260 I-1 INTRODUCTION PERCENT ERROR AND PERCENT DIFFERENCE

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1 60 I- INTRODUCTION PERCENT ERROR AND PERCENT DIFFERENCE A percet error hould be calculated whe a eperetal value E copared to a tadard or accepted value S of the ae quatt. We epre the dfferece betwee E ad S a a percet of the tadard value S: E - S PE 00 S Th forula eld a potve reult f the eperetal value hgh coparo to the tadard value, ad a egatve reult f t low. A percet dfferece hould be calculated whe two eperetal value, E ad E, are copared to each other, ad there o tadard value for coparo. I th cae we epre the abolute value of the dfferece betwee the eperetal value a a percet of the average of the two value: () PD E - E Percet dfferece alwa potve. 00 E - E ( E E ) ( E E ) 00 () THE METHOD OF DIFFERENCES The Method of Dfferece ued whe oe varable beleved to chage b equal aout ucceve eaureet. Th ethod eld the average chage the varable per terval. A phcal eaple of uch a cae the tretchg of a prg b a force whch creae b equal aout ucceve terval. Aother eaple the peed of a dee fallg bod eaured at equal te terval. A a cocrete eaple to llutrate the Method of Dfferece, uppoe that we wat to eaure the wdth of oe of a uber of detcal floor tle a roo. Oe ethod to place a eaurg rod dow, eaure a tle, ove the rod, eaure aother tle, ad o o. We could the fd the average of the dvdual eaureet. However, ovg the rod creae the eperetal error. A ore prece ethod to place the rod dow ol oce, ad the to take a et of readg of the poto of ucceve crack (aued to be of eglgble thcke). For eaple, ucceve readg would pa fve tle. Let the readg be a, b, c, d, e ad f. There are everal wa whch thee uber could be cobed order to eld a gle tle wdth.

2 60 I- A poor ethod would be to ue the forula f - a w 5 The reult would be approatel correct, but le prece tha t could be, ce we have ued ol two of the readg. A ethod that look better at frt to calculate the fve wdth, b-a, c-b, d-c, e-d, ad f-e, ad the average the. The equato for th procedure w ( b - a) ( c - b) ( d - c) ( e - d) ( f - e) 5 Cloe pecto of th equato, however, how that t reduce to equato (), o we have gaed othg b all of our etra work. The Method of Dfferece ue each of the readg oce, ad o readg cacel out. I order to ue t, we ut have a eve uber of readg. For a odd uber of readg, ether the frt or lat readg ut be dcarded. The, we dvde the readg to two et. I our eaple, et oe would cot of readg a, b ad c; et two would cot of readg d, e ad f. We ca get oe etate of the average b ug readg a ad d, d - a. Here, the dtace (d-a) pa three tle, o we have dvded b three. We ca get two other etate ug the par b ad e ad c ad f, e - b ad f - c The ethod of dfferece ue the average of thee three etate w d - a e - b f - c. ( d - a) ( e - b) ( f - c) I the geeral cae, uppoe that we have ucceve readg, A to A ad B to B, of oe varable S. The the average chage S per terval gve b ΔS ( B - A ) ( B - A ) L ( B - A ) () (4) (5) (6)

3 60 I- UNCERTAINTY IN DATA AND CALCULATIONS I. Ltato o Preco of Meaureet No eaureet eact. The preco of a eaureet lted b the ature of the eaurg truet telf, b the codto of eaureet, ad b the kll of the pero ug the truet. For eaple, the preco of our eaureet of the legth of a board, ug a eter tck, lted b the fact that the allet dvo o the tck a lleter. We ca etate the legth of the board to a preco of the earet ffth or teth of a lleter b etall dvdg the lleter to aller dvo, but oo we reach a lt. Eve f we were to ue a agfg gla a effort to dvde the lleter to eve aller part, we would evetuall be lted b rregularte the ar o the eter tck. A coo ource of error ug a eter tck paralla error. Th error caued b the le of ght ot beg precel perpedcular to the tck. For eaple, eaurg the legth of a heet of paper b placg the tck dow flat o the paper eal lead to paralla error; t would be better to place the eter tck o edge o that the ar touch the paper. Paralla error ca occur a tuato; t occur, for eaple, readg a electrcal eter whe the poter ove a dfferet plae fro the cale. II. Abolute ad Relatve Ucertat Suppoe that we eaure the legth of a pece of paper a 0.00 c, b ug a eter tck. The, after coderg the proce of eaureet, we decde that we ght be error b a uch a 0.0 c (oe-ffth of the allet dvo of the tck). We uuall epre th b ag that the legth 0.00 ± 0.0 c. I th cae, 0.0 c would be the abolute ucertat the eaureet or the poble abolute error the eaureet. We defe the relatve ucertat a eaureet or the poble relatve error a eaureet to be the rato of the abolute ucertat to the actual eaureet; for our eaple, 0.0 c/0.00 c %. Note that the abolute ucertat ha the ae ut a the eaureet; wherea the relatve ucertat utle ad ofte epreed a a percet. Ucertat ad poble error are o. Ucertat ore ofte aocated wth eaureet. Poble error ore ofte aocated wth reult calculated fro eaureet.

4 60 I-4 III. Ue of Dfferetal to Repreet Ucertate Let the legth of the paper ecto II be called L. The we repreet the abolute ucertat L b the bol dl. We repreet the relatve ucertat L b the rato dl/l. Sce relatve ucertate are uuall rather all, the cocept of a dfferetal ueful deterg how two or ore ucertate cobe together whe eaureet are ued atheatcal operato. Th wll be llutrated the et ecto. IV. How Ucertate Data Affect Calculated Reult Coder a forula F(A) evaluated for oe eaured quatt A whch ha a df df ucertat da. If da all, F(A ± da) F(A) ± ( da ) da, where da the dervatve of the forula wth repect to A evaluated at the eaured value of A. Note: Th forula pl the frt two ter of the Talor Epao of F at A. The ucertat df the calculated value for F the df da da. Th ae relato ca be ued for a forula F, f A a calculated quatt. Eaple: Let F(A) A ad A be eaured a.54 ² ± 0.07 ² df da /(.88 ) A.54 df df da da ² F.88 ± 0.0 Now, coder a forula F(A, B, C) evaluated for the depedet quatte A, B ad C whch have ucertate da, db ad dc, repectvel. If da, db ad dc are all, F F F F(A ± da, B ± db, C ± dc) F(A, B, C) ± ( A ) da ± ( B ) db ± ( C ) dc, where the dervatve are evaluated at the eaured value of A, B ad C. The ucertat the F F F calculated value for F the df da B db C dc. Let u ow coder the ot frequet operato we wll perfor calculato: addto, ubtracto, ultplcato, dvo, ad rag to a power. A. Addto Let S be the u of two depedet quatte, S A B. S S ad B. Therefore, ds da db. Eaple: (5.0 c ± 0. c) (0.0 c ± 0.5 c) 5.0 c ± 0.7 c For calculato volvg depedet quatte, refer to ecto IV-F.

5 60 I-5 B. Subtracto Let D be the dfferece of two depedet quatte, D A - B. D D ad B -. Therefore, dd da db. Eaple: (5.0 c ± 0. c) - (0.0 c ± 0.5 c) 5.0 c ± 0.7 c C. Multplcato Let P be the product of two depedet quatte, P AB. P P B ad B A. Therefore, dp B da A db. Dvdg both de b P A B, we fd dp/ P da/ A db/ B. Eaple: (5.00 ± %)(8.00 ± %) 40.0 ² ± 4% 40.0 ² ±.6 ² (.6 ² obtaed b calculatg 4% of 40.0 ².) D. Dvo Let Q be the quotet of two depedet quatte, Q A/B. Q Q /B ad B -A/B². Therefore, dq /B da A/B² db. Multplg both de b / Q B / A, we fd dq/ Q da/ A db/ B. Eaple: (.00 g ± %)/(.50 c³ ± %).00 g/c³ ±4%.00 g/c³ ± 0.08 g/c³ Note: The geeral rule for addto ad ubtracto that the abolute ucertat a u or dfferece equal to the u of the abolute ucertate the quatte added. The geeral rule for ultplcato ad dvo that the relatve ucertat a product or quotet equal to the u of the relatve ucertate the factor. Spl put: If F A ± B, df da db. If G A B or G A/B, the dg/ G da/ A db/ B. E. Epreo Cotag Ol Power, Multplcato ad Dvo Sce power are pl repeated ultplcato, we have, effect ol ultplcato ad dvo. Fro the reult of part C ad D, we ee that we ca pl u the relatve error. For eaple, f A B p C q /D r, the da/ A p db/ B q dc/ C r dd/ D The det of a coe V, where the a, r the radu, ad h π r h ρ the heght. The relatve ucertat the det dρ/ρ d/ dr/r dh/h, where d, dr ad dh are the abolute ucertate the a, radu ad heght repectvel.

6 60 I-6 F. Epreo cotag depedet quatte. Oe ut be careful to ol appl the rule above to epreo cotag depedet quatte or ele error ca be double or trple couted. Two quatte calculated fro the ae eaureet are depedet, a are a eaured quatt ad a value calculated fro t. Oe ple geoetrc eaple of the latter cae the calculato of the dfferece betwee the crcuferece ad daeter of a crcle. Coder a crcle wth a daeter eaured to be 0.0 c ± 0. c (0.0 c ± %). It crcuferece C πd. Sce π a cotat, t relatve ucertat zero ad the crcuferece wll have the ae relatve ucertat a the daeter, %. The reult C.4 c ± 0.6 c. Ug the rule above, we would fd C d.4 c ± 0.8 c. The ucertat the dfferece actuall le tha 0.8 c. To calculate the actual value we ut put together the two forula above, C d πd d (π-)d. Sce (π-) a cotat, t relatve ucertat zero ad the dfferece wll have the ae relatve ucertat a the daeter, %. Thu, C d.4 c ± 0.4 c. A eaple of the frt cae would be the rato of the volue of a phere to t urface area. Coder a phere wth a daeter eaured to be 0.0 c ± 0. c (0.0 c ± %). The urface area of the phere A πd. Sce π a cotat, t relatve ucertat zero ad the urface area wll have twce the relatve ucertat of the daeter, 4 %. The reult A 4 c ± 4 %. The volue of the phere V /6 πd. Sce π/6 a cotat, t relatve ucertat zero ad the volue wll have three te the relatve ucertat of the daeter, 6 %. The reult V 54 c ± 6 %. Ug the rule above, we would fd V/A.66 c ± 0 %. The relatve ucertat the rato actuall le tha 0 %. To calculate the actual value we ut put together the forula above, V/A (/6 πd )/( πd ) d/6. Sce 6 a cotat, t relatve ucertat zero ad the rato wll have the ae relatve ucertat a the daeter, %. Thu, V/A.66 c ± %. A how the two eaple, whe a epreo cota depedet quatte, t ecear to algebracall plf the epreo before calculatg the relatve ucertat. G. Cople Epreo Cople epreo a be hadled b repeated ue of the prevou reult or b the ethod outled at the begg of th ecto (ecto IV). BC Eaple: Let A DE F where B 50.0 ± 0.5, C 0.0 ± 0., D 00 ±, E.00 ± 0.06, ad F 50.0 ±.0 ². Soluto I: B C DE F ( 0.0 ) ( 00 )(.00 ) ( 50.0 )

7 60 I-7 BC C DE F D E F - EBC ( DE F) - DBC ( DE F) BC ( DE F) ( )( 0.0 ) ( 00 )(.00 ) ( 50.0 ) (.00 )( 50.0 )( 0.0 ) ( 00 )(.00 ) ( 50.0 ) ( 00 )( 50.0 )( 0.0 ) ( 00 )(.00 ) ( 50.0 ) ( 50.0 )( 0.0 ) ( 00 )(.00 ) ( 50.0 ) A A A A A da B db C dc D dd E de F df da ( )(0.5 ) (.00 4 )(0. ) ( 0.40 )( ) ( 8.00 )(0.06 ) ( )(.0 ²) da.8 S ( 50.0 )( 0.0 ) BC A ± DE F 0.0 ( 00)(.00) ( 50.0 ).8 Soluto II (repeated ue of earler reult): BC A DE F A A ( 50.0 ± 0.5 )( 0.0± 0.) ( 00 ± )(.00 ± 0.06) ( 50.0±.0 ) ( 50.0 ± % )( 0.0 ± % ) ( 00 ± % )(.00 ± % ) ( 50.0±.0 ) ( 50.0 ± % )( 0.0 ± % ) ( 00 ± % ) ( 50.0±.0 ) ( 50.0 ± % )( 0.0 ± % ) ( 50 ± 0 ) ( 50.0 ± % )( 0.0 ± % ) ( 00 ± 9.0 ) ( 50.0±.0 ) ( 50.0 ± % )( 0.0 ± % ) A A 0.0 ± 9% 0.0 ±.8 ( 50 ± 4% )

8 60 I-8 SIGNIFICANT FIGURES AND DECIMAL PLACES Ofte, we wh to quckl etate the preco of our calculated reult wthout applg the rgor of the prevou ecto. To do th, we ue rule that relate the uber of decal place or gfcat fgure we ca keep a calculated value to the uber of decal place or gfcat fgure our data. I. Defto The uber of decal place a uber the uber of dgt to the rght of the decal pot. The uber of gfcat fgure a uber the total uber of dgt, ecluve of leadg zero. II. Eaple The followg table how the uber of decal place ad the uber of gfcat fgure fve uber. Nuber Nuber of Decal Place Nuber of Sgfcat Fgure , or 4 The abgut the uber of gfcat fgure the lat eaple eal reoved b ug cetfc otato three gfcat fgure. III. Rule for Roudg Off Calculated Reult I addto or ubtracto, keep a a decal place the reult a the allet uber of decal place foud a of the uber beg added or ubtracted. Eaple: Notce that the uber of gfcat fgure the reult ca be ore tha the uber of gfcat fgure ether uber or le tha the uber of gfcat fgure ether uber. I ultplcato or dvo, keep a a gfcat fgure a the allet uber of gfcat fgure foud a of the uber beg ultpled or dvded.

9 60 I-9 Eaple: 8.75(49.86)/ or 90, but NOT (.7)(4.9) (.7) (4.9) (4.9) 80 or.8 0 The reaoablee of thee rule a be etablhed b ug the ethod dcued the prevou ecto. THE METHOD OF LEAST SQUARES Whe two varable, ad, are kow or beleved to have a lear relatohp to each other, the cotat ad b of the equato b a be obtaed fro the eperetal data b plottg a graph. The cotat are, repectvel, the lope ad Y tercept of the graph. Subjectve judget requred drawg the le that bet ft the eperetal data, f the pot are oewhat cattered. The Method of Leat Square allow the calculato of the lope ad tercept for the leat-quare le. The leat-quare le the le that ze the u of the quare of the vertcal dtace of each data pot fro the le. Let u a that we have data pot: (, ), (, ),, (, ). The dagra o the followg page how the le we are ee, alog wth oe tpcal data pot P wth coordate (, ). Let P' be a aocated pot o the le, havg the ae coordate. Sce P' le o the le b, t coordate b. Let Δ be the dtace betwee thee two pot. Let S be the u of the quare of all uch dtace. Th the u that we wll ze. 5.5 P P 0 P' Δ 7.5 P' The dtace Δ Thu we have gve b the dfferece coordate of the pot P ad P'. ( b) b Δ.

10 60 I-0 The u S the gve b ( ) ( ) Δ b S ( ) b b b b b b S a fucto of two depedet varable ad b, ad cotat,,,...,,. I uch a cae we ca fd a relatve u (there o relatve au) b ettg both partal dervatve to zero. b S 0 b b S 0 Solvg thee equato ultaeoul, we fd ad b. The followg a uercal eaple, ug the four data pot (.5,.7), (.87, 6.07), (9.47,.7), ad (.7,.6) Su Su * The lat row how the u to the correct uber of gfcat fgure. It hould be oted that the uber of gfcat fgure each u creae whe ore data pot are preet. If there were three te a a data pot, the u of the -coordate would creae b about a factor of. Th would reult a u that would eceed 00, but would tll be accurate to the earet 0.0. The u would have 5 gfcat fgure. Subttutg the u to the equato for ad b, ad reeberg that four th cae, we obta

11 60 I ( 6.58) ( 6.58) b Our equato for the leat-quare le the, The X tercept, f eeded, a be calculated fro b , ad.4. WARNING: The equato for ad b the leat-quare ethod ted to gve zero dvded b zero. I order to obta eagful reult, oe eed to have a data pot. MICROSOFT EXCEL NOTE: The lope ad tercept of the leat-quare le ca be foud Ecel ug the SLOPE ad INTERCEPT fucto. Oe ut be aware of gfcat fgure whe ug thee fucto, ce Ecel doe ot take the to coderato. The ta for the two fucto SLOPE(kow_',kow_') INTERCEPT(kow_',kow_') Kow_' the depedet et of obervato or data. Kow_' the depedet et of obervato or data. MODIFIED LEAST SQUARES I oe cae, we wll ecouter a relato that ot ol lear, but a drect proporto. If ad are related b a drect proporto, the b wth b 0. Wth th cotrat, the u S gve b S ( Δ ) ( ) For th cae, the u S deped o ol oe ukow,. We ca fd the value of that ake S u b ettg the dervatve of S wth repect to equal to 0. 0 Solvg for, ds d

12 60 I- Sce there are o ubtracto th forula, oe doe ot ted to get zero dvded b zero wth the Modfed Leat Square. MICROSOFT EXCEL NOTE: Mcrooft Ecel progra alo cota a bult fucto that wll evaluate th forula, LINEST. The ta LINEST(kow_',kow_',FALSE) Kow_' the depedet et of obervato or data. Kow_' the depedet et of obervato or data. LINEST wll alo retur a arra cotag ad b for the regular Leat Square le. Th ca be doe b replacg the FALSE wth TRUE. LINEST ca alo be ued to obta regreo tattc. RESISTOR COLOR CODES The oal retace of a retor pecfed b the color of three bad prted o the retor. If preet, a fourth bad dcate the tolerace ad a ffth bad gve a relablt ratg. Whe ol three bad are preet the tolerace udertood to be 0 %. Bad the bad earet a ed of the retor. Bad ad pecf the frt two dgt of the retace, A ad B, repectvel. Bad pecfe the epoet C the forula R AB 0 C. A, B ad C have teger value foud the table below. Note that A ad B are ot ultpled, but are the dgt of a two dgt uber wth A the te place ad B the ut place (ee eaple below). Color lver gold black brow red orage ellow gree blue volet gre whte Bad or Bad Bad 4 0 % 5 % % % 0.5 % 0.5 % 0. % 0.05% Eaple: Bad Bad Bad Bad 4 A B C Tolerace Retace brow black brow lver 0 0 % 00 Ω 0 % red red orage 0 % kω 0 % volet ellow gold gold Ω 5 % Note: Mot retor wth a tolerace of % or le have four bad that pecf the oal value. We wll ot ecouter thee th cla.