Error Aalyss Preamble Wheever a measuremet s made, the result followg from that measuremet s always subject to ucertaty The ucertaty ca be reduced by makg several measuremets of the same quatty or by mprovg the expermetal techque But for fudametal reasos ose sources such as thermal ose or quatum fluctuatos caot be elmated ad the ucertaty caot be reduced to zero Types of ucertates Radom ucertaty ad systematc error A batch of dgtal watches was purchased from a factory wth the expectato that they were all set to the correct tme before dspatch Oe mght mage that some weeks later a varety of tmes would be dsplayed by a selecto of the watches There could be a umber of reasos for the dffereces the tme dsplayed, such as a) the crystal oscllators are rug at dfferet rates; b) the watches were tally correctly set; c) the crystal oscllators mght be ustable; d) the tme stadard at the factory was correct Radom (statstcal) ucertates ca be observed whe repeated measuremets of the same quatty gve rse to dfferet values May radom factors cludg vbratos, varatos temperature ad electroc ose, usually combe to form the 'radom ucertaty' a expermet Systematc error refers to a costat factor whch flueces all readgs equally durg a expermet Such errors ormally arse from the effect of a fte umber of dsturbaces Sources of systematc error could be aythg from the cotact resstace of a wre a electrcal crcut, to a stopwatch rug too fast ad the use of a mproperly calbrated strumet I some expermets t mght be dffcult to see whether errors are due to radom or systematc effects Wheever possble, sources of systematc error should be elmated or corrected for Such sources of error are, however, ot easly detfed ad the expermeter may ever be aware of them I practce, the systematc ad radom ucertates are smply added up to obta the maxmum ucertaty estmate: u ( ) u ( ) u ( ) u( ) max sys ra Estmate of Ucertates Physcal Measuremets Measured value of a physcal quatty : x xbest u( ) x best u() best estmate (mea) for ucertaty or error the measuremet (systematc, radom or statstcal) Relatve ucertaty (error): u ( ) fract x best Expermetal ucertates should almost always be rouded to oe sgfcat fgure (dgt): measured g=(9800) ms -
The last sgfcat fgure ay stated aswer should usually be of the same order of magtude ( the same decmal posto) as the ucertaty Propagato of Ucertates If varous quattes,,, are measured wth small ucertates u( ) [u( )/ < 0] ad the measured values are used to calculate some quatty Y, the the ucertates the varous cause a ucertaty Y Ths ca be estmated by a Taylor expaso to frst order Example : Ucertates Sums ad Dffereces If Y s the sum or dfferece Y ( or), the u( Y) u( ) (maxmum error estmate) u (depedet radom errors) u( Y) ( ) Example : Ucertates Products ad Quotets If Y s the product ad quotet Y uy Y ( ) u ( ) m k k, the k, (maxmum error estmate) k, k, uy Y ( ) u ( ) k, k, k, (depedet radom errors) Example 3: Ucertaty a Power If Y s a power, Y= m, the u( Y) u( ) m Y If Y = Bx, where B s kow exactly, the u( Y) B u( ) Example 4: Ucertaty a Fucto of Oe Varable If Y s a fucto of a sgle varable (measured quatty), Y(), the dy u( Y) u( ) d
Geeral Case: Ucertaty a Fucto of Several Varables If Y s a fucto of several varables (measured quattes),,,, the Y u( Y) max u( ) (maxmum error estmate) Y u( Y) u( ) (depedet radom errors) Y Y u( ) u( ) (always) Remark: I some examples the dfferece u( ) betwee the results of expermetal measuremets ca also be used as a measure of ucertaty (error) Examples for the Maxmum Ucertaty Estmate from Physcs Determato of the desty of a cyldrcal metal rod measured quattes mass of the rod m legth of the rod l radus of the rod R m m Equato to calculate the desty V R l Relatve maxmum ucertaty estmate u ( ) ( ) ( ) ( ) max u m u l u R m l R Determato of the specfc heat capacty c f of a sold by a calorc expermet measured quattes temperature of the hot sold f tal temperature of the flud fl mxg temperature m mass of the flud m fl mass of the sold m f heat capacty of the calormeter C K specfc heat capacty of the flud c fl
( cfl mfl CK)( m fl) Equato to calculate c f cf m ( ) x f f m Maxmum ucertaty estmate ( m fl) ( cfl mfl CK) u( c f ) cfl u( mfl) u( CK) u( mf ) mf ( f m ) mf ( cfl mfl CK) ( m fl) ( f fl) u( fl) u( f ) u( m ) mf ( f m ) ( f m ) ( f m ) Statstcal Aalyss of Radom Ucertates Let us cosder data pots of a measured quatty wth measured values x k, k =,, 3, These values of wll be dstrbuted about ther mea value, whch s some termedate value ot ecessarly cocdet wth ay of the data values The mea value of the data set s wrtte as x ad s defed to be x () Subtractg the mea value from ay data value produces a resdual We ca equally well vew the data set as a collecto of resduals (devatos from the mea), x x x, () ad we see that the mea s chose so that the sum of the resduals s always zero x ( x x) 0 (3) Sce the trasformato s lear the shape of the dstrbuto of data s ot altered The resduals wll have a mea value of zero A secod pece of formato from the data set, besdes the mea, s related to ts spread Ths s gve by a quatty called the stadard devato (s x ) ad ts square, called the varace Data showg a large stadard devato wll have the mea value poorly determed Coversely, f the stadard devato s small, the data cluster closely, ad the mea s well-determed The stadard devato for ay data pot s ofte called ts error, the sese of 'ucertaty' The better the data are determed the more detal they reveal The stadard devato (ad the varace) deped o the scatter of the data about the mea ad from t we fd the ucertaty of the mea The varace of the data set s defed as the mea value of the squares of the resduals : var ( x x) (4) The stadard devato of the sample, s x, s the square root of the varace of the sample,
sx ( x x) (5) Ths root-mea-square result for the stadard devato s tutve; t descrbes the ucertaty of every sgle measuremet pot The mea value, however, s better defed wth a smaller stadard devato Ideed, the stadard devato of the mea value s gve by sx s x x (6) x ( ) ( ) I expermetal practce we adopt for the measuremet ucertaty approxmately twce the stadard devato of the mea: u( ) s x (7) Normal or Gaussa Dstrbuto I addto to the measuremet value ad ts ucertaty oe should specfy the probablty that the true value falls to the terval x u( ) I order to do so we assume that the data are dstrbuted accordg to a ormal dstrbuto (Gaussa fucto) ( x x) px ( ) exp u u (8) Ths fucto s show the fgure Its wdth s gve by the ucertaty u = u() Sce the Gaussa fucto s ormalzed, the probablty that the true value falls the rage x ku( ), where k s a umercal factor, s gve by the area uder the curve wth ths rage Ths leads to the cofdece level of the measuremet, where the cofdece level smply s the probablty for the true value to fall the rage x ku( ) As see from the fgure the cofdece level depeds o k ad s gve by 683%, 955% ad 997% for k =,,3