13 ERRORS, MEANS AND FITS

Transcription

1 3 ERRORS, MEAS AD FITS Ths chapter contans a chose of essental nformaton on the analyss of measurng uncertantes, and the treatment of large numbers of data. Startng wth a dscusson on "errors", we arrve at a technque of judgng the probablty of the truth of a fnal result. 3. ERRORS In dctonares the word error s defned as the dfference between the approxmated value as the result of an observaton or measurement, or a calculaton- and the true value. The problem s that n general we do not know the "true value", as ths s generally the result of a measurement or calculaton. Therefore, we have to fnd a way of estmatng the "relablty" of our result. The name "errors" s not well specfed as such. We thus have to become more specfc about the defnton. Errors can be classfed as follows: ) Blunders or mstakes n measurement or calculaton are usually apparent as beng far from expectatons. They are to be dealt wth by repeatng the measurement or calculaton. ) Systematc errors are more dffcult to detect. They are reproducble dscrepances, often the result of a falure n the nstrumentaton or a consstent mathematcal nsuffcency. They are to be found (and corrected for!) by repeatng the analyss wth dfferent equpment or by recalculaton (by a colleague or by dfferent means). 3) Random errors are the most common type of errors. They are the result of the unavodably lmted qualty of our nstrumentaton. They can only be partly overcome by refnng the nstrumentaton or the analytcal method, and by repeatng the measurements (such as readng a temperature or ph) or extendng the observaton tme (for nstance, of radoactvty). 3. PRECISIO AD ACCURACY 3.. DEFIITIOS It s mportant to dstngush between precson of and accuracy. ) The precson of a result s a measure of the reproducblty of an observaton, of how well the result can be determned, rrespectve of how close the result s to the "true" value. The assocated "error" s better referred to as the uncertanty of a result. 43

2 Chapter 3 ) The accuracy s a measure of the correctness of an observaton, of how close the result s to the "true" value. The two sets of defntons we have gven are smply related. Precson s a measure of the sze of random errors. If we are able to reduce random errors, for nstance by better equpment or procedures, the precson of the measurement s better, the result s more precse, and the analyss s more reproducble. Increasng the precson by reducng random errors s a task of each sngle laboratory. On the other hand, a systematc error drectly effects the accuracy of the measurement; avodng or elmnatng systematc errors makes the result more accurate and trustworthy. Increasng the accuracy of a result s often the goal of nternatonal ntercomparsons by a number of laboratores analysng the same set of samples, and by routnely runnng standards. For studyng and eventually reducng systematc errors t s mportant to dspose of data wth small random errors, wth relatvely hgh precson. On the other hand, t s a waste to spend much effort on ncreasng the precson, f the systematc error s large. Fg.3. llustrates the dfference between precson and accuracy. 3.. SIGIFICAT FIGURES AD DIGITS When reportng numbers t s a common rule to ndcate the uncertanty by the fgures and dgts of the number gven. If a dstance s gven as 5000 km, t s common sense to trust only the leftmost fgure. However, f one s also certan about the next fgure (the leftmost 0), t s better to wrte km. In general, t s preferable to wrte numbers n the scentfc notaton,.e. an argument n decmal notaton wth a number of dgts, multpled by a power of 0. The rghtmost dgt contans the uncertanty. As a rule the precson of the uncertanty (.e. the degree of certanty of the uncertanty) s not better than 0% of the uncertanty. For example, f the radoactvty of a sample s determned as 3.56 Bq, t may go wth an uncertanty of or , but gvng an addtonal dgt n the uncertanty lke would exaggerate the "certanty of the uncertanty". The uncertanty also determnes the number of dgts quoted. For nstance, t s rght to quote Bq, but t s not consstent to wrte Bq. In computer calculatons all dgts are to be kept; roundng off s done only wth the fnal result. evertheless, results wrtten down n the course of a mathematcal calculaton are to be gven wth the number of dgts that can be justfed. The entre calculaton tself, however, s to be carred out wthout ntermedate roundng off. 44

3 Errors, Means and Fts A B Fg.3. Example to llustrate precson and accuracy, showng two seres of results from 9 measurements of the same radoactvty. A. The data are mprecse but accurate, gvng the proper average value of 3.56 Bq. The grey area refers to the confdence level;.e. 68% of the data should be wthn ths range (example n Sect.3.5.). B. The data are precse, but naccurate probably because of a systematc error, as the average value s now 3.50 Bq, nstead of the "true" value of 3.56 Bq UCERTAITIES There are two dfferent types of uncertantes. ) Instrumental uncertantes, due to fluctuatons n the result of any nstrumental observaton, whether t concerns measurng the outdoor temperature or weghng your letter on a letter balance, or applyng fancy equpment to measure tme n the laboratory. 45

4 Chapter 3 An estmate of the sze of the uncertanty can be obtaned by an "educated guess", or by repeatng the measurement and observe the dstrbuton of the results. ) Statstcal uncertantes, due to the fact that certan processes even theoretcally show fluctuatons. A proper example here s the process of radoactve decay. Even (nonexstng) deal equpment would observe the fluctuaton n the actvty measurement, or the "statstcal spread" n the results. In cases lke these procedures exst to determne the uncertanty beyond doubt. 3.3 ISTRUMETAL UCERTAITIES 3.3. MEA VALUES The mean or average value of the result of a number of measurements s defned as the sum of the results dvded by the number of measurements: x x (x x x 3 x ) / (3.) s the number of measurements, stands for the seral number of an arbtrary measurement and x s the parameter measured. We wll often omt and =, and smply wrte x. The number of measurements s always lmted. However, f we would be able to ncrease ths number to nfnty, we would end up wth a better value of the mean, then defned as lm ( x ) (3.) The medan s now defned as the value of the set of data such that the results of half the measurements s smaller, half s larger than the medan value. For a symmetrcal dstrbuton the mean and the medan values are equal. We wll later use the devatons of a sngle result from the mean (or medan), x x ; by defnton the average devaton of the results from the mean value equals zero: x x (x x) x x x x 0 (3.3) 3.3. DISTRIBUTIO OF DATA Results of a number of measurements can be presented n a hstogram, a graph showng the number of tmes (y-axs) certan results ndcated along the x-axs were obtaned (Fg.3.). It s obvous that the probablty of obtanng results further removed from the most frequent 46

5 Errors, Means and Fts P 99.7% 95% 68% (x x)/ Fg.3. Hstogram (block-shaped), ndcatng the rregular dstrbuton of measurement results wthn a range of x (x).e. between x and x + x around a mean value; the devatons from the mean value (x ) rather than the actual results are gven along wth the number of observatons (y-axs) for values wthn a certan range. The smooth curve llustrates the Gaussan dstrbuton, whch s the hypothetcal result of an nfnte number of measurements. It also represents the probablty dstrbuton (P) of data around the mean value. The devatons from the average value are gven n terms of the standard devaton (). On top of the graph the ntegral or summed probabltes are shown: the probablty of observng values between x + and x s 68 %, between x + and x the chance s 95 % and fnally between x + 3 and x %. result s less and less probable. The hstogram (or block dagram) consstng of columns representng the number of tmes ( ) results x (x) wthn a certan range x and x + x have been observed, s referred to as the sample dstrbuton. The mean value s: and x x ( = x) (3.4) 47

6 Chapter 3 If n makng the hstogram x or the class wdth s chosen to be too large, nearly all data may be wthn one column, suggestng a good statstcal certanty, but a bad resoluton; f x s made too small the resoluton s ncreased, but few data fall wthn one column and the relablty appears to be small (scatterng hstogram). The more measurements have been carred out, the better s the mpresson of the dstrbuton of data around a certan mean value. For an nfnte number of results wth random errors the sample dstrbuton s represented by a bell-shaped normal or Gaussan dstrbuton, n whch the probablty of observng a value of y = y at x = x s: y f (x ) P exp (3.5) y s the measured value of the dependent varable y, f(x ) s the y value calculated for the value of the ndependent varable x, s the standard devaton of y, to be defned later. The most probable value to be observed, the mode, corresponds to the peak of the dstrbuton,.e. to the top of the smooth curve. For data wth random errors the dstrbuton s symmetrcal around the top. Fg.3. shows the Gaussan curve together wth the hstogram resultng from a lmted number of measurements STADARD DEVIATIO PRECISIO OF DATA It s obvous that f random errors are small, the values of the devaton (x x) are small and the dstrbuton of the results around the mean s narrower. The average devaton s a measure of the spread of the data around the mean or the so-called dsperson of the data set. Eq.3.3 has shown that we can not use the smple average of all devatons, a consequence of the defnton of the mean. The average of the absolute values of the devatons,.e. rrespectve of ther sgn, characterses the dsperson better: d x x (3.6) For mathematcal reasons however, usng absolute values s not approprate. Therefore, the squares of the devatons are taken to characterse the dstrbuton. The resultng value s referred to as the varance: lm (x ) lm x (3.7) 48

7 Errors, Means and Fts The varance s thus the average of the squares mnus the square of the averages. The quanttatve measure of the sze of the random errors,.e. of the statstcal spread of the data around the mean, or n other words of the precson s now gven by the standard devaton, whch s the square root of the varance. The smaller the standard devaton, the better the precson, the narrower the Gaussan curve. If we now consder the real set of measurements, the standard devaton of the set s: (x x) (3.8) The fact that nstead of s nserted n the denomnator s dscussed n standard texts on statstcal analyss. The necessty can be apprecated by consderng the extreme case of only one measurement. A sngle measurement can gve no dea on the precson of the measurement. Therefore, the fracton may not be a real number. The possblty of calculatngx and s ncluded n modern pocket calculators. In Fg.3. varous confdence levels are ndcated. The probablty that a random result of a measurement les betweenx + andx s calculated to be 68%. Ths means that a repeated measurement wll result n a new result wthn of the mean: the standard devaton s the 68% confdence level, s the 95% confdence level, and 99.7% s the 3 confdence level PRECISIO OF THE MEA In the precedng dscusson we have been dealng wth the precson of data, as charactersed by the standard devaton. It s equally mportant to report the uncertanty n the fnal outcome of a number of measurements. Therefore, we have to calculate the precson of the mean value or, more specfcally, the standard devaton of the mean. Below we wll brefly dscuss the propagaton of errors;.e. the overall uncertanty obtaned from a number results, each wth ts own uncertanty. The concluson s that the varance of a mean s the varance of the data set dvded by the number of measurements: x x (x x) (3.9) ( ) The standard devaton of the mean s then: x x (x x) (3.0) ( ) 49

8 Chapter 3 As an example we wll calculate the mean and standard devatons of the data shown n Fg.3.A and Table 3.. All data are assumed to have the same uncertanty/precson. Table 3. Set of data correspondng to Fg.3.A. r. x x x r. x x x r. x x x Mean x = 3.56 Standard devaton x = {(x 3.56) }/8 = of mean = x /9 = STATISTICAL UCERTAITIES Statstcal uncertantes, defned n Sect.3..3, arse from the random fluctuatons n the number of events, for nstance the number of radoactve dsntegratons per unt tme, rather than from a lmted precson of the measurng equpment. For these statstcal fluctuatons the theory of statstcs provde the mathematcal technque for descrbng the data dstrbuton and the standard devaton. The result s that the standard devaton of a number of counts M detected durng a tme nterval t s smply: = M (3.) For the countng rate R,.e. the number of counts per second, the standard devaton then s: R R M Rt (3.) t t t The relatve uncertanty of the countng rate s gven by: R R Rt (3.3) 50

9 Errors, Means and Fts It s obvous that the relatve precson s better the hgher the countng rate and the larger the measurng tme. The matter of confdence levels wth observng data wth statstcal uncertantes s smlar to the nstrumental uncertantes as dscussed n the prevous secton. The probablty that a "true value" observed n an nfnte perod of tme les between x + and x of the measured value s 68%: the standard devaton s the 68% confdence level, s the 95% confdence level, and 99.7% s the 3 confdence level. 3.5 ERROR PROPAGATIO 3.5. STADARD DEVIATIO We often want to determne a quantty A that s a functon of one or more varables, each wth ts own uncertanty. The uncertanty of each of these varables contrbutes to the overall uncertanty. We wll lmt ourselves to gve the mathematcal expressons for n varous cases. The equatons are based on the general relaton for the functon: A = f (x, y, z) In the case of statstcal uncertantes the standard devaton of A depends on the ndependent varables x, y and z as: σ A A A A σ x σ y σ z (3.4) x y z If the uncertantes are estmated nstrumental uncertantes, smlar equatons are to be used for calculatng the uncertanty n the fnal result. For the general relaton: A = f(x, y, z) wth the estmated nstrumental uncertantes x, y and z, the uncertanty n A s: A A A A x y z (3.5) x y z resultng n the equvalent equatons for A as for A n the Eqs In these examples a and b are constant coeffcents, x and y are the ndependent varables, A the dependent varable. 5

10 Chapter 3 ) A = ax + by and A = ax by wth uncertantes x and y ; n both cases: A = a x + b y (3.6) ) A = a xy and A = a x/y A A x x y y (3.7) bx 3) A = a e A /A = b x (3.8) 4) A = a ln( bx) A = a x /x (3.9) 3.5. WEIGHTED MEA Tll now, when calculatng average values, we have consdered all numbers to have the same precson and thus to have the same weght. If we assgn to each number ts own standard devaton, the mean s then to be calculated accordng to: x x (3.0) whle the standard devaton of the mean s obtaned from: x resultng n: x (/ ) (3.) The weght of each result s nversely proportonal to the square of the standard devaton; / s referred to as the weghtng factor. 5

11 Errors, Means and Fts If the standard devatons are equal, the expresson for of the mean reduces to Eq.3.0: x = / (/ ) = / [(/ ) ] = / or x = / 3.6 LEAST-SQUARES FIT A measured quantty s often related to some other varable, for nstance y = f(x). Ths functon may have any form such as lnear, quadratc, harmonc, arbtrary, et cetera. The subject of ths secton s to brefly dscuss methods to obtan the most probable graphcal and algebrac adjustment of a functon to the data. The major nterest s the adjustment of a straght lne to a number of data, whch are expected to be lnearly related FIT TO A STRAIGHT LIE. The prncple of least-squares fttng s to mnmse the sum of the squares of the devatons of the dependent varable (y) (the uncertanty n x s assumed to be neglgble) from the straght lne wth coeffcents a and b: y = a + bx (3.) The devaton from any value of y (y ) to the straght lne s gven by y = y f(x ) = y a bx (3.3) Mnmsng the sum of these devatons results n: y = 0 (3.4) whle applyng the absolute values of y does not result nto a useful mathematcal procedure. Therefore, we are lookng for a procedure to fnd the coeffcents a and b, charactersng the straght lne, by whch the sum of the squares of the devatons: (y ) = (y a bx ) (3.5) s mnmsed. The matchng condtons are: and (y a (y b a bx ) a bx ) 0 0 (3.6) The resultng values of a and b are: 53

12 Chapter 3 x y x x y a (3.7a) and x y x y b (3.7b) whle x x (3.7c) If the standard devatons of y are equal, the values of a and b are: a = (x y x x y ) / b = ( x y x y ) / = x (x ) (3.8a) (3.8b) (3.8c) Many modern pocket calculators have the possblty of calculatng the least-squares ft to the straght lne. An example s shown n Fg.3.3.The standard devatons of the coeffcents a and b are: a x (3.9a) and b (3.9b) 3.6. FIT TO O-LIEAR CURVES The least-squares ft to, for nstance, quadratc or second-degree polynomal, harmonc, exponental and arbtrary curves wll not be treated quanttatvely. These can be calculated analytcally, but the normal routne s to apply the proper computer programmes. These also exst for composte curves, consstng of the superposton of more than one curve. Even through data ponts, whch are not theoretcally related to the ndependent varable, curves can be ftted. One example s the cubc splne. In prncple cubc (3rd order) fts through successve group of data are adjusted to each other. The resultng curve can be chosen to ft all data ponts, as well as chosen to smooth rregulartes as much as wanted. 54

13 Errors, Means and Fts 3.7 CHI-SQUARE TEST The least-squares ft s based on mnmsng the exponental of the Gaussan probablty functon of Eq.3.5,.e. of the sum of the (quadratc) devatons between the observed y values (y ) and the y values calculated from the relaton between y and x: y = f(x ). 6 y y = x x Fg.3.3 Lnear least-squares ft through a number of (x,y) data sets, of whch the ndependent and dependent varables are related by the equaton y = x. All y values have the same precson. For the straght lne ft ths comes to calculatng the coeffcents of the lnear relaton y = a + bx for whch the sum (y a bx ) s mnmal (Eq.3.5). In vew of the defnton of the probablty, P, t logcal to take the same sum of the square devatons n relaton to the standard devaton as a measure for the goodness of ft: y a bx (3.30) For the set of data n Fg.3.3 and Table 3. the s calculated. In summary, the optmum ft to data s that whch mnmses. The method by whch s mnmsed s the least-squares method. The result of a test s reassurng, f the dvded by the number of data (=9) mnus the degrees of freedom (= the number of parameters to be determned, here = ) s about equal to one (n ths case =/8). 55

14 Chapter 3 Table 3. Seres of (x, y ) data sets obeyng the relaton y = a + bx, resultng n a value of. The graph for these data s shown n Fg.3.3. The optmum ft to the data s that whch mnmses. x y y = x [y f(x )] /