Combining Multiple Averaged Data Points And Their Errors

Transcription

1 Combiig Multiple Averaged Data Poits Ad Their Errors Ke Tatebe August 10, 005 It is stadard practice to average measured data poits i order to suppress statistical errors i the fial results. Here, the error is typically cited as the stadard deviatio of the poits divided by the square-root of the umber of poits averaged. That is ε = σ 1 where is the umber of poits averaged, σ is the stadard deviatio of the poits, ad ε is the resultig error. The sample stadard deviatio of the poits is defied as σ = 1 S i 1 S where S i is the value of each data poit i the set, S is the average of the data set, ad 1 is the umber of degrees of freedom. I the evet that oe has a cotiuous strig of measuremets it is sometimes appropriate to bi the data for example by time. I this case istead of beig over all poits, the sum i Equatio would be over oe bi of data. If it becomes desirable to further average the data the the sesible thig would be to take the origial data, bi it ito larger bis, ad re-average. Sometimes the origial data is o loger available. I this case we must seek a way to combie the already averaged data ad determie the proper error to associate with it. To make our discussio clearer let us take the example of biig oe large data set ito two sub-sets of data. Assume that we had at some poit a large set of poits S i where i rages from 1 to, the total umber of poits i the set. If we kew the values of every S i we could simply average them to fid S ad use Equatio 1 to determie the error. This, however, is o loger the case as we have already bied the data S i ito two groups a i ad b i cotaiig a ad b = a poits respectively. We are assumig the origial data is uavailable so we do t have ay kowledge of the values of the idividual measuremets. The problem before us is to determie S ad ε S give ā, b, ε a, ε b, a, ad b. That is, the two averaged data poits, their associated errors, ad the umber of poits averaged to create each data poit. To do this we will first determie the value of the composite poit S. The we will determie the appropriate error to associate with this ew poit. 1

2 1 Determiig S We kow the averages of a i ad b i are ā = 1 a a a i ad b = 1 b b b i 3 Similarly S is give by S = 1 a 1 b S i = a i + b i a + b 4 To fid S i terms of kow quatities we write ā + b = 1 a a i + 1 b b i 5 a b ad put these over a commo deomiator,, the deomiator i Equatio 4. This gives us a ā + b a 1 a b = a i + a = 1 a b a i + b i = 1 b 1 b b b i 6 7 S i 8 = S 9 yieldig a compact equatio for S i terms of ā, b, a, ad b. a b S = ā + b 10 This is simply a weighted average of the poits.

3 Determiig ε S ow that we have a expressio for S we ca proceed to fid the error of our ew composite poit. From Equatios 1& we ca write the error as ε S = 1 S i S 11 As stated above, if we kow the value of all the data poits S i, the solvig for the related error is trivial usig Equatio 11. I our case, however, we oly have kowledge of pre-averaged sub-sets of the data. I terms of the two sub-sets of data, the error of the composite poit is ε S = 1 a a i S b + b i S at this poit we make a chage of variables ad defie for readability. Similarly, we will use a a a ad b b b. If we substitute Equatio 10 ito Equatio 1 we get ε S = 1 a a i a ā b b b + b i a ā b b ow, we ca add zero to the terms i each sum, i the form of 0 = ā ā ad 0 = b b, ad simplify. ε S = 1 = 1 a a i + ā ā a a i ā + ā1 a ā b a b b + b i + b b b b a ā b + b i b + b1 b b b a ā 15 At this poit it gets a bit cumbersome. We ca regard the argumet of each sum effectively as oly two terms as follows A + B = [ a i ā + ā 1 a b ] b 16 where A a i ā ad B ā 1 a b b 17 At this poit, oe should ote that B is a kow costat, as it is composed etirely of kow quatities. We ow expad this argumet ad restore the summatios. Let us look at each of these sums i tur. A + B = A + AB + B 18 First, we look at the sum over A. This is simply a A = a i ā = ε a a 19 So far so good. We are well o our way to expressig the error of S i terms of the errors of a ad b. 3

4 Lookig at the AB term, we fid it equals zero. Sice B is a costat it may be take out of the sum. A few calculatios yield: a AB = B a i ā 0 = B = B = B = B a a a i ā a a i a ā a 1 a i a a a a i a i a a a i = B0 5 = 0 6 Great, so we ca igore that term. The fial term, B, is a bit messy, but evetually simplifies thusly, B = b B 7 = a B 8 a b ] = a [ā 1 b 9 [ a b ] = a ā b 30 [ b b ] = a ā b 31 [ b b] = a ā 3 b = a ā b 33 OK, ow we re i the home stretch. We ca, of course, follow the same lie of logic for the terms i the secod sum of Equatio 15 by expadig it i similar terms of C + D. This lie of logic evetually yields: Substitutig the above expressios ito Equatio 15 gives us C = ε b b 34 CD = 0 35 D a = b b ā 36 4

5 a ε S = 1 a b b a i ā + ā1 b + b i b + b1 = 1 a b A + B + b i b + b1 b a ā = 1 a a a b A + AB + B + b i b + b1 b = 1 ε b b a a a ā b + b i b + b1 b = 1 ε b b a a + a ā b + C + D = 1 ε b b b b a a + a ā b + C + CD + D = 1 ε b a a + a ā b + ε a b b b b ā Here we ote that ā b = b ā ad perform some algebraic simplificatio b a ā a ā a ā ε S = a ε a + b ε b + 1 a b [b ā b + a ā b ] 44 = a ε a + b ε b + 1 a b a + b ā b 45 a b = a ε a + b ε b + 1 ā b 46 = a ε a + b ε b + a b ā b 47 This gives us our fial expressio for the error ε S : ε S = a ε a + b ε b + a b ā b 48 3 Coclusio The above treatmet allows oe to combie two data poits, each created as a average of multiple measuremets. Equatios 10 & 48 will give the true average ad error of the composite poit. I this presetatio oly two poits are combied for clarity of the derivatio. If more tha two averaged poits eed to be combied, the formulae may be used repeatedly to combie multiple sets. For example, if oe has three averaged data poits ā, b, ad c oe may use Equatios 10 & 48 to combie ā ad b. This result ca the be combied with c to get the fial composite poit. 5

6 Addedum August 11, 005 Occasioally, a ad b may ot be kow but are ofte approximately equal. I this case oe might assume equal weightig with oly mior cosequeces. I this case Equatio 48 reduces to 0 ε S = ε a ε ā b b where 0 is the umber of poits comprisig either of the averaged data poits. Iterestigly this still depeds o the umber of measuremets per data poit. I the limit of a large umber of measuremets, eve if the umber of measuremets are oly approximately equal, we ca largely igore the last term uder the radical i Equatio 49. This meas that i a pich, where little is kow about the data, if oe wishes to combie two data poits the approximate error of the ew data poit will be ε S ε a + ε b 4 50 Ideally the two poits would be close together, i.e. ā b, so that ā b 0 further suppressig the last term i Equatio 49. This is how oe would expect two errors to add uder such coditios, with two poits beig combied ad the resultig error beig suppressed by a factor of. From this we ca be reasoably cofidet that our more geeral expressio i Equatio 48 is correct. Further, umerical simulatios have verified the validity of Equatio 48. ote For those desirig a eve more simplified expressio tha Equatio 50 we ca assume that ε a ε b. This, with our previous assumptio that a b 1 will allow us to make the followig simplificatio ε a ε b ε a + ε b ε avg 51 which ca be substituted ito Equatio 50 to get ε avg + ε avg ε S ε avg 4 5 That is, the average of the errors reduced by a factor of. = 4 53 = ε avg 54 = 1 εa + ε b 55 6

7 Addedum August 16, 005 A more basic method of fidig the error of a poit that is the result of combiig other poits with errors is to simply igore the errors of the costituet poits ad cite oly the stadard deviatio of the poits that are combied. If two poits are combied, ad we igore their errors Equatio 1 yields ε S = 1 ā ā + b + ā ā + b 56 which simplifies to ā b ε S = 57 Iterestigly this matches Equatio 48 if we set ε a = ε b = 0 ad = 1. This results i oly the first term survivig, givig us ā b ā ε S = = b 58 Clearly, i additio to iformatio lost by assumig = 1 our error estimate will also be off by a ε S = ε a + b ε b 59 which will be egligible if the error are small. This meas that this approximatio should oly be used whe the errors are small but the separatio betwee poits is large. Of course i this case it is probably best ot to average poits as the two poits likely represet two distict measuremets rather tha statistical fluctuatios i a costat sigal. 7

8 4 Appedix : Total umber of poits i both averaged data poits. : a : umber of poits i data set a. i.e. the maximum idex i, for a i. b : umber of poits i data set b. i.e. the maximum idex i, for b i. a : a a b : b b = 60 = a + b a + b 61 = a + a b + b a b 6 = a + b + a b 63 8