6 Random Errors in Chemical Analysis

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1 6 Rndom Error n Cheml Anl 6A The ture of Rndom Error 6A- Rndom Error Soure? Fg. 6- Three-dmenonl plot howng olute error n Kjeldhl ntrogen determnton for four dfferent nlt. Anlt Pree Aurte 4 Tle 6- Pole omnton of four equl-zed unertnte Comnton of Unertnte Mgntude of Rndom Error umer of Comnton Reltve Frequen + U + U + U + U 4 +4U / U + U + U + U 4 + U U + U + U 4 + U + U U + U 4 +U 4 4/ U + U + U U 4 U U + U + U 4 + U + U U U 4 + U U + U U 4 U + U U + U 4 U + U + U U 4 + U U U + U / U U U U 4 U + U U U 4 U U + U U 4 U 4 4/60.50 U U U + U 4 U U U U 4 4U /

2 Fg. 6- Frequen dtruton for meurement ontnng ( 4 rndom unertnte; ( 0 rndom unertnte nd ( ver lrge numer of rndom unertnte. 6A- Dtruton of expermentl dt Tle 6- Replte Dt on the Clrton of 0-mL Ppet Trl ml Trl ml Trl ml Trl ml Trl ml Trl ml Men volume 9.98 ml Spred 0.05 ml Medn volume 9.98 ml Stndrd devton ml Tle 6- Frequen Dtruton of Dt from Tle 6- Volume Rnge, ml umer n Rnge % n Rnge 9.969~ ~ ~ ~ ~ ~ 9.987~ ~ ~ Fg. 6-5

3 6B Stttl Tretment of Rndom Error 6B- Smple nd Populton 6B- Properte of Gun Curve Fg. 6-4 orml error urve. The tndrd devton for urve B twe tht for urve A; tht, σ B σ A. ( The the devton from the men n the unt of meurement. ( The the devton from the men n unt of σ. Thu, the two urve A nd B re dentl here. Populton Men μ nd mple Men x Populton tndrd Devton (σ ( x σ μ z (x μ σ Are under Gun Curve μ ± σ 68.% μ ± σ 95.4% μ ± σ 99.7% Smple tndrd devton ( ( x x ( d ( x x 6

4 Stndrd Error of the Men: m Vrne ( ( x x ( d Coeffent of Vrton (CV, Reltve Stndrd Devton (RSD CV x 00% (% RSD Spred or Rnge (w Ex. 6- The followng reult were otned n the replte determnton of the led ontent of lood mple: 0.75, 0.756, 0.75, 0.75 nd ppm P.. Clulte the men nd the tndrd devton of th et of dt. Smple x x.77 x x ( x ( x.77 x pp P ppm. Clulte the ( vrne, ( reltve tndrd devton n prt per thound, ( oeffent of vrton nd (d pred. ( ( ( RSD 0.008/ ppt 5.0 ppt ( CV 0.008/ % 0.50 % (d w ppm P 7

5 Ex. 6- Gluoe level re routnel montored n ptent ufferng from dete. The gluoe onentrton n ptent wth mldl elevted gluoe level were determned n dfferent month petrophotometr nltl method. The ptent w pled on low-ugr det to redue the gluoe level. The followng reult were otned durng tud to determne the effetvene of the det. Clulte pooled etmte of the tndrd devton for the method. Tme Gluoe onentrton, mg/l Month 08,, 075, 099, 5, 08, 00 Men Gluoe mg/l Sum of Squre of Devton from Men S. D Month 99, 975, 0, 00, Month 788, 805, 779, 8, Month 4 799, 745, 750, 774, 777, 800, Totl o. 4, Totl um of qure Sum of qure of Month ( ( ( ( ( ( ( pooled tndrd devton: pooled mg/l 6C Stndrd Devton of Clulted Reult Tle 6-4 Error Propgton n Arthmet Clulton Tpe of Clulton Exmple Stndrd Devton of + or or / ( + ( Exponentton x x( Logrthm log Antlogrthm ntlog ( 8

6 6C- Stndrd Devton of Sum or Dfferene * (± S + (± S - (± S (± (± (± ( ± 0.0 the um hould e reported.6 (± ( ± 0.0 6C- Stndrd Devton of Produt or quotent + ( ± ( ± ( ± * ( / ; ( ±?.97( ± 0.04 ± 0.06 ( + ( + ( ± 0.0 ( 4.0 ± ( ± (.97 ( ( (0.00 ± (± (± ± The nwer (± Ex 6-4. Clulte the tndrd devton of the reult of ± ± ± [80( ± ( ± 5] 4.( ± 0.4 [ 4.( 0..6( 0.] 0.050( , ( ± 0. + ( ± 0. ± , ( ± 0 + ( ± 5 ±..7( ± ( ± ( ±.] 4.( ± ( ±? 0 ± 0.8 (.7 ± ( 0.05 ±. + ( 850 ± ( S (± 0.07 ± round the nwer to.7(±

7 6C- Stndrd Devton n Exponentl Clulton x S S x Ex. 6-5 The tndrd devton n meurng the dmeter d of phere ± 0.0 m. Wht the tndrd devton n the lulted volume V of the phere f d.5 m? SV V 4 4 d 4.5 V π π π r 5.0 m Sd , S V , V 5. (± 0. m d.5 Ex. 6-6 The olult produt K p for the lver lt AgX 4.0 (± The molr olult of AgX n wter Solult (K p / ( / M Wht the unertnt n the lulted olult of AgX n wter? olult, K p, x ½ S , S Solult.0 (± M 0.05, S C-4 Stndrd Devton of Logrthm nd Antlogrthm Logrthm log Antlogrthm ntlog 0.0 Ex. 6-7 Clulte the olute tndrd devton of the reult of the followng lulton. ( log [.00(± ] ±? (± ( ntlog [.00(± 0.00] ±? 5.8 ± , S ( ntlog [45.4(± 0.] ±? (± , S

8 6D Reportng Computed Dt 6D- Sgnfnt Fgure ll of the ertn dgtl + the frt unertn dgt *Rule for gnfnt fgure. Dregrd ll ntl zero.. Dregrd ll fnl zero unle the follow deml pont.. All remnng dgt, nludng zero etween nonzero dgt, re gnfnt. *Sum nd Dfferene *Produt nd quotent *Logrthm nd Antlogrthm. In log. of no., keep mn dgt to the rght of the deml pont there re gnfnt fgure n the orgnl no.. In n ntlog. of no., keep mn dgt there re dgt to the rght of the deml pont n the orgnl no. Ex. 6-8 ( log , ( ntlog.5 0 ( log Ex. 6-9 A.484-g mple of old mxture ontnng enzo d, C 6 H 5 COOH (. g/mol, w dolved nd ttrted wth e to phenolphthlen end pont. The d onumed 4.6 ml of 0.8 M OH. Clulte the perent enzo d (HBz n the mple. 4.6 ml 0.8 mmol/ml. mg/mmol % HBr 00%.484 g 000 mg/g.749 % ( uret : ± 0.0 ml the S.D. of the volume wll e [(± (± 0.0 ] ½ ± 0.08 ml the reltve unertnt (± 0.08/ ppt ± 0.68 ppt

9 ( nltl lne : unertnt ± g ± 0.000/ ppt ± 0.09 ppt ( the A.U. n the molrt of the regent oln ± ± 0.000/ ppt ± 0.4 ppt (4 the R.U. n the molr m of HBz everl order of mgntude mller thn tht of the three expermentl dt nd of no onequene. (5 00 % nd 000 mmol re ext numer The nwer rounded o tht t reltve unertnt le etween 0. nd tme the lrget reltve unertnt of the nput dt. (0.68 ~ 0.( ~ 0.6 ppt.7 reltve unertnt (0./ ppt ppt.75 reltve unertnt (0.0/ ppt 0. ppt

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