Ed Eyler 003 otes on epermental uncertantes and ther propagaton These notes are not ntended as a complete set of lecture notes, but nstead as an enumeraton of some of the key statstcal deas needed to obtan quanttatve results from the analyss of straghtforward measurements We wll gradually cover these ssues n much more detal as the course progresses I References A number of good tets on data analyss for the physcal and engneerng scences are avalable One of the best s Data Reducton and Error Analyss for the Physcal Scences, Thrd Edton, by Phlp R Bevngton and D Keth Robnson (McGraw Hll, 003) Much of ths short book s accessble to a second-year student, but t nevertheless presents a wealth of technques and nformaton that are suffcent to see many scentsts through ther entre careers Most of the numercal technques are accompaned by eamples n the form of computer programs n Fortran, or n the Web supplement to the tet, n C++ At a more ntroductory level, there are numerous books, partcularly An Introducton to Error Analyss, Second Edton, by John R Taylor (Unversty Scence Books, 997), Ths books s orented towards the student laboratory, so t s easer to follow, but has less to say about the analyss of real-world eperments II Populaton Statstcs It s useful to devse a few statstcal measures of the propertes of epermental data that are ndependent of any presumptons about the model to be used n analyzng the eperment The most wdely-used such measures are the mean or average and the standard devaton The mean can be determned whenever a mnmum of two ndvdual measurements of the same quantty are avalable, accordng to mean: The standard devaton s a measure of the scatter of repeated measurements, and can be estmated only f you take at least three measurements under nomnally dentcal condtons Thus n a quanttatve eperment subject to random fluctuatons t s usually wse to take each data pont at least ths many tmes The sample standard devaton s often denoted by s to dstngush t from the (populaton) standard devaton σ of the underlyng probablty dstrbuton, though sometmes σ s used for both It s best to keep them dstnct, snce s s actually an epermental estmate for σ, whch becomes eact n the lmt of a large number of measurements It s calculated accordng to, standard devaton: s ( ) Most calculators have ths or a smlar formula pre-programmed Sometmes a factor of / nstead of /(-) s used; the verson gven here s preferable when the mean has been determned from epermental data and not from prevous knowledge of the underlyng parent dstrbuton The value of s gves an estmate of the uncertanty σ assocated wth each ndvdual measurement To determne from ths the uncertanty of the mean or any other derved quantty, we must use the technques for error propagaton descrbed n secton V
III Statstcal dstrbutons-- parent vs epermental statstcs The populaton statstcal measures n secton II only descrbe the fluctuatons of the measured values To say anythng more we must make some assumptons about the statstcal dstrbuton that descrbes these fluctuatons By way of llustraton let s look at a partcular eample One of the most common types of physcal measurement s a countng measurement, where we measure the pulses from a detector, or some other sgnal that depends on the number of electrons or photons arrvng at a partcular pont Such a process, where the tme of arrval of the net count s ndependent of when the prevous count arrved, s called a Posson process The probablty of observng a partcular number of counts n a unt tme nterval s governed by the Posson probablty dstrbuton wth mean µ, µ µ P( )! e The mean µ s the average number of counts that are observed f the measurement s repeated many tmes Thus f we take a sample of epermental data and fnd the sample mean as n secton II, we determne an appromate value for the underlyng mean µ, whch wll become ncreasngly accurate as the sample sze s ncreased A smlar result apples for the standard devaton It can be shown that f data are accumulated under dentcal condtons untl counts are epected, the standard devaton n the number of counts s gven by σ In partcular, f we sample for a unt tme nterval we epect µ counts, so we say that the standard devaton of the Posson dstrbuton s µ If the populaton standard devaton s determned as descrbed n secton II, t should yeld a result close to the actual value of σ Because t s estmated from fluctuatng data, t wll not gve an eact result, although for large the estmate wll become qute good In dscussng ths stuaton we say that the Posson dstrbuton s the parent dstrbuton, whose parameters are eact constants and must be clearly dstngushed from the populaton statstcs obtaned from epermental results sampled from ths dstrbuton It s worth pontng out that the fractonal accuracy of a measurement, / σ /, mproves only as the square root of the number of counts accumulated If t takes one day to acqure data gvng a measurement accurate to 0%, t wll take 00 days to reach an accuracy of %! Ths dscouragng property s an unavodable accouterment of lfe as an epermentalst Other dstrbutons also arse commonly n physcs and engneerng One s the bnomal dstrbuton, whch characterzes con flps and many thermodynamc processes Another s the Lorentzan dstrbuton, whch descrbes atomc lne shapes It has the unque pecularty that there s no fnte value for the standard devaton, so t s nstead characterzed by ts full wdth at half-mamum! Most mportant of all s the Gaussan dstrbuton, whch s a lmtng case of both Posson and bnomal dstrbutons IV The Gaussan or normal dstrbuton Ths s by far the most commonly used statstcal dstrbuton, snce t s the lmtng case of other mportant dstrbutons It has two parameters, the mean µ and the standard devaton The Gaussan probablty functon s gven by P() σ π ep µ σ Ths s one of the most ubqutous functons n physcs and engneerng An ndcaton of ts mportance s gven by observng that ths functon and ts graph are prnted on the German 0-Deutschmark bll! It
s a contnuous probablty dstrbuton, so the probablty of observng a value for lyng wthn a gven range s gven by ts ntegral More specfcally, the probablty dp of measurng a value wthn a range d around s gven by, dp P()d Just as for the Posson dstrbuton, the populaton statstcs for the mean and standard devaton can be used to estmate the underlyng mean and standard devaton f they are not already known The standard dstrbuton s a measure of the wdth of the dstrbuton The probabltes are appromately 68% and 95% that a measurement wll fall wthn one and two standard devatons of the mean, respectvely Though ths dstrbuton s eceedngly common, t can be overused: t s sometmes ncorrectly assumed that vrtually all random varables obey Gaussan statstcs! Thus we often see absurd dscussons of quanttes lke IQ scores or economc fluctuatons n whch detaled predctons are made by assumng strct adherence to the functonal form gven above V Propagaton of uncertantes to derved quanttes Assume that we have measured a number of quanttes u, v, and that we know the standard devatons σ u, σ v, for each Ordnarly we know these standard devatons by estmatng them from the populaton standard devatons s as descrbed n secton II, although we may know ther values n some other way How do we fnd the standard devaton σ for a derved quantty f ( u, v, )? Ths can be solved by usng the error propagaton equaton, whch can be derved usng a combnaton of the chan rule for dervatves and a Taylor epanson The resultng formula, the most mportant result gven n ths entre dscusson, s σ σ u u + σ v v + + σ uv u v + Terms lke the last one are called covarance terms, and are only needed when the varables u and v are not ndependent of one another Ths type of correlated fluctuaton usually does not arse n a smple eperment The covarance σ uv of two varables can be estmated by takng repeated samples much lke the standard devaton The estmate s gven by Eamples: s uv (u u )(v v ) Sums: au ± bv, where a, b are constants From the above, we fnd that u v uv σ a σ + b σ + abσ Ordnarly only the frst two terms wll be needed Ths s the sngle most common stuaton that s encountered Standard devaton of the mean (sometmes called the standard error of the mean) Ths s a specal case of a sum, so we apply the treatment above Assume we take ndependent measurements of a randomly varyng quantty Snce the measurements are ndependent and they scatter randomly, the covarance between them s zero Snce they are taken under dentcal condtons,
they all have the same standard devaton σ If we then calculate the mean and propagate the uncertantes, we fnd that, f, then σ σ Thus, as you mght epect, the mean s better-determned than the ndvdual measurements In most real eperments, we use the populaton standard devaton s to estmate the standard devatons n the ndvdual measurements, so the estmate for the standard devaton of the mean wll be gven by s / In analyzng the results of an eperment, one typcally uses the mean values calculated from the measurements to fnd derved quanttes lke focal lengths or magnfcatons, so the uncertantes of these mean values are the ones that should be propagated In summary, f you use the mean values to determne some other derved quantty, then the standard devatons you nsert nto the error propagaton formulas for sums, products, and so forth should be the standard devatons of the mean values They are smaller by a factor / than the uncertantes of the ndvdual data ponts 3 Products (or dvson): ± auv or ± a u v Ths gves the result, σ σ u σ v σ uv + +, u v uv where the + sgn for the covarance term must be changed to a - sgn for the u/v case 4 Powers: au ± b Ths general power law gves the error propagaton formula, σ σ u b u A partcular common case s the square, D In ths case we just obtan σ Dσ D 5 A real formula In optcs, the focal length of a lens can be determned from two measured lengths, the object-mage dstance L and the separaton D between the two postons at whch a lens wll form a focused mage of the mage at the object plane The focal length s gven by f L D 4L Ths s a partcularly nce eample because t has some nuances If you break t nto two terms and attempt to use the formulas -4 for dfferences, squares, and dvsons, you quckly notce that both terms depend on L, so there s a non-zero covarance between them because they aren t ndependent Ths s typcal wth a formula of any complety, and makes a real mess out of somethng that ntally seems smple You can avod ths problem by nstead usng the error propagaton formula drectly, so everythng s done n terms of the ndependently measured quanttes L and D The necessary dervatves are:
f D f D + L 4 4L and D L The result for the standard devaton of f s then, σ f 4 + D 4L σ L + D 4L σ D