Measurement Uncertantes Reference Introducton We all ntutvely now that no epermental measurement can be perfect. It s possble to mae ths dea quanttatve. It can be stated ths way: the result of an ndvdual measurement of some quantty s the actual value and an error. We assocate the se of the error wth the partcular method of measurement some technques have nherently smaller errors than others (e.g. a mcrometer vs. a yardstc.) We wll use the terms uncertanty and error nterchangeably. An mportant part of understandng an eperment and reportng ts results s beng able to determne the measurement uncertanty. A practcng scentst or engneer needs to understand measurement uncertantes both for the nterpretaton of measurements made by others, and for desgn of future measurements. To fnd an uncertanty, t s necessary to fnd the range n whch an ndvdual measurement s lely to le. For eample, f we measure the speed of lght to be 3.0 0 0 cm/sec, and a study of the measurement system ndcates that the ndvdual measurements are lely to le between.98 and 3.04, we would quote the result as c = ( 3.0 ±0.03) 0 0 cm /sec. The uncertanty s usually eplctly dsplayed wth error bars when epermental results are graphed, as n Fgure.. Random and Systematc Errors Measurement errors fall nto two categores: random and systematc. The random error n an ndvdual measurement s not predctable n eact magntude or sgn. However, the average of random errors over many repeated, ndependent measurements of the same quantty s ero. Thus, the average of several measurements s generally closer to the actual value than any of the ndvdual measurements. The uncertanty n the case of random errors should ndcate the range of results that would be obtaned from many measurements. Ths should be a characterstc of the measurement apparatus. In fact, one common method for estmatng the range of random errors for a partcular measurng devce s smply to repeat the measurement many tmes. The other type of error s systematc. Here the magntude and sgn of the error s always the same. The measurement result s shfted, up or down, by the same amount, even n repeated measurements. Thus, averagng many measurements wll not mprove results. Typcally a measurement has both random and systematc errors. Sometmes they can be ndependently estmated and quoted. If ths were the case for the speed of lght measurement, above, the result mght be quoted: Here the last term s the systematc uncertanty. c = ( 3.0 ±0.03 ±0.0) 0 0 cm /sec. There s actually a thrd type of error: a mstae,.e., an error that s caused by the person performng the measurement, not by the apparatus. These are preventable or caught wth attenton and some art. One of the functons of wrtten epermental reports
s to show readers, by descrptons and nternal consstency checs, that ths type of error has probably been elmnated. Systematc errors are usually unque to specfc technques, and so the dscusson below wll concentrate on general methods for estmatng random errors. The RMS Devaton One way to specfy the range of a set of values s the RMS devaton. Ths stands for Root Mean Square devaton. Mean n ths contet, means the average. Gven a set of values of, the average, or mean, denoted by s = = For the same set of s we can also defne the RMS devaton from the mean, denoted by, through the equaton = () = ( ) The quantty s a measure of how dspersed are the values of away from ther average,. If all s have the same value then s ero. The RMS devaton s frequently called the standard devaton. Strctly speang, s the standard devaton f the s come from a partcular nd of dstrbuton of errors called the normal or Gaussan dstrbuton. Another equvalent epresson for n Eq () s gven by: = (3) If the number of measurements s ncreased, the values of and tend toward lmtng values whch are ndependent of. The value of s determned by the physcal quantty beng measured. The value of s determned by the measurement apparatus. It s common practce to use as the epermental uncertanty. () A more detaled study of errors nvolves consderaton of the probablty of obtanng a partcular error n a measurement. One of the useful results of such a study s the followng chart whch gves the odds that an ndvdual error wll devate from the mean by a specfed number of standard devatons,, for the normal error dstrbuton.
Fgure : Standard Devatons of a Gaussan Dstrbuton So t s very unlely for an error to eceed -3, but not so rare to eceed. These results are useful n comparng dfferent measurements. What matters n determnng whether two measurements agree or dsagree s not the absolute amount by whch they dffer, but the number of s by whch they dffer. The mportance of ths s llustrated n the Fgure, whch dsplays two hypothetcal measurements of the same quantty, together wth a predcted value (from some theory, say.) The quantty s plotted along the vertcal drecton. Eperment A devates from the predcton by slghtly more than. It cannot be sad to dsagree, sgnfcantly wth the predcton. It s also evdent that A and B agree wth each other. Eperment B, however, devates by over 3.5 from the predcton. Ths s a sgnfcant dsagreement, even though B s closer to the predcton than A! Agan, what matters s not the absolute dfference, but the dfference n unts of uncertanty. Smlarly, a calculaton of the percentage devaton between two measurements, or between a measurement and a boo value, yelds a correct but meanngless number, unless the uncertantes are taen nto account. 3
Fgure : Graph of Data Ponts A and B Error Estmaton n Indvdual Measurements As descrbed above, one way to estmate the value of a certan measurement process s to mae repeated, ndependent measurements and compute the RMS devaton from the mean. In certan cases, smpler methods may be used. A common source of random error s the readng of a dal or scale. For such readngs, a reasonable estmate of s one half the smallest scale dvsons. One must be careful though, many effects can cause measurement errors. For eample, the dgtal stop watch may dsplay tme to hundredths of seconds, however the human reacton tme to start and stop the watch s often 0.05 seconds or more mang the tmng error at least 0.05 s nstead of the 0.005 s based only on the dsplay. Alternately, some wobble n the apparatus mght cause the tmng of the apparatus to vary, even though one mght measure tme to great accuracy. Identfyng the domnant source of epermental errors often requres careful reflecton. Error Propagaton Often t s necessary to combne measurements of dfferent quanttes, and to fnd the overall uncertanty. For eample, f we measure a dstance moved,, durng a tme, t, for an obect ntally at rest wth a constant appled force, then the acceleraton s a = ( ) / t. If we now the uncertantes and t, what s the uncertanty n a,.e. what s a? The general form of ths queston s: gven a functon = f (,y), what s, gven and y? For those who now calculus, the answer can be shown to be, for the case of ndependent measurements of and y, 4
Snce calculus s not requred for ths course, we wll use some formulas for specal cases whch are derved from the above formula:. Sum or Dfference: If the quantty s the sum or dfference of and y,.e. = ± y, then y = + (4) ote that the operaton between the s n Eq.4 s + for both summaton and subtracton, so that errors always get larger (see the eample below). Tang the square root of the sum of the squares s the way that the magntudes of two vectors that are at rght angles to each other add. It s called addton n quadrature. Thus for addtons and subtractons, the errors add n quadrature. A lttle epermentng on the calculator or by loong at geometrc vector addton wll show you that n ths form of addton, f one number s much larger than the other the result s a lttle larger than the larger number. If they are about equal, the result s about.4 tmes ether one. For eample: f a =.3 cm ± 0.3 cm and b =.3 cm ± 0. cm, then the dfference s gven by: ote that t s conventonal to round to one sgnfcant fgure n the uncertantes and round the number tself (not the uncertanty) to the same place as the smallest sgnfcant fgure of the uncertanty (tenths n the above eample).. Product or quotent: If the quantty s the product or quotent of and y,.e. = y or = /y then % % + % y = (5) where % = ( /) 00% and therefore, = (% )/00%. In ths form we say that for multplcaton and dvson the percent errors add n quadrature. For eample f v = 0.04 m/s ± 0.00 m/s and t = 5.80 s ± 0.5 s, then the dstance traveled s gven by: 5
ote that we frst convert the uncertantes (or errors) to percent uncertantes before addng them n quadrature. We usually convert them bac to uncertantes n meters (or whatever unts are beng used) at the end. 3. M of the Above Two Types: If a calculaton has a mture of the two types n and above, such as addton and dvson then one needs to handle the frst calculaton, and then the second. That s, do the addton as n above, and then do the dvson as n : 4. Averagng: If the quantty s the average of and y,.e. = ( + y)/ and = y, then usng the same procedure as above wth = 0, we can show that = The general result for averagng ndependent measurements of, all wth the same s avg = (6) 5. Multplcaton by a constant: If the quantty s a constant value tmes a measurement,.e. = a, where 'a' s a constant wth no uncertan (e.g. π) then 6. Tang somethng to a power: If = n, then % = % (7) ( ) % = n % (8) that s, the percent error of s n tmes the percent error of. Ths holds for n =, 3, ½ (square root), and other nteger and fractonal eponents. 6
7. Specal functons: For a specal functon, such as sne, cosne, log, ep, then follow the procedure below: If = sn and = 0.90 ± 0.03, then sn(0.93) = 0.80 sn(0.90) = 0.78 sn(0.87) = 0.76 Therefore, = sn(0.90) = 0.78 ± 0.0. Here we try varous values to see how much changng the angle changes the sne of the angle. Least Squares Lnear Graph Fttng: After data has been collected, graphcal analyss of the data s desred. To nterpret data, graphs are often compared to an epected shape or curve from a theoretcal equaton. Most lely, data collected wll be graphed n such a way as to produce a straght lne. Ths s desred snce a lne has a constant slope. Snce we now n advance that the data wll not determne a perfect straght lne, we must use some technque to acqure the best straght lne for the data collected. Statstcal analyss provdes an equaton, the least squares ft (assume Gaussan populaton), that wll mnme error and determne the best slope and y-ntercept for a collecton of data ponts (,y) for a straght lne, y = m + b. The least squares appromaton for the slope s descrbed by the equaton: m = ( y ) y The least squares appromaton for the y-ntercept s descrbed by the equaton: b = y ( y ) Usng the above equatons an appromaton for a lne whch best fts the data ponts can be determned. In Ecel, the 'trendlne' opton n graphng wth a choce of 'lnear' ft wll calculate the above 'm' and 'b' values for you and produce a lne followng the prescrbed equatons. Intally, t s a good dea to chec Ecel to ensure you understand what calculatons t s mang for you. You may have Ecel dsplay the equaton for you, thus allowng you to compare these values to theoretcal values of the slope and ntercept. (9) (0) 7
Percent Dfference: To calculate the percent dfference between an accepted value and the epermentally determned value, use the equaton below. Remember that ths value s a percentage and therefore requres a %. AcceptedValue EpermentalValue % PercentDfference = 00 () AcceptedValue 8