample: Find uncertaint in v, where v v a a t + t.1 9.8 v at with a 9.8 ± 0.1 m/s, t 1. ± 0.1s 0 0.1 + 3.4 ( 0.010) + ( 0.09) 0.031 or 3.1% Notice that since the relative uncertaint in t (.9%) is signiicantl greater than the relative uncertaint or a (1.0%), the relative uncertaint in v is essentiall the same as or t (about 3%). Graphicall, this is like the Pthagorean theorem: 3.1% The total uncertaint is the length o the hpotenuse o a right triangle 1.0% with legs the length o each uncertaint component..9% Timesaving approimation: A chain is onl as strong as its weakest link. I one o the uncertaint terms is more than 3 times greater than the other terms, the root-squares ormula can be skipped, and the combined uncertaint is simpl the largest uncertaint. This shortcut can save a lot o time without losing an accurac in the estimate o the overall uncertaint. 1 Universit o North Carolina

Introduction to Measurements & Error Analsis The Upper-Lower Bound Method o Uncertaint Propagation An alternative, and sometimes simpler procedure, to the tedious propagation o uncertaint law is the upper-lower bound method o uncertaint propagation. This alternative method does not ield a standard uncertaint estimate (with a 68% conidence interval), but it does give a reasonable estimate o the uncertaint or practicall an situation. The basic idea o this method is to use the uncertaint ranges o each variable to calculate the maimum and minimum values o the unction. You can also think o this procedure as eamining the best and worst case scenarios. For eample, i ou took an angle measurement: θ 5 ± 1 and ou needed to ind cosθ, then: ma cos(6 ) 0.8988 min cos(4 ) 0.9135 0.906 ± 0.007 Note that even though θ was onl measured to signiicant igures, is known to 3 igures. B using the propagation o uncertaint law: sinθ θ (0.43)(π/180) 0.003 The uncertaint estimate rom the upper-lower bound method is generall larger than the standard uncertaint estimate ound rom the propagation o uncertaint law. The upper-lower bound method is especiall useul when the unctional relationship is not clear or is incomplete. One practical application is orecasting the epected range in an epense budget. In this case, some epenses ma be ied, while others ma be uncertain, and the range o these uncertain terms could be used to predict the upper and lower bounds on the total epense. Signiicant Figures The number o signiicant igures in a value can be deined as all the digits between and including the irst non-zero digit rom the let, through the last digit. For instance, 0.44 has two signiicant igures, and the number 66.770 has 5 signiicant igures. Zeroes are signiicant ecept when used to locate the decimal point, as in the number 0.00030, which has signiicant igures. Zeroes ma or ma not be signiicant or numbers like 100, where it is not clear whether two, three, or our signiicant igures are indicated. To avoid this ambiguit, such numbers should be epressed in scientiic notation to (e.g. 1. 10 3 clearl indicates two signiicant igures). When using a calculator, the displa will oten show man digits, onl some o which are meaningul (signiicant in a dierent sense). For eample, i ou want to estimate the area o a circular plaing ield, ou might pace o the radius to be 9 meters and use the ormula area πr. When ou compute this area, the calculator might report a value o 54.4690049 m. It would be etremel misleading to report this number as the area o the ield, because it would suggest that ou know the area to an absurd degree o precision - to within a raction o a square millimeter! Since the radius is onl known to one signiicant igure, the inal answer should also contain onl one signiicant igure. A more truthul answer would be to report the area as 300 m ; however, this Department o Phsics and Astronom 13

Introduction to Measurements & Error Analsis ormat is somewhat misleading, since it could be interpreted to have three signiicant igures because o the zeroes. The better wa to report the number would be to use scientiic notation: 3 10 m. From this eample, we can see that the number o signiicant igures reported or a value implies a certain degree o precision. In act, the number o signiicant igures suggests a rough estimate o the relative uncertaint: The number o signiicant igures implies an approimate relative uncertaint 1 signiicant igure suggests a relative uncertaint o about 10% to 100% signiicant igures suggest a relative uncertaint o about 1% to 10% 3 signiicant igures suggest a relative uncertaint o about 0.1% to 1% To understand this connection more clearl, consider a value with signiicant igures, like 99, which suggests an uncertaint o ±1, or a relative uncertaint o ±1%. (Actuall some people might argue that the implied uncertaint in 99 is ±0.5 since the range o values that would round to 99 are 98.5 to 99.4. But since the uncertaint here is onl a rough estimate, there is not much point arguing about the actor o two.) The smallest -signiicant igure number, 10, also suggests an uncertaint o ±1, which in this case is a relative uncertaint o ±10%. The ranges or other numbers o signiicant igures can be reasoned in a similar manner. Use o Signiicant Figures or Simple Propagation o Uncertaint B ollowing a ew simple rules, signiicant igures can be used to ind the appropriate precision or a calculated result or the our most basic math unctions, all without the use o complicated ormulas or propagating uncertainties. For multiplication and division, the number o signiicant igures that are reliabl known in a product or quotient is the same as the smallest number o signiicant igures in an o the original numbers. Eample: 6.6 ( signiicant igures) 738.7 (5 signiicant igures) 48369.4 48 10 3 ( signiicant igures) For addition and subtraction, the result should be rounded o to the last decimal place reported or the least precise number. Eamples: 3.64 5560.5 +54 + 0.008 78 5560.5 14 Universit o North Carolina

Introduction to Measurements & Error Analsis I a calculated number is to be used in urther calculations, it is good practice to keep one etra digit to reduce rounding errors that ma accumulate. Then the inal answer should be rounded according to the above guidelines. Uncertaint and Signiicant Figures For the same reason that it is dishonest to report a result with more signiicant igures than are reliabl known, the uncertaint value should also not be reported with ecessive precision. For eample, i we measure the densit o copper, it would be unreasonable to report a result like: measured densit 8.93 ± 0.4753 g/cm 3 WRONG! The uncertaint in the measurement cannot be known to that precision. In most eperimental work, the conidence in the uncertaint estimate is not much better than about ±50% because o all the various sources o error, none o which can be known eactl. Thereore, to be consistent with this large uncertaint in the uncertaint (!) the uncertaint value should be stated to onl one signiicant igure (or perhaps sig. igs. i the irst digit is a 1). Eperimental uncertainties should be rounded to one, or at most two, signiicant igures. To help give a sense o the amount o conidence that can be placed in the standard deviation, the ollowing table indicates the relative uncertaint associated with the standard deviation or various sample sizes. Note that in order or an uncertaint value to be reported to 3 signiicant igures, more than 10,000 readings would be required to justi this degree o precision! N Relative Uncert.* Sig.Figs. Valid Implied Uncertaint 71% 1 ± 10% to 3 50% 1 ± 10% to 4 41% 1 ± 10% to 5 35% 1 ± 10% to 10 4% 1 ± 10% to 0 16% 1 ± 10% to 30 13% 1 ± 10% to 50 10% ± 1% to 10 7% ± 1% to 10 0.7% 3 ± 0.1% to *The relative uncertaint is given b the approimate ormula: 1 ( N 1) When an eplicit uncertaint estimate is made, the uncertaint term indicates how man signiicant igures should be reported in the measured value (not the other wa around!). For eample, the uncertaint in the densit measurement above is about 0.5 g/cm 3, so this tells us that the Department o Phsics and Astronom 15

Introduction to Measurements & Error Analsis digit in the tenths place is uncertain, and should be the last one reported. The other digits in the hundredths place and beond are insigniicant, and should not be reported: measured densit 8.9 ± 0.5 g/cm 3 RIGHT! An eperimental value should be rounded to an appropriate number o signiicant igures consistent with its uncertaint. This generall means that the last signiicant igure in an reported measurement should be in the same decimal place as the uncertaint. In most instances, this practice o rounding an eperimental result to be consistent with the uncertaint estimate gives the same number o signiicant igures as the rules discussed earlier or simple propagation o uncertainties or adding, subtracting, multipling, and dividing. Caution: When conducting an eperiment, it is important to keep in mind that precision is epensive (both in terms o time and material resources). Do not waste our time tring to obtain a precise result when onl a rough estimate is required. The cost increases eponentiall with the amount o precision required, so the potential beneit o this precision must be weighed against the etra cost. Combining and Reporting Uncertainties In 1993, the International Standards Organization (ISO) published the irst oicial worldwide Guide to the Epression o Uncertaint in Measurement. Beore this time, uncertaint estimates were evaluated and reported according to dierent conventions depending on the contet o the measurement or the scientiic discipline. Here are a ew ke points rom this 100-page guide, which can be ound in modiied orm on the NIST website (see Reerences). When reporting a measurement, the measured value should be reported along with an estimate o the total combined standard uncertaint o the value. The total uncertaint is ound b combining the uncertaint components based on the two tpes o uncertaint analsis: Tpe A evaluation o standard uncertaint method o evaluation o uncertaint b the statistical analsis o a series o observations. This method primaril includes random errors. Tpe B evaluation o standard uncertaint method o evaluation o uncertaint b means other than the statistical analsis o series o observations. This method includes sstematic errors and an other uncertaint actors that the eperimenter believes are important. The individual uncertaint components should be combined using the law o propagation o uncertainties, commonl called the root-sum-o-squares or RSS method. When this is done, the combined standard uncertaint should be equivalent to the standard deviation o the result, making this uncertaint value correspond with a 68% conidence interval. I a wider conidence interval is desired, the uncertaint can be multiplied b a coverage actor (usuall k or 3) to provide an uncertaint range that is believed to include the true value with a conidence o 95% or 99.7% respectivel. I a coverage actor is used, there should be a clear eplanation o its meaning so there is no conusion or readers interpreting the signiicance o the uncertaint value. You should be aware that the ± uncertaint notation might be used to indicate dierent conidence intervals, depending on the scientiic discipline or contet. For eample, a public opinion 16 Universit o North Carolina

Introduction to Measurements & Error Analsis poll ma report that the results have a margin o error o ±3%, which means that readers can be 95% conident (not 68% conident) that the reported results are accurate within 3 percentage points. In phsics, the same average result would be reported with an uncertaint o ±1.5% to indicate the 68% conidence interval. Conclusion: When do measurements agree with each other? We now have the resources to answer the undamental scientiic question that was asked at the beginning o this error analsis discussion: Does m result agree with a theoretical prediction or results rom other eperiments? Generall speaking, a measured result agrees with a theoretical prediction i the prediction lies within the range o eperimental uncertaint. Similarl, i two measured values have standard uncertaint ranges that overlap, then the measurements are said to be consistent (the agree). I the uncertaint ranges do not overlap, then the measurements are said to be discrepant (the do not agree). However, ou should recognize that these overlap criteria can give two opposite answers depending on the evaluation and conidence level o the uncertaint. It would be unethical to arbitraril inlate the uncertaint range just to make a measurement agree with an epected value. A better procedure would be to discuss the size o the dierence between the measured and epected values within the contet o the uncertaint, and tr to discover the source o the discrepanc i the dierence is trul signiicant. To eamine our own data, tr out the Measurement Comparison tool available on the lab website: http://www.phsics.unc.edu/labs Measurements and their uncertainties 0 0.5 1 1.5.5 A B A 1. ± 0.4 B 1.8 ± 0.4 These measurements agree within their uncertainties, despite the act that the percent dierence between their central values is 40%. These measurements do not agree since their uncertainties do not overlap. Further investigation would be needed to determine the cause or the discrepanc. Perhaps the uncertainties were underestimated, there was a sstematic error that was not detected, or the dierence ma be real! Measurements and their uncertainties 0 0.5 1 1.5.5 A B Department o Phsics and Astronom 17

Introduction to Measurements & Error Analsis Reerences: Talor, John. An Introduction to Error Analsis, nd. ed. Universit Science Books: Sausalito, 1997. Baird, D.C. Eperimentation: An Introduction to Measurement Theor and Eperiment Design, 3 rd. ed. Prentice Hall: Englewood Clis, 1995. Lichten, William. Data and Error Analsis., nd. ed. Prentice Hall: Upper Saddle River, NJ, 1999. Bevington, Phillip and Robinson, D. Data Reduction and Error Analsis or the Phsical Sciences, nd. ed. McGraw-Hill: New York, 1991. ISO. Guide to the Epression o Uncertaint in Measurement. International Organization or Standardization (ISO) and the International Committee on Weights and Measures (CIPM): Switzerland, 1993. NIST. Essentials o Epressing Measurement Uncertaint. http://phsics.nist.gov/cuu/uncertaint/ 18 Universit o North Carolina