b g (17) c n m 1 [see (1)], and by (2) and (3) s 2 and s are both zero since standard error = s (16) JUNE 1999 THE AUSTRALIAN SURVEYOR Vol. 44 No.

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1 JUE 999 A OTE O STADARD DEVATO AD RMS R.E.Deak D.G.ldea RMT Uverty GPO Box 476V MELBOURE VC 3 AUSTRALA TE AUSTRALA SURVEYOR Vol. 44 o. TRODUCTO The am of th paper to clarfy ome cofug omeclature of ome tattcal term frequetly ued the urveyg lterature. Thee term are Stadard Devato ad Varace, Root Mea Square (RMS), Mea Square Error ad Stadard Error. Much cofuo are from a lack of apprecato of the dfferece betwee populato value ad etmate of thee populato value baed o ample. Moreover, there cofuo a to how the tattcal cocept of ubaede whe aocated wth the uual dvor of - the defto of the ample varace lead to a baed etmate of populato tadard devato. The reult are ot ew ad ca be foud ay elemetary textbook (or rather ay collecto of everal elemetary textbook), but we demotrate from example draw from the urveyg lterature that the cofuo deftely real. POPULATO QUATTES AD MATEMATCAL EXPECTATO Populato a potetal et of quatte that we wat to make ferece about baed o a ample from that populato. Coder a fte populato, uch a the examato mark m of a group of tudet a gle ubject. th cae calculatg tattc preet o problem ce we have complete formato about the populato. The mea m, varace ad tadard devato of the fte populato are, m m b g m -m () () bm - mg (3) ote that the varace the average quared dfferece of a member from the populato mea m. or large fte populato (uch a all tudet a coutry), we may wh to etmate the populato average baed o the ample average wth ome dea of the accuracy of etmato. ow coder urveyg meauremet, draw from fte populato wth the attedat dffculte of etmato ce populato average ca ever be kow. To deal wth th, probablty dety fucto have bee troduced to model fte populato. urveyg, ormal (Gaua) probablty dety fucto are the uual model. 74

2 TE AUSTRALA SURVEYOR Vol. 44 o. A probablty dety fucto a o-egatve fucto where the area uder the curve oe. or f( x)³ ad + f( x) dx The probablty a member of the populato le the b terval a to b f( x) dx. The populato mea m a defed a (reyg 97, p.77) + the value of f( x) are ot probablte. m xf ( x ) dx (4) where f( x) replace ummato (). ad the tegral replace the By aalogy wth (), the populato varace defed a the mea quared dfferece from m (reyg 97, p.8) + ax-mf f( x) dx (5) where f( x) replace ad the tegral replace the ummato (). Smlarly to (3), the populato tadard devato defed a + ax-mf f( x) dx (6) or the famly of ormal probablty dety fucto x-c ad > f the populato mea - d f( x; c, d) e d p m gve by (4) c, the populato varace gve by (5) d ad the tadard devato d. Th expla the uual preetato tattc text of the famly of ormal dtrbuto a p.7). e p - x-m (e.g. reyg 97, f we deote a potetal member of the fte populato a X, aother ame for the populato mea m the expectato of X ad uually wrtte a Ekp X. m Ekp X (7) We ca coder more geeral expectato ElgX afq where (reyg 97, p.83) EgX lafq + gx affxdx ( ) (8) other word gaf x f( x) ubttuted for xf( x) the rght had de of (4). tattc text X called a radom varable. ece, ug the otato of expectato, the populato varace + ax- mf f( x) dx EoaX-mft (9) JUE 999 ESTMATES O POPULATO QUATTES AD TE DEAS O UBASEDESS AD DEGREES O REEDOM Suppoe that we have a ample of potetal member of a ormal populato X, X, L X. Throughout, we wll aume that thee member wll be obtaed depedetly of oe aother. urthermore, we have a rule for etmatg oe of the three populato quatte m, or, whch we wll deote geercally a k. We wrte th rule a, T h X, X,L X b g () ET kp ad { b - kpg}. A overall meaure of the a -kf The T alo a radom varable wth mea m T varace T E T E T accuracy of T t mea quare error m E T. t tur out that (ee Appedx) ob gt { c lqh } c lqh m E T κ E T E T + κ E T (a) Th the equato gve by Gau (8-8, p. ) for what he termed m the mea error or mea error to be feared. The term k - ET kp (a) the dfferece betwee the populato quatty k ad the average value of the etmatg rule T ad termed the ba of T. ece, (a) ca be expreed word a mea quare error varace + ba a f (b) Whe the ba ero, T ad to be ubaed ad mea quare error equal varace. The uual rule for etmatg m the ample mea X, T X X () The uual rule for etmatg the ample varace, c h (3) T X - X - The uual rule for etmatg the ample tadard devato, T X - X - Other rule are poble; e.g. we could ue a a rule for etmatg, T e X -X c h (4) c h (5) t wa Sr Roald Aylmer her who demotrated that hould be preferred to e for etmatg (Box 978, pp. 7-3), ad th lead hm to the dea of a -f degree of freedom for etmatg a ample of depedet obervato. 75

3 JUE 999 A mple llutrato may expla the mportace of the dvor - rather tha. Coder a populato of oly, called m. By defto, the populato mea jut m [ee ()], ad by () ad (3) ad are both ero ce there o varato the populato. owever, f we have a ample of e oe, from a fte populato.e., we mut ue th gle obervato to etmate m. Clearly, we have o further formato th ample to etmate varato the populato. That we have ued all our degree of freedom to etmate m, hece, our etmate of (or ) mut be udefed. owever, f we ue e (5) our etmate ero but f we ue (3) the awer udefed (dvo by ero!) a we hould expect. See edall ad Bucklad (97) for a excellet dcuo of the cocept of degree of freedom. t ca be how that for the rule T X, T ubaed for m ad t varace (reyg 97, p. 75), hece the tadard devato of X ad referred to a the tadard error of X. The uual rule for etmatg the tadard error of X, tadard error (6) Th formula how why, tutvely, we thk that the larger the ample e the more prece our determato are. A mple way to remember th to ote that to double the preco of meauremet we mut quadruple the umber of meauremet take. t alo ca be how that a ubaed etmator of but that a baed etmator of. Th wa kow a early a 9 by her (9, p. 76), log before the cocept of ubaede wa formally troduced a a term theoretcal tattc. Referrg to the formulae above, the acto of takg a quare root chage the property of ubaede. Th more a accdet of mathematc rather tha a caue of faulty etmato but t ot well apprecated geeral. Th reult could fact be ued a a argumet for ug baed etmator. owever, t ca be how that the approprate dvor c for ubaed etmato of gvg where * x - x c b g (7) TE AUSTRALA SURVEYOR Vol. 44 o. R U G c S - V (8) G T W ad Gaf x a gamma fucto (Spegel 968). A proof of th equato gve the Appedx ad t teretg to ote that c udefed whe thu * alo udefed for a ample of oe. Value of c are gve Table c Table. Value of the dvor c for ubaed etmato of So far, the empha ha bee o defg rule for etmatg populato quatte. Whe we ubttute meaured value to thee rule for etmator, we obta a gle umber called a etmate. We are tereted three dtct defto for the populato quatte. or example, the mea ca have three dfferet cootato. The frt the theoretcal defto of m gve by (4); the ecod the rule we apply to etmate m gve by (). The thrd the umber, or etmate, we obta whe we ubttute a partcular ample that rule. Th dtcto hould be bore md whe ug the term mea ad average to decrbe the ame quatty or proce. The ame ubtle dfferece apply to the uage of tadard devato ad mea quare error. SOME EXAMPLES ROM SURVEYG LTERATURE Cofuo of termology ad cofuo of ueage appear qute frequetly urveyg lterature ome example follow: [] STADARD ERROR. The Geodetc Gloary (GS 986, p.77) ay: error, tadard Equvalet, for the mot part to tadard devato. owever, tadard error alo ued to mea a umber of thg uch a the tadard devato of the mea ad the tadard devato calculated from large ample. t evdet from the foregog ecto that tadard error o way equvalet to tadard devato ay ee. We have how that tadard error ha the populato ee of whch the tadard devato of the rule 76

4 TE AUSTRALA SURVEYOR Vol. 44 o. X for etmatg m. f we whed to etmate (tadard error) we would ue the rule ad would obta a gle umber for ay partcular ample, whch we could alo call tadard error. The mportace of tadard error fdg a cofdece terval. [] ROOT MEA SQUARE (RMS). The Geodetc Gloary (GS 986, p.77) ay: error, root-mea-quare A quatty meaurg devato of a radom varable from ome tadard or accepted value; t value determed by b g x -x lq the et of radom varable, ad lq x where x the correpodg et of accepted value. Th at frt glace look lke the rule above. Cofuo prg from the term lq x ce x almot uverally accepted a a mea value. fact th defto, t refer to a et of accepted value ad the term wth the ummato are fact dfferece. The equato would be better expreed a RMS x -a b g (9) where a refer to accepted value. Th fact how RMS ued geod modellg where the accepted value ofte ero (eathertoe et al 997, Table ). Whe the accepted value ay ample a (a cotat) ad the mea of the ample x, the (9) become (ee Appedx) armsf R b g a f S - T U x x V W + x - a (a) or word armsf etmate of varace + aetmate of baf (b) ad th the ample aalogue of (b) above. To llutrate thee three formulae the frt row of Table eathertoe et al ha a mea of.37, a tadard devato of.66 ad rm of.894 baed o a ample of 59. The accepted, or target value ero. the rghthad-de of (a) ax- af ax- f ad b g.. 58 x - x JUE 999 The left-had-de of (a) whch actually Th mea that ther colum, headed t.d, doe ot correpod to the uual defto of tadard devato but rather to our e, gve by (5). Clearly, the dfferece of o practcal mportace to the terpretato of ther reult. [3] A EXAMPLE O CLEARLY DEED ROOT MEA SQUARE (RMS) Rapp (997) a paper o geod modellg preet data Table. There are two colum labelled RMS, each cae relatg to dfferece. t clear that our formula (9) ued wth the accepted value a equal to ero. Aother colum the table gve the mea of the dfferece betwee the two model. The quare of thee mea correpod to the ba-quared term our equato (a) ad (b). [4] EXAMPLES O POORLY DEED ROOT MEA SQUARE (RMS) Grejer-Breka et al (998) a paper o GPS error modellg tate: Example of etmated tadard devato (Root- Mea-Square, RMS) for poto, velocty, ad oretato are plotted g Clearly, etmated tadard devato are meat ad there are o accepted value preet. Lagley (99) a paper o the mathematc of GPS tate: We ca determe expermetally by makg a large umber of obervato ad calculatg the um of the quare of the error the obervato dvded by oe le tha the umber of obervato made. t th method of computato that gve t ala of root-mea-quare error (rm) error. Th correpod to our formula (9) wth accepted value a all equal to ero ad the dvor - tead of. t wll oly be a good etmator of whe the ba ero. e ha ued a hybrd of gve (4) ad RMS gve (a) ad (b). [5] SCETC CALCULATOR UCTO EYS. May calculator, e.g. the ewlett-packard P3S, have two fucto key for calculatg tadard devato, labelled ad. The key correpod to the of (3) ad therefore applcable where we have complete formato o a fte populato. Sce t rare that the etre populato could be ampled ay urveyg exerce, th value would be approprate. The other fucto key correpod to the of (4), whch a etmator of a fte populato defed by (6). The dfferece the two value that may are mply due to the dvor, ether or -. 77

5 JUE 999 Clearly, th dfferece wll oly be mportat mall ample. Tradtoally a ample e 3 regarded a mall. COCLUSO We have hghlghted the dtcto betwee populato value ad ample value of tattcal parameter ad retated well-kow rule for the calculato of mea ad varace or ther etmator. the proce, we have demotrated that tattca regularly ue a baed etmator of. her (9, p. 366) h ummary tate: Durg the rapd developmet of practcal tattc the lat few decade, the theoretcal foudato of the ubject have bee volved great obcurty. Adequate dtcto ha eldom bee draw betwee the ample recorded ad the hypothetcal populato from whch t ha bee draw. We have how that th lack of dtcto betwee populato value ad ample value rema a ource of cofuo. Our paper hould alo clarfy the dtcto betwee Root Mea Square (RMS) ad tadard devato. RMS value clude compoet of varace ad ba ad we have demotrated that the cocept of RMS ha a populato defto ad a correpodg etmate a ample from that populato. Ug ample RMS ad mea ca dcate ba mathematcal model a clearly demotrated the paper by Rapp (997) ad eathertoe et al (997). APPEDX Proof that mea quare error varace + abaf Mea quare error defed a the average quared dfferece of a member T of a radom populato from ome populato quatty k. Ug the otato of expectato we may wrte mea quare error E T-k (A) We may wrte a E T- k E T- ET + ET -k f a f { b kpgbkp g} ET { b - ET kpg + bt-et kpgbet kp- kg+ bet kp-kg} E{ bt- ET kpg} + Em bt-et kpgbet kp- kgr+ E{ bet kp-kg} TE AUSTRALA SURVEYOR Vol. 44 o. ow, ce ElT- ET kpq ad E cotat cotat k p, we may wrte a f ob kpgt ob kp EobT - Ekp Tgt+ bk - Ekp Tg varace + abaf E T - k E T - E T + E E T -k gt Proof that * a ubaed etmator of the populato tadard devato. (A) or depedet ormal radom varable X, X,L X each wth wth mea ad varace the um of the quare, uually deoted by c (ch-quared), ha a probablty dety fucto - x f( x) x e G - where faf x whe x < ad a potve teger kow a the degree of freedom. (A3) f X, X,L X are depedet ormal radom varable wth mea m ad varace the (reyg 97, p. 8) a f T - (A4) ha a c - dtrbuto wth a probablty dety fucto f() t - G - t - -3 t e (A5) where - ha replaced (A3). ow from (A4) we may wrte equal to ome fucto of T or - gatf a f - ad by the geeral defto of expectato, we may wrte the expected value of a a f E kp EgT l q gt a f ftdt a f a -f ftdt a f - (A6) 78

6 TE AUSTRALA SURVEYOR Vol. 44 o. a but - t (A5) mplfe to f E kp G whch whe ubttuted to (A6) wth G - t - t e dt G - t - t e dt G (A7) ad the tegral (A7) evaluate to uty. Th mplfe to kp E G - - G k Sce k a cotat, the rule of expectato ca be ued to gve * R E ST U k VW E ad * a ubaed etmator of the populato tadard devato. Proof that RMS etmate of varace + ba whe accepted value a a cotat a f a f rom (9) wth a a (a cotat) we have armsf b g x -a We may wrte armsf bx - xg+ ax-af bx xg ax afbx xg ax af (A8) bx xg ax afbx xg ax af b g ow, ce x - x ad acotatf acotatf we may wrte R armsf S b - g a f T x x V W + x - a REERECES U JUE 999 Box, Joa her., 978, R.A. her, The Lfe of a Scett, Joh Wley & So, ew York. (A9) eathertoe, W.E., Glllad, J.R. ad earley, A..W., 997, Data preparato for a ew Autrala gravmetrc geod, The Autrala Surveyor, Vol. 4, o., March 997, pp her, R.A., 9, A mathematcal examato of the method of determg the accuracy of a obervato by the mea error ad by the mea quare error, Mothly otce of the Royal Atroomcal Socety, 8, pp her, R.A., 9, O the mathematcal foudato of theoretcal tattc, Phloophcal Traacto of the Royal Socety of Lodo, Sere A, Vol., pp Gau, C.., 8 8, Theory of the Combato of Obervato Leat Subject to Error: Part Oe, Part Two, Supplemet. A tralato of Theora Combato by G.W.Stewart, Socety for dutral ad Appled Mathematc (SAM), Phladelpha, 995. Grejer-Breka, D.A., Da, R. ad Toth, C., 998, GPS error modellg ad OT ambguty reoluto for hghaccuracy GPS/S tegrated ytem, Joural of Geodey, Vol. 7, o., ovember 998, pp edall, Maurce G. ad Wllam R. Bucklad, 97, A Dctoary of Stattcal Term, 3 rd ed, Olver & Boyd, Edburgh. reyg, Erw, 97, troductory Mathematcal Stattc, Joh Wley & So, ew York. Lagley, R.B., 99, The mathematc of GPS, GPS WORLD, Vol., o. 7, July/Augut 99, pp GS, 986, Geodetc Gloary, atoal Geodetc Survey, Rockvlle, MD, September 986. Rapp, R.., 997, Ue of potetal coeffcet model for geod udulato determato ug a phercal harmoc repreetato of the heght aomaly/geod udulato dfferece, Joural of Geodey, Vol. 7, o. 5, Aprl 997, pp Spegel, Murray R., 968, Mathematcal adbook, Schaum Outle Sere Mathematc, McGraw-ll, ew York. 79