Error- Obscrvcd valuc- Truc valuc
CHAPTER
II
ERROR ANALYSIS
Wc saw in Chaptcr 4'that thc crrors associatcd with cxpcrimcntal
mcasurcmcnts or with guantirics calculatcd from cxpcrimcntal data can bc
cxprcsscd quitc simply in tcrms of significant 6gurcs. Thc rulcs govcrning
thc usc of signiFcant figurcs amount to an clcmcntary lorm of crror.analysis, which is sufficicnt for most of our purposcs in gcncral chcmistry. Ordinarily, rhc quantitics that wc mcasure in thc gencral chemistry laboratory
are bascd upon a singlc trial or, at mostr upon duplicatc detcrminations.
Such cxperimcnts hardly justify a more sophisticatcd rypc of crror analysis
than that dcscribcd in Chaptcr 4.
Wc may, howcvcr. havc thc opporrunity to carry out morc cxact cxperiments whcrc it is imporunr thar r.,c bc ablc to cstimatc quitc accuratell, thc unccrtaintics associatcd with individual mcasuremcnts or with
rcsults dcrivcd from scvcral successivc mcasurcmcnts. Lct us suppose, for
cxamplc, that wc arc askcd to dctcrminc rhe pcrccntagc of chlorinc in a
sample of sodium chloridc, givcn thc cquipmcnt neccssary ro dctcrminc
thc pcrccntagc to four significant figurcs and rhc timc to makc fivc succcssivc dctcrminations. Wc might obtain thc dara listcd in Tablc I I .l .
Thcrc arc scvcral qucstions wc mighr raisc conccrning rhis data.
I . Since wc wcrc ablc, iri principlc, to obtain thc pcrccntagc of chlo'rine
to +0.01 pcr ccnt, why didn't wc gct thc samc vatuc in cach
trial
?
2. Should we rcjcct any o[ thcsc rcsults? (Trial 2 looks a littlc
dubious.)
3. 'What should wc rcport for lhc pcrccnragc of chlorinc? Thc avcragc
o[ all fivc trials? Thc avcrage omitting trial 2? Somc othcr
4.
.
answcr?
Should wc rcpcat thc cxpcrimcnt a fcw morc timcs to gct a
lcr" value lor the percentage ofchlorinc?
"bct'
ERNOfl ANAI,TS'S
5- How confidcnt can wc bc of thc valuc thar wc rcpon lor thc pcr_
ccnragcof chlorinc? What are (hc chaoces thar it wiil bc
-ithio
*0.01 olthe rruc valuc? *0.05?
Thcsc arc rypical of rhc kinds of qucstions that wc will arrcmpr to nna
.*_*:o for in rhis chaptcr. Bcforc procccding funhcr. ir will bc ,,.."r=rry
to defioc ccnain tcrrns that are uscd rcpcatcdly in crror analysis.
II.I
ACCURACY AND PRECISION
. . T-hc aicurecy of a mcasurcd quanrity indicatcs rhc rxrcnt ro
egrccs wirh ihc truc valuc. It is dcscribcd in terms o[thc crror:.
Error
Thcimaller
th-c
-
lrror, ihc
Obscrvcd valuc
-
(r
Truc valuc
more accirratc a mcasurcmcnt
which it
is.
r.l)
f
- Thc prccision of a mcasurcmcnt altor..s us ro estimarc its icproduci_
- b-di,y- It is dcrcribcd by thc dcviutioa, which is thc diffcrcn..
L.,r...r,
rhc cibscrircd valuc and thc averagc vatue. obtaincd from a ,..i.,
of .rr.rlltrcmcnis:
Dcviation
-
Obscrvcd value
-
Averagc vatuc
(
r 1.2)
Thc smatlcr thc dciiation, thc morc prccisc rhc mcasurcmcnr isTo illustratc rhc distinction bctwccn prccision and accuracy, considcr
ttrc data givcn in Tablc t l.l for rhc p.r..r,,"g. of chlorinc ln
sodium
dloridc- Thc avcrage valuc,
r4rqsr which
wtllgll t5
is olIe
oftcn rcfcircd ro as thc ariihmetic
is rcadily found ro bc 60.50 pcr ccnr.
Arithmcricmcan
r
60.50
+
60.41
+
60.5J
+
60.5a
+
60-52
5
-ry-6o.sb
Thc true valuc for thc pcrccnragc of chlorine in sodium chloride is,
ro
significant figurcs,'60-66 perccnr. (This numbcr is bascd upon
ih.
wcights of sodium and chtorinc, which havc bccn vcry
carctully
164
CHAPIER
dctcrmincd roy'uasignificanr 6gures.) Conscqucntly, tor
.t
l.l.
Equarions I l.t and I t.2 bccome:
Error =
valuc
Obscrvccl
'Deviation = Obscrvcd valuc
-
I'
thi dau otTablc
60.66
66.50
Thcse cquations can bc used to cxpress thc prccision and accuracy of tach
mcasurcmcnt o[ thc pcrccnragc of chlorinc (Tablc I t.2).
:
fobtc t
1.2
P*cirion ond Acrurocy.of hcorured perccntogci of Chtorinc
Trirt o-t::**
Eluc
I
60.50
2
60.4 I
:3
'4
l'
60.53
.
16
-0.2t
-0.13
-O-12
.
-0.
:
-
60.54',
S.
Err.or i.a.iio;:
60-52.'
-0-t4
ii
NoCl.
Pcrccnlr''
1Pc1c91t i.ir::
'.. dcviation
crrori i.- 1:
0.00
-0.09 ,"."
*0.03;':'
*0.04
. -:'
+0.02. ',.'
Clcarly, the prccision o[ thcsc mcasurcmcnrs is bcttcr than their accuta.cy. In evcr,v rrial, thc crror is gr.catcr in magnitudc than thc deviation.
ln gcncral, wc can always cxpcct thc prccision of mcasurcmcnts to surpass
thcir.accuracy. Onlv if the arirhmctic mcan coincidcs cxactly with rhc truc
va[uc will our mcasurcmcnts bc as accuratc as they are pricisc.
'
Frcgucntly, r,c rcfcr ro.,'h..para.nt..ror, o, thc perccnt.dcviation.
Thesc quanrirics arc dcfincd as lollows:
Pcrccntcrror==4"
value
I ruc
perccnt dcvia.tion
'
1gg
(11.3)
x too
= Antnmettc
: ,,:Pc'i1,tion
mcan
Ttrchumbcrs in ihc lasr rwo cotumns ar rhe right of Tablc
culatcd in this manncr.
(t 1.4)
tt.2 wcrc
Exomple I l.l Four diffcrcnr srudcnts mcasurc thc boiling point
o[ a ccrtain organic liquid at 760 mm Hg pressurc. Their rcsults
54.9'C. 54.4'C, 54.I'C, and 54.2'C
cal-
.
ERROn
'
ANAIYSIS
I6s
Thctruc boiling point, to-rhrcc slgnificanr 6gures, is 54.0'C. Dctermine. lorcach mcasurcmcnrr thc crror, dcviation pcrccnt crror.
and pcrcent dcviation.
Solulion Thc cmor can bc calculated dircctly from Equation
l.l.
I
In ordcr to usc Equatiori 1.t.2 ro calcularc rhc dcviarion, rze
must nnrt obtain thc arirhmcric.mcan. Wc could do rhis by adding
thc lour numbcrs and dividing thc sum by
amount of arithmaig wc might wrirc:
Arithmctic mcan
-
5.i.0 +
.
54.0
+
0.9
[our. To reducc rhc
+ 0.4 + 0.1 +
!9 -
0.2
i+.1
_4
For thc 6rsr measurcmcnr,
J. nur.,
Error = 5't-9 - 5{-0
..
Pcrccnt
:
".ro. . .il
Dcviarion
-#
Pcrccnt dcviation
Procccdir.rg
.
' 5.r.9
x fio - a!/o
-'54.1
-
,. lN =
0-j;
ag.|yo
Pcrccnt
:. Error
's.r.+
54.t
54.2
0-9;
similarly for cach mcasurcmcnr:
Obscrvcd
value
54.9
=
-
crror
0.9
0.4
0.1
0.2
Pcrccnt
Dcviation
dcviation
+2
0.5
+0.7
+0.2
+0.4
0-0
0.0
-0.1
-0.2
-0.6
-0.4
+0.9
EXERC'SES
l.
in
('K - 'C + 271-2')-
Rcpcat thc calculations
prcsscdin'K
Examplc
ll.l
for .rcmpcraiurcs
e:(-
- 2- In 1894, I-ord Raylcigh dcrcrmincd thc mass of nirrogcn gar filling
a ccrtain containcr a( a known prcssurc and rcmpcrarurc. Thc rcsults o?
succcssive wcighings wcrc as follows:
2.31029, 2.30999,
2.3103
2.3101
g,
2.31
g,
l6
2.3t0Og. 2.3102g, 2.)l0t
g, and
2.3096 g
g.
r66
CHAPIER TI
Thc nitrogcn hc uscd was prcparcd from air by rcmoving oxygcn, *atcr
vapor, and carbon dioxidc. Whcn hc pcrforincd similar cxpcrimcnts wirh
chcmically purc nilrogcn, he obtaincd a mass of 2.2997 g. Taking this to
bc thc "rruc" mass, .calculatc rhc dcviations and crrors o[ cach of thc
masscs listcd abovc.
I
I.2
TYPES OF ERRORS
'
Thc most scrious crrors that studcnts makc in thc chemistry labora-'
tory (or in lilc, for that mattcr) arc oncs which could cithcr bc avoidcd or
corrcctcd [or- Thcsc arc callcd dctcrminatc crrors. As an cxamplc, consider a studcnt.who is attempting to analyzc a mctal oxidc by hcating it
in a srrcam of hydrogcn to form rhepurc mctal.
Mo(s) + H,(s).- MG) + HrO(g)
lf he spills pan o[ his samplc, his rcsult is likcly to show a rathcr largc
crror. This crror could bc avoidcd by bcing morc carcful; thc only way to
"corrcct " for it would bc to rcpcat thc cxpcrimcnt.
As an. cxamplc of a dctcrminate crror which could bc corrccicd for
quirc rcadily, considcr thc dilcmma o[ a studcnt who determincs tirc dcnsity o[ bcnzcnc at 20'C by wcighing samplcs of thc liquid issuing lrom a
l0 ml pipcttc. Hc obtains thc following masscs:
8.681 g.8.678 g,8.683 g, and 8.678
S
Finding thc arithmctic mcan o[ thcsc masscs to bc 8.680 g, hc dividcs by
t0.00 ml and conhdcntly rcpons a dcnsity of 0.8680 g/ml, rusting that his
exccllent precision insures high iccuracy. Unfortunatcly, thc truc valuc is
0.8790 g/ml; thc srudcnt has madc an crror of morc than t pcr ccnt. Hc
failcd to rcatize that a l0 ml pipcttc docs not dctivcr cxactly tcn ml of
liquid. Had he takcn thc trouble to calibrate his pipctte with distillcd
warcr, he would havc lound that it dclivcrcd about 9.90 ml. Hc worjld thcn
havc rcponcd:
.
dcnsity bcnzcne
=
8.680 9/9.90
mI =
0.977
t7.,
with an crror of about 0.2 pcr ccnt.
Eiomplc
I1.2
A studcnt dctcrmincs thc gram cquivalcnt wcight
of a mcral by rcducing ir whh hydrogcn. . Hc makes rhrcc wcighinss:
"
tcst lubc
mass = I
tcst tubc
tcst tube
4 mctal oxidc
+ mctal
maSS
: I
mass = C
EnnoR.ANAtySrs
...
167
. and calculates thc gram cquivalcnt wcighr using rhc cquation:
c-E.w.
,
-
g.0oogo
I
massmctal
(c - -{)
= 8.oooso
-. x (g _ C)
f,assoxygen-
How will cach o[ thc following dctcrminarc crrors affccr thc acwill thcy makc ir largcr or smallcc rhan
curacy of his rcsutt (i.c.,
thc truc valuc) ?
a. ln wcighing thc tcs( tubc. hc rccords a masi ooc gram
.
grtarcr than rhc truc niiss-
b. After weighing rhc mcrat oxidc
of the oxidc.
c.'Hc fails ro conyen all
plus tcsr tubc, hc spiils pa rr
o[thc mcral oxidc (o mcral.
Solurion
a- Mass-l is too targc;
8 and C erc prcsumably
corrccr.
From thc cquaiion for G.E.W., wc scc rhar rhc catculatcd
mars of mctal, C
- A, will
bc too smatt. Hcncc. thc
- .{). witt bc
G.E-W., which is dircctly proponional to (C
'
too small-
b. Hcrc,
I
and 8 will bc corrcct. Miss C will bc too srhall.
rhird rvcighings. Looking ar ihc cquarion for G.E.W.. rvc scc lhar
(C -..1)will bc roo small, (B - C) will bc too largc, and
thc calculatcd G-E.W. wilt bc too smalt.
c. Again, A aad B arc corrc(; C is roo largc bccausc not all
of rhc samplc is rcduccd. Following thc rcasoning of parr
(b), rvc dcducc thar thc calcularcd G.E.W. will bc too
bccausc samplc is losr bcrwccn thc sccond and
largc.
Dctcrminatc crnors account hr thc facr that thc accuracy o[ cxpcriments rarcly cquals their prccision. Wc 6nd, howcvcr. rhat cvcn.whcn a[
dctcrm-rnatc crront arc ctiminatcd, a scries o[ mcasurcmcnts shows a ccr-
tain amounr of scarrcr, reflcctcd in dcviations lrom thc mcan. Thcsc dcviations arc duc to what arc callcd indctcrminate crrorr. Thc a-djcctivc
l'indctcrtninarc" implics rhat thcsc crro6 cannor bc corrccrcd for. In orhcr
words. wc cannot in any rational way adjusr our rcsutts to compcosatc for
or to ctiminatc crrors of this typc.
Indctcrminatc crrors rcsutt from inhcrcnt impcrfccrions in thc insrrumcnts and tcchniques uscd to carry ou( mcasurcmcnrs. To i[ustratc this
point, considcr thc studcnt who dctcrmincs rhc density of bcnzcnc using a
carcfu-Ifycalibrarcd pipctrc and a high quality analytical balancc. In four
trials, hc might obtain dcnsitics o[:
0-8782
g/ml,
0.8794
g/ml,
0.8785
g/ml, and
0.8779
g/ml
r6E
CHAPTER
I'
Thc indcrcrminatc crrors thar arc responsibtc
for dcviations in this scries
(a) slight
flucr-uations in tcmpcraiure;
with tcmpcraturc.
thc dcnsity o[ a riguid varics
(b) Evaporation ola smail amounr of bcnzcnc bcforc
(c) Failurc ro makc rhc benzcnc rcvcl in rhc pipctte it is weighcd.
.oin"ial .r"",ty
with thc graduation
mark.
(d)???
Evcn though indctcrminarc
crrors cannot bc corrcctcd [or, thcy can bc
rrcarcdstaristically to reil us how thlf
1c likcly to affcct thc rclijbili,y;;
our mcasurcmcnts. This rypc of analysis
is bascd on the so-callcd normat
crror curve shown in Figurc I l.l. This curvc tclts us
rhc rctativc
t .qu.n"y
o[ thc various dcviarions rhat wc can cxpcct ro 6nd
if wc makc a t"rgc num._
bcr of mcasurcmcnts. Thc 6grrc allo*i us to make
somc gcncral observations about the magnirudc of indcterminatc crrors.
. l.
Sincc rhc curvcs arc symmctric about thc midpoint. rcprescnting
rhc arithmcric mcan, positivc and ncgatirc dcriations
.q;;ii;
likcly.
";.
2. Sinct thc curvcs risc to a maximum ar rhc midpoinr,
sma[ dcvia-
.
rions occur morc frcqucntly than large dcviarions.
shapc of rhc curvc is dctcrmincd by rhe inhcrcnt
prccision o[
the measurcmcnt. l[ the instrumcnd or techniqucs thai *c
usc arc
3. Tlre
z
o
tr
.o u.t
o
E
tr
o
o
z
ul
f,
o
t
u,
o
lr
o
o.2
u,
E
C)
tr.
ut
f
o
u,
tc
lr
-2-1
0 1
-2 -1
2
DEVIANON
(a)
FIGUIE tf
.l.
fhccrrcrcsrc:
0
1
DEVIATION
(b)
(c,r.
lor prccirioq
(6L higlr prccirion.
2
A^{ALYS'S
t
69
rdativcly crudc or "sloppy," wc can cxpcct to havc an crror curvc
of thc type shown at thc lcft o[ Figurc I l.l, wirh a rclativcty high
fr,cquencT of largc dcviatioos. As rvc rcfine our mcasurcmcnrs to
hrprovc their precision, wc would cxpc€r ri, appmach ahc crror
disribudon shown ar rhc right of Figurc I l.l, whcrc large d6vi3tions arc highly improbablc.
l.
A studcnr dctcrmining thc G.E.!V. of a metal makcs thc following
cruciblc
crucibtc
-
-
+ mctal
sulEdc
n'"" t!l'l
l6.0ss
- * mass sulfur= l6.0ss
" ,.!4
(C -
cruciblc
c.E.w.
-I
mass - I
mass - C
mass
+
mctat
J.lA)
cach o[ rhe following dctcrminarc errors affccr rhc accuracy o[ h is
rG.E.W.?
a- Afrcr wcighing thc cruciblc plus mcral, he spills somc of thc
metal.
b. Somc of thc mctal sulfidc (MS) is oxidizcd to sulfatc (lvtsO. )c- Thc mass of thc crucibtc plus mcral sulfidc is incorrcctty rc.
cordcd as 16-900 g; it should hasc bccn 16.100 g.
Using Figurc I l.l, cstimatc rhc rclarive frcqucncy o[ obtaining dcof0-5 vs 1.0 for mcasurcmcnrs associarcd wl1[ inhcrcnrly
a. low prccision (Figurc I l.la).
b- high prccision (Figurc I l.lb).
A dass of 20 studcnts mcasurcs thc lcngth of a ccnain spcctrat linc
lollowing rcsults:
zo5o.i
zo5o
A
2054
A
2052
i
2049 205t
2041 20.t9
2o5t 2048 2050
2043
2053 2052 205t
20{5
2o5o A
2049
205t
2050
a- Find thc arirhmctic mcan and rhc.dcviation o[ cach mcasurement-
b. Consrruct
'
an "crror curvcl' by plorring thc lrcqucncy (numbcr)
o[cach dcviation vs thc magnirude ofthc deviation.
I7O '
I
CHAPTER
,,
I.3
MEASNES OF PTECISION
Tro q-c &ril qrrffs ert uscd ro dcscribc
rhc acartcr o[ cx-
er:lc &rilh t rtzdilyEcutarcd by takiig rhc sum of
dcvirirc, drrn !Edtn &uo rhc arithmetic mcan, and dividing
it by thcrotd dcrotdeaarins-
viation.-Thi
thc
a-whcrc a is thc
erege ddatioa,
dividual dcviedoo,
D
ldl
,
lll
(r r.s)
r,cprtscnts rhc magnirudc of an inof triats.
nt3lqi'g is-:fun, and a is thc numbcr
Exomple I l3 Frc sudcrrts rcpon thc fotlowing pcrcentagcs of
chlorinc in a lamplc:
t9-8a 1957, t9.68, l9-7t, and t9.75
Calculatc thc arithrnctic mcan o[ thcsc rcsults and thc avcr:rgc
deviation.
:'
Solution To obtaio rhe arithmctic mcan, it
I
9.50
as.
is convcnicnt to takc
our.basc numbcr and writc:
Arithmcric mcan = 19-50
+
0'32
+ 0'07 +
0'-18
)
+ 0'21 +
0'25
-le.5o*+_te.zl
(Somc pcoplc prcfcr to c:lrry onc cxtra digit in thc mian; in this
case, they would writc t9.706 for thc mean. Wc shall round off ro
t9.71 to simplify thc arithmctic-) Wc procccd ro calculatc rhe individual dcviations, d aid the magnitude o[ cach dcviadon, I / | .
1234
PcrccntagcolCl 19.82 19.57 19.68
d
+0.1r -0.r4 -0.03
0.1 I
0.14 0.03
t
I I
Trial
a-
o.il + 0.14 4
0.03
5
19.7t
0.00
0.00
+ 0.00 +' 0.04
--- : ooir.
t9.75
+0.04
0.04
ERROR
ANA(rstS
Thc arrnsc
I7I
dcrhtln girts
ofour data. Unfortuner*n
stendard dcvietioa
t
us e quetiterir 6.imarc of thc prccision
hes ao iirccr stadstical significancc- -Thc
i
a oooc signiEczor quantity in that ir dcrcrmincs
thc shapc of thc crnor crrrE q,
bc .!ci.t a *itt, s..i., o[ mcasurcme
nrs.
rhc.vcragc dcviilirq ir caorpt bc
"
calcularcd
.Unlikc
cxacdy from a limitcd arnounc of crpcriocanl daaIt
i;;;;.;
the approximatc
-r,
rderbn:
bc cstimatcd from
(t
At
1.6)
whcrc c is rhc srandard dcyiarion
and, as bcforc. d is ao individual dcwia_
tion and n is thc numbcr of triats-
Exomple I 1.4 Esrimate
givcn in Examplc t l.J-
Sotution Wc rccall rhat thc arithmctic
mcan is 19.71. Calcutating dcviations
from rhc rncan, wc havc:
Trial
ofCl 19.82 tg.5t t9.68 19.71 19.75
+0. t I
{
-0. t4 -0.0i +o.OO +0.04
dz
o.0r2r 0.0re6 0.000e 'o.o6oo -J.*,u
Pcrccntagc
xd2
-
9.9121+ 0.0t96 + 0.0009
+
O.O0O0
+ 0.00t6 _
0.0342
-l-5-t-4
Dd211n
- t) -
o.o34zl.r
-
0.0086
Comparingrhe answcrs to Examptcs
ll.l and 1t.4, wc scc thar thc
standard dcviarion. 0.092,
is grcalcr'rhan',L.
"r.r"g. dcviation, 0.066.
*:+".::'u'u|*::';:*ii;*iyi:,:xi;i,rf
prccision of rhcir rcsutrs! r,
.rn
of trials, thc standard dcviation
,ri".1,..n*rs.r.
i.'r-r,'oJn.ii:;';.":'r.1ii[iffi::
approachcs ,/. of rhe avcragc
dcviation:
o
-1
o,as n
+ .o
(n
.-
numbcrof rriels)
(r r.7)
t72
CHAPIER
'I
Thc imponancc thar wc attach to thc standard dcviation rcflccts thc
'influgncc
ir has on thc shapc of thc crror curvc- It can bc shown (sec
Excrcisc 4 ar rhc cnd of rhis sccrion) that a smail vatuc bf a corresponds to
a sharp, stccply rising curvc, on which thc vasr majority o[ dcviaiions arc
vcry closc to zcFo-' Convcrscly, a largc o tcads to a broad, squat crror
curvc, on which largc dcviations havc a rclarivcly high probabiliry.
In Figurc ll-2, wc havc shown a gcrciar lorm of the crror curvc similar ro thosc shown in Figrrrc I t.l, cxcept that thc divisions along thc horizonral axis arc cxprcsscd as mulriplcs o[ rhc standard deviatio-n, c
_
lx
-3a, -2o, -o,O,o,2o,3al. To intcrprct this curvc, lct us took at
rhc
shadcd arca, which is boundcd on thc lcft and righr rcspectively by rhc
r - -o
-
vcnical lincs
and x i.'rThis arca is proporrional to Uc jnbiOtt;E
ol obrcruing o dcoiatioa within ont unii o! o of thc arirhmctic mcan, llc"tcd ar
thc midpoinr o[the curvc. Thc shadcd arca compriscs a littlc more than
of thc total arca undcr thc crror curyc. This mcans thar if wc werc'to
makc a largc numbcr of trials, wc would cxpcct about ,[ of them to fa[
I
within thc rangc:
U:otoM.4o,orM*.o
z
o
tr
!r.l
o
lt
o
o
lr,
)o
lrl
E
u-
-3c -2a -o
FIGURE
0
o
DEVIATION
11.2. Normolcrror(vrc.
2t
la3
ERflOR .AI{AI,Ys's
whcrc rl{ is rhc arithmcric mcan and a is thc standard dcviation. slightly
would fall
lcss than f. of thc rriats would rhow larger dcviations and hcncc
outsideo[rhis range.
lcr us
As wc movc finhcr e*ay [rom thc midpoinr o[ thc cfror curvc.
rcgion
Thc
arca'
rotal
o[
thc
ponion
greatcr
a
still
s y io *2n,wc cnclosc
b"t*l"or - -2aand i - +2a comprises about 91'pcr ccnt o[ thc total
arca. Wcintcrprct this to mcan that i[wc madc a largc numbcr o[mcasurcmcntl wc would cxPcct about 95 pcr ccnt of rhcm-to lall in thc rangc :
M-2a
to M+2a, or lt-2o
wotrld
Orily about 5 pcr ccnr, or I in 20. would fall outsidc this rangc; i'c''
show a.dcriation grcatcr in magnitudcthan 2c.
Onc featurc of Tablc ll.3 is panicularly notcwonhy' It can bc sccn
that hatfofrhc dcviations can bc cxpcctcd to fall in thc rangc:
M-
O-674a
to M +
O-674a
('t/ *
0'674o)
Thc quandty 0-6744 is somctimcs rcfcrred lo as thc probablc dcviation:
(l l-8)
p - 0-614a
with rhc undcrsranding that rhcrc is a
choscn
5G.50 chancc
that a dcviation,
et random, -o,rtd occcd thiptobablc deviation, p'
Exomph I
..
l5
Rcfcrring to Examplc I1.4 and Tablc
l-l'l:
Ass,r-ing rhat a largc numbcr o[ studcnts dctcrminc thc
pcncclntagc
of chtorine in a samplc, within what rangc
woutd you expcct S) Pcr ccnt of thc rcsults to fall?
174
CHAPIER
b. What
pcrcenragc o[ thc class woutd you
rcsults grcatcr rhan t9.80 pcr ccnt?
II
.*p.., ro ,.pon
t_
Solution
'l
a. 90 pcr ccnr of thc rcsulrs shoutd falt in thc rangc (Tlblc
r
r.3):
M - l.64ia to ll + 1.64'>o (M *
1.645a)
In Examplc 11.4, wc found rW = 19.71, o
rangc musr rhcn bc:
19.7t.
-
,
t.645 (0.092)
=
0.092.
riir
= r9.7t + 0.t5
In other words, 90 per ccnt o[ thc studcnts *ould bc cxpcctcd to submit rcsults bctwccn 19.56 and 19.86.
b. t9.80 falls ibout onc standard dcviation aboua thc mcan.
Wc would cxpcct about 32 pcr ccnt of thc studcnts te !c:
porl results dcviating lrom'ihc"incan.by morc than bnc
' sta.ndard dcviation. Of thcsc, half would bc on thc liigh
sidc ( > 19.80) and half on thc low sidc ( < I 9.62). Wc dcducc that about onc halfot32 pcrcenr, or 16 pcr cenr o[
the studcnts would rcport'results highcr rhan t9.80 pcr
ccnt.
fil
E
A word ofcaution is in ordcr conccrning thc kind of prcdictions made
in Examplc 11.5. Likc any predicrions madc on thc basis o[thc lar^s of
probability, rtq con fu crpcclid to bc ualid on$ whca uc arc dcaling with a largc
aumbcr oJ obscntarionr. With only a fcw rcsults to work with, wc can cxpccti
at bcst. an crror pattcrn which crudcly approximates that shown in Figure
I 1.2 or Tablc I I .3. Indccd, if wc havc only a lcw obscrvations,.we cannot
expcct ro obtain thc rruc a?irhmctic mcan, M, on which thc 6gure and
tablc arc based. .h is highly likcly that if morc rcsults wcrc availablc. thc
apparcnr mcan would changc to a ncw valuc, hopcfully approaching morc
and morc closcly thc rruc mean of an infinitc numbcr of trials.
EXERC'SES
.
Calcutarc a and s for thc data in Table I t .l .
czrclrol cxpcrimcntcr obtains thc following vatucs for thc atomic
wcight of cadmium: t t 2.25, I I 2.36,-t 12.32, ll2.Zl, t t2.30, and I t2.36.'
a. What are thc arithmeric mcan and rhc standard deviation?
I
2. A
l7s
ERNOI ANAI,YS's
.' b. lf tha
c.
oak6 onc monE mcasuncmcna, what is thc
litclihood thrt it wilt fall within +0.06 of t'hc mcan?
If many morc trials err madc, within whar rangc would you
cxpcct 50 Pcr
ccil
of thcm to fall
?
3. Rcfcrring to Eamplc ll-5:
a. Vfithin what rangc would you Gxp€ct 99 pcrccnt o[ thc rcsults
.
to tal!?
b. Wlietpcnctntagcof eclass would you cxpcct to rcporr rcsults
.
lcss than 19.53 pcr ccnt?
4- Thc cquation of thc crror curvi is:
r-G
Plota curvco[;r vs: in thc rcgionr
e-c-t
b-u-0.5
lli4
= -2 tox - *2
whcn:
REtlABltlTY OF THE MEAN
WhcB. rir€ carry out a quantitativc cxpcrirncnt. wc arc always rcstrictcd to a li-mitcd numLcr o[ rriats, oftcn as fcw as two, scldom rnorc
6vc dr six.. Thc valuc which wc rcpirn is ordinarily thc arirhmctic
mcanofthc rcrcrat trials. The gucstion ariscs as to how much conEdencc
we can placc'in rhis valuc. It wc could makc an infinitc numbcr o[ dctcrmiaations, irow ctosc could wc rcasooably cxpect tic truc mean to comc
thin
to thc mcan calculatcd on thc basis o[a fcw rrials?
Qucstions such as this can bc answcrcd statistically in tcrms o[ what
are known as confidcncc lcvcls. Considcr. for cxamplc, Excrcisc 2 at rhc
qnd of Scction t lJ, in which wc showcd that thc mcan valuc o[ rhc atomic.
wcight o[ cadmium, bascd on six riats. was 112.30. It can bc shown statisticelly (scc Excreisc I at thc cnd of this scction) that thc truc mcan wilt
fall within +0.02 of t t2-30 at thc "50 pcr ccnt ionfidcncc lcvcl," which wc
intcrpret to mcan that thcrc is a 5G,50 chancc that thc truc mcan would
fall in rhc rangc lt2-28-ll2-32- Again. wc can show thet thc truc mean
wilt hll within *0.05 of t 12-10 at thc "90 pcr ccnt conGdcncc lcvcl." I;
othcrwords..thc chanccs arc 9 in t0 rhat rhc truc mcan, which coutd bc
dctcrmincd only by making an infinitc numbcr of triats. would fatl bctwccn
ll2-25 atd l l2.l5.
:
Rctiability timits cao bc assigncd to e olcula(cd mcan by using the
so-catlcd "t-tesql! rvhich makcs usc o[thc cquation:
.
n{ -
Calcutatcd mcan
+
talrf
,t
(l1.9)
176
CHAPIER
II
nbcrc M is thi rruc mcan, , is.thc standard dcviarion, n is.rhc numbcr of
trials, and I is a statisrical paramctcr which can be obtained from tablcs
such as Tablc t t.4.
.90pcrccat
99 pcr.ccn1.'-r'
63.657 r .i; ..i
2
9-9?5 .,.: ..
3
4
5.841
',';::.:;
. l.ool i;i.i
5
''
6.
,
8
l0'
4.032.
;r,i;
..3.5s9
.;ri..:a'i
2.576
':,!'i
. .-3.250 .:!,:!ii
To il.lustrarc how Eguation li.9 is used. lcr us considcr a.spccific
cxarnpic.' Supposc a studcnt, askcd to dercrminc the molccular weight of
an organic solutc, mikcs thrcc trialslvith the folloil,ing.rcsijlii:
t59.0, l6l.O, and 160.0
.Clcarly, rhc calculated mcan is 160.0. The standard dcviation (Equation
I
t.6) is:
o_tr_y@@=,0
Subslituting in Eguatibn I l:9, wc obtain i
ru
-
160.0
+ t(t.0)l\/1-
Reading across Tablc t t.4 ar n
160.0
- 3:
+ t.otlt.1 = 160.0 +
0.59,
.
l! = 160.0 + O.AfefOjS)
- 160.0 + 0.5; r:rnge - 159.5 90 pcrc-cnt lcvcl: ,.:
,90-O * 2.920(0.59)
t60.0 * 1.7;
= t58.3 50pcrcdnt lcvcl:,
t60.5
.
nranSc
99 pcr ccnr
lcvcl: M =
160.0
ll|n
*
.
9.925(0.59)
a a
t6l-7
ENSOR ANATYs'S
t77
This calculadoa tclls us th:r il an infinitc numbcr of trials wcrc rnadc,
thcrc is e 50-!0 charrc rhet thc rruc mean woutd tatt within *0.5 of I 60.0.
9 chancc in t0 rh:t k rould fatt within * l-7 of 160.0, and 99 chanccs
in
Ifi) that it woutd bc yirhi; *5.8. Purring,ir anothcr rvay. thcrc is onc
chance in nvo drat rk mrc mcan witl tall ouuidc thc rangc 159.5:t 60.5;
t chancc in t0 th:t it ritl htl outsidc thc rangc t58.3-t61.7, and onty ooc
chancc in tfil thr ir sill bc lcss rhan 154.2 orgrcarcr than 165.8.
As rhis cnmplc tluintcs, thc highcr thc confidcncc lcvcl wc dcrnand.
the lcss prcciscly wc an
thc value if thc ruc mcan. To choose rwo
cf,tnemc crses: wc can bc 0 pcr cent conEdcnt that r.hc truc mcan coincidcs
pify
cxacdy with rhc calculatcd mcan (,U - 160.0 + 0.0). bul wc can
pcr ccnt confidcnt that the truc mcan falls bctwccn * - and
-*!
be
Exomple I 1.5 Thcobscrvcil mcanofa scrics of dctcrminarions
thc dcnsity of a liquid is t-69.g/rnl. Calculatc rcliability limirs
thc 90 pcr ccnt confidcncc livcl if:
.;2-'. " . - 2
cl,. i:0.05ia - 5
a. o:=
b. c -
0.10, a
0.05; a
...
o[
at
:
l
- .:'
S'olution'
a. FromTablc ll-4. wcfind rhat for a
confidcncc lcvct,
.
b.
100
M
:
-
t-
- Z irthe
90
pcrccnt
6.314
1.69*6'i(oj.lo) = r.69*0.45
\/2
- r.6e+ryiggt- r.6e*0.22
c.t-z-rrz- M -t.6s*2'l(Los) -
t.69+0.05
Wc scc from Examplc ll.6 (and from Equarion ll.9) rhat rhc rcliability or thc mcan dcpcnds upon thc magnirudc of rhc srandard dcviarion.
Laric valucs o[o imply poor prccision and lcad to a rcrativcly unrctiabrc
mcan. Thc reliability of thc catculatcd mcan is also a function of the numbcr'of obscrvations upon which it is bascd. Thc morc trials wc make. rhc
more coofidcncc wc havc that thc mean wc calculate is a good approximation to thc truc mcan. ln Examptc 11.6, wc found that whcn r incrcascd
STIAPIEh IT
178
.
lrom 2 to 5, thc limia of M , at thc 90 pcr ctnt conndcne lcvcl, rharpcncd
from *0.22 ro +0.05It should be notcd, howcvcr, that bcyond a ccnain point.thcrc is littlc
to bc gaincd by increasing the numbcr of obscnrations. Rdcrring again
whcn c - 0'05'
to ExJrnplc t t.'6, lct us see what happcns ifwc makc n - 6
Hcncc
lcvcL
Gent
ar
rhe
90
I
2-Ol5
wc
6nd
Pcr
From Tablc I l.{,
M
- r.oc *
zo(ojls)
=
a/6
r.6e +.0.04
Comparing this rcsult with that calculatcd in Examplc I l'6, part J"l'-I"
,". ,i r. in-...rsing n from 5 ro 5, had very littlc cffccl on thc rcliability
timits of M (+0.04 instcad of *0.05).
wc
Strictllspcaking, Equation tl.g tctli ds only thc dcviatibn which
wc
can
If
mcan'
truc
and
obscncd
the
..rror,"bly c*pict bctwccn
-"y
should
assumc thal thcrc arc no detcrminatc crrors invotvcd, thc trilc mcan
this
coincidc with rhc truc vatuc o[ tirc quantity wc arc mcasuring' In
with
associatcd
to
bc
crror
crr.,, Eqrurion I 1.9 should tctl us thc cxpcctcd
thc n1can. Thqs io Examplc ti-6, iran (c), if th.e prccision and accgfircy
dt*tg,::
of thi dcnsity -."rr*-.nt, arc thp samc, wc could rcport tt
x'oulo lrc
bc 1.69 * 0.05 with 90 pcr ccnt conhdcncc that the rue density
within thc indicarcd timits.
is known
Equation 11.9 can bc uscd in this manncr to catculatc what
mcan'
of
thc
crror
probablc
as thc
P.E.=
+
!n
(50 pc, ccnl confidcncc lcvcl)
(r.10)
thar thc crror of thc mcan will bc lcss
arc conthan thai catcutarcd by Equadon I l-t0. In Pnicticc' thc chanccs
indtcatco
thc
within
will
fall
valuc
truc
sidcrably tcss than cvcn that thc
ccrp.irrrr.ily bccausc thc accuracy of thc mcasurcmcnt is almost
ln principlc. rhcrc
is a 50'50 ctrancc
.rrrg.,
tain ro bc poorcr than thc piccisionby
Thc cxprcssior, fo. ti. probabtc crror is oftcn furthcr simplificd
Equarcwriting
and
(Tablc
ll'4)
n
rhat
ukin! thc t valuc ro bc
tion
ll.t0as:
",
l,-L- =
0-67o
(I.ll)
{n
-
mcan
givcs us an cstimatc of thc probablc crror of-thc
for:
I
lincc
which is cvcn monc optimisric than'that in Equadon lt't0'
0'67'
finitc numbcr olobscrvations will bc grcatcr than
Equation
lt.lt
t
j
a
ERROn
17,9
ANAI'YS,S
EXERC'SES
t- comidcr rhc data of Ercrci; 2, Sccrion 11.3. What arc thc rcliand 99 per ccnt
ability limits of rhc mcan rt the 50 PGr ccnt, 90 pcr ccnt'
confidcncc limits?
for thc pcrA student, in two trials, finds thc vatucs 16'6 and l7'2
sqr-c o[ his
Gcnt
99
to
bc
wishcs
Pcr
ccntageotchlorinc in e'samptc- If hc
rcpons?
hc
valuc
on
thc
put
hc
what limits rhould
wcight
"n.rvJ.,
3. 1 studcnt fin& thc following valucs for thc gram.cquivalcnt
Z
o[ao acid:
202-{, 199.8,201'7. and 2fl)'8
Gcil confidcncc
Calculate thc rcliability timir of thc mcan at the 50 Pcr
II ' tl'
Equation
from
calculatcd
crror
thc
io
lcrcl and comparc
Probablc
I
I.5
REJECTION OF
A
RESULT
that a studcnt in the gcncral chcmistry laboratory'
makcs 6ve dctcr'
askcd to nnd tirc molcctlar wcight of an organic liquid'
Lct us
suPPosc
mir.rations with thc [ollowir-rg rcsults:
.,.-.
:
.
$4.2,t+g-0, tsr.:.'ts2.9. and
154'5
run was carricd out in rhc samc manner- Thcrc
numbcr
was no obvious dctcrminatc crror in any of rhc trials' .Yct' thc
thc
calculating
in
rcjcctcd
it
bc
shouid
linc;
o[
out
to
bc
148.0 appcars
So far as hc knows, cach
mcan?
Thc qucstion wc arc rcally asking is"'Docs thc valuc 148'O rcflcct
i d.t..*in"t" crror of which thc studcnt was unawarc?" lf this is thc casc.
What
it should bc rcjcctcd. To answcr this qucstion, we must isk anothcr'
up-normally
turn
weuld
arc thc chanccs that a dcviation of this magnitudc
in carrying out a scries of expcrimcnts? Unfortunatcly' rhcrc ari no hard
.nd f.rt irlcs which will givc us dcfinitivc answcn to thcsc gucstions'
of thrce
Chcmists and othcr scicntiits frcqucntly cmploy onc or anothcr
cmpirical rulcs to dccidc whcn to rcjcct a doubtful rcsult'
l.Ilrc2Jorulc.Thcvatucisrcjcctedi[itsdcviationfromthcrrial
2'5 timcs
mcan, catculatcd by ignoring thc doubtful valuc, is grcatcr than
wcight
data
thc
molccular
lo
rutc
this
aAfplying
dcviation.
thc avcragc
discusscdlaborc, wc 6rst rakc thc mcaq ofthe four"rcliablc" valucs:
154.2+ 153.5+ 152.9+ t54.5
4
-
l5i.B
C}'APTER I
0-4+OJ+0.9+0.7
Os
-
'
0-6
Thc dcviation of rhc suspcacd valug 148.0, from thc mcan, 153.g is 5.g.
Sincc
I
.
5.8 > 2.5(0.6)
-
t.5
this rulc would clcarly rcjcct thc valuc 148.0.
2. The 1o rule. This rulc is simitar io the rulc givcn in (t), cxccpt
thar rhc critcrion for rrjcction is 4a rathcr than 2.5a. Ctcarln if we usc rtis
rulc insrcad of (t), wc arc lcss likcly io rcjca doubtful vatucs. If thc dcviation o[thcsuspccred value wcrc J rimes. the avcrage deviarion, it would bc
rctaincd ifwe uscd the 4a rulc and rcjccrcd ifwc used the 2.5a rulc.
In thc casc o[thc molccular wcighr data. thc 4a rule would advisc us to
rcjcct thc numbcr t48.0.
s.8:4(0.6) -2.4
3. Ihe Q test. This rcst, which is uscd morc frcguently today than
cither of the prcceding rulcs. is bascd on thc following proccdure:
(a)- Arrangc thc rcsults in asccnding order:
'
Exarnptc: ilg.o, t52.9.153.5,154.2,154-5
(b). Obtain
rhc diffcrcncc bctwccn thc suspcctcd valuc and thc vatuc
nearcst to it:
152.9-148.0=4.9
(c). Calcularc.rhc rangc, i.e., rhc diffcrcncc bcrwecn the highcst and
lowcst valuc, including the suspcct:
tic.s-r4g.o=6.5
(d)- Obtain a quoticnr, Q by dividing the answcrobtaincd in stcp (b)
by thc answcr obuincd in (c):
Q-4.el6.s=0.75
(c). Comparc to rhc valuc
o[ Q found in Tablc t 1.5, or similar rablcs.
Q cxcccds rhat givcn in rhc rablc, rcjccr thc suspccted
valuc. Looking arTiblc t t.5, wc find that eae6 forn - 5 {5 obscrvations)
is 0.64- Sincc 0.75 is grcatcr rhan 0.64, wc would rcjecr the numbcr 14g.0.
If thc calLulatcd
Etfoi
ANA{.fSrS
This rulc rcnds to
the rcnsc that
ir is
bc. morc
lcss
,rringinr thao cirhcr rulc ( l) or rulc (2) in
ritcty to
Examplc I1.7).
advi-sc us
to.rcjcct a suspcctcd vatuc (scc
Each ofthc rulcs that wc bavc srarcd is bascd
upon sratisricat consid_
.T!i1la
of onc rypc or :norhcc
q rcr,, i"
can bc applicd
3,.rcjc-ctcd |-"ni"ut"r,
with 90 pcr ccnt con'dcn'e rhat rhc
varuc is rigniica".ry diF;;;;
from thc othcrs. Thar iq g0 pcr ccnt if rhe ,atu.s
rcjcctcd on rhc basis of
thc Q tcst rcflcct dctcrmin ti..ro- rathcr than
normar rtarisricat ffucruations' Each of rhc rurcrsuffco r.o- o* trnJ"-..,"r
dcfccc with a
rirn-,
ited numbcr of obscrvations, wc arc nor in a position
to accuratcly csrima tc
thc truc mcan, and hcncc thc ruc dcviation,tf,h.
,urp..,.d value. Li thc
care of the molctular wcight dacrminations,
for cxample, it is quitc possiblc that if wc wcrc to czrry out tcn dctcrminations
rathcr rhan 6vc. urg
might obain thc following rcsults:
t,19-6, t52.0,
l{8.{. t54.8, 15t.2
inwhichcasc the trial mean would shift trom
l5l.g to t52.i, and onc can
rcadily show (sec Excrcisc I ar thc cnd
of this sccrion) rhat each of thc
thrcc iulcs would advisc us to ralain thc
numbcr 14g.0.
Exomptc tI.7 A srudcnt obtains rhc
folowing rcsuks lor thc
molariry of an NaOH sotution: 0.504,
0-510, O]:i.t, ana o.Sfa.
Applythc {c rulc and thc tcst ro at.ia. *t.ri.r
Q
rhc
numbcr
0.538 should bc rcjcctcd.
Solulion
'4a rulc:
-
triatmcan
- zltl
+ 0.5t0 + 0.514
l
0.005+0.00t+0.00s
g--
a
0'504
-
ffi-0.00a
Dcviarion ofsuspccrcd valuc
0.029
> i(0.004)
-
-
0.029
0.016. R.ict
-
0.509
Q rcst:
cHAfiee
tt
0.02{
-o-50{ o-5ro o-5t1-t.518
0.034
Q-0-024/0.034-O-Zr
From Tablc I t.5, wc find Qu.. - 0.ZO for a 4.
Sincc Q < Q.., wc shoutd rakiz rhc valuc 0.53g.
Not infrcqucntly' wc find.rhaq wi(h a rimircd numbcr of riais, rhc
e
rcst adviscs us to rc..in a valut which the 2.5a rule or thc 4d rutei..,odi
rcject- Whai do u,e do in such a ca;i? The obvious answdr is to carry out
morc dcrcrminatioos, in which casc the thrcc rutcs become
-o.. ,"ii"bt"
and niorc ncarly consiitcnt. If wc cannot do this, wc havc to accept thc
lact rhar sratistical considcrations arc no( vcry hcrpful when wc arc limited
ro a fcw trials- wc ma1 wcll find oursclvcs wirh the dilcnima of choosing
betwccir thc Qrist, irhich is likcly ttr retain invalid rcsuts, aird the 4a anj
2-5o rulcs. which art succcssiyely morc likely to rcjccr vatid rcs,itts.
EXERC'SES
1.. 4pply tfrq !.5a.ru|c, thg 4a rutc, and thc Q resr to rhc tcn molccular
wcight ualucs givcn on p. l8l to scc whcthcr l48.0should bc rcjcctcd.
. 2- A srudcnr dcrcrmincs thc pcrccntage bf iron.iri an'oic, obtaining
six valucs:
41
.O, 55-2,49.4, 5Q.l , 49.6, and 50.5
Apply,rhc Qtcsrsucccssivcty ro
any of thcm should bc rcjictedncccssary. 50.5, and so on.
I
I.6
ERROR OF
A
it. tigh.st and lowcst numbcrs to scc if
That is, test 552 6rst. thcn-47.0, thcn. if
CALCUTATED RESULT
Most of thc quantitics that we dctcrminc in thc raborarory arc bascd
upon calculations involving morc than onc mcasured quantity. Associatcd
with cach individual mcasurcmcnt. thcrc witt bc a ccnain crror which may
bc csrimatcd, pcrhaps by a formula such as that givcn by Equation I t.t 1..
Hovr should thcsc errors bc combincd to cstimatc rhe totat crror ro bc associated with a calculatcd rcsult?
In chaptcr 4, wc dcscribcd horv this qucstion courd bc ans*crci in an
approximatc way by using thc rulcs o[ significant figurcs. A.morc cxact
trcatmcnt lcads to lhc rulcs givcn in Tablc t 1.6.
ERROR /ANAIYSrS
, Thccquations listcd in column I of Tablc tl.6 givc lhc crrorr to bc
8. I E l, and
expcctcd in a calculatcd rcsult whcn thc crrors in '{ and
with. the
wc
arc
conccrncd
(Notc
that
I A I arc dctcrminatc in origin.
addirion
[n
signs.)
with
thcir
not
and
crrors
of
thcsc
absolutc magnitudc
and subtraction, lhc total crror is t\c sum of thc crrors in thc individull
quantitics- For multiplication and division, thc fractiooal (or pcrccnt)
crror ih rhc surir or producr is cqual ro the sum of thc lrcctioaal (or pcr-
ccnt) crrors in
I
and
I
(Examplc I l-8)-
Eromptc lI.8 Thc dcnsity of a liguid is dctcrmincd by 6nding
thc mass and votumc of a samplc and applying thc dcfining
cquation:
p=m/Y
I[thc
mass is known tp bc
in crror by 0-l pcr ccnt and thc volumc
by'1.0 pcr ccnt, *hat is thc pcrccnt crror in p?
Solrrlion I[thcpcrccnterrorin
Similarty:
?E
m
is0.l, thcn:
!: -
O.OO!
= o.oro
Hcncc: E,/p
- 0.001 + 0.010 = 0.011
Pcrccnt crror
- l0o$r/p -
l.l7o
Whcn thc crrors associatcd with mcasurcd quantitics arc indctcr'
minate in origin, wc usc Column 4 of Tablc I1.6 to cstimatc ihc crror in a
deiivcd rcsult.
CHAPIER IT
Exoraple IIJ A studcnt dctcrmincs thc gram cquivalcnt wcight
of iron by rcducing a wcighcd sample o[ an gxidc of iron to tbc
mctal and using thc cquation:
.
G.E.w.Fc
-
8.fi)0gO x
-I!lL
wr. oxygcn
His wcighings, along with thc cstimatcd iadctamiaabcrror, arc
as
follows:
cmpty tcst
rcsr tube
tcst
tubc:
+
tube'*
iron
oiidc:
iron:
*
*
t4.076 *
12.602
0.002 g
l{.709
0.002g
0.0029
Hc calculates thc grarn cquivalcnt wcight to bc:
8.ooogx#H=18.63s
Estimatc thc crror in:
. i. fnc wcight of.iron" .
b. Thc wcight of oxygen.
c. ThcG.E.W.ofiron:
Solution
a.
Using rhc last column in Tablc I 1.6:
'
*
x l0-')r+ (2 x l0-')2
- 4 x l0-'+ 4 x l0-'- 8 x l0-'
E =,(8 x lO-r;ur = 2.8 x t0-rg
Ez
(2
b. Hcrc again, two wcighings. cach with an crror of 2 x
l0-'g, arc involvcd:
Hcncc: E=2.8 x l0-tg
c. Thc gram cquivalcnt wcight is catculatcd by taking thc
ratio o[ rwo weights, cach of which is in crror by 2
/ r \' =\/z.a x ro-'\' /2.8 x ro-'\2
,ar-l "\-t=r/.
\'*r/
= 3.6 x
t0-'+
20
x l0-'
= 24
x l0-'g:
x l0{
Ennoa.lfl
ttTn
:
185
E -izt x l0-r;rn - l.s r
tE-63
E
- :It'will
-
(o.m1e)ir8.6l)
bc mtcd-from Examplc
-
ll.9
to-'
o.oe
thar thc'fractional crror in thc
cntircly to tha(
gram cquivalcnt ucighr (E/18.61) can bc attributcd almost
x t0-'/0.633); thc smattcr lractional crmr in
x l0-!/t.{74) madc a rctativcly small cootribution
inrhewciglrtotort3cn(2-g
rhc v,cight of iron (2"8
crtor. It is gencratly truc that whcrc indctcrminatc crrorr arc
involtc4 it is thc crror in thc lcast accuratc quantity'rhat dctcrmincr thc
megnitudc of thc ovrrall ermr. This justiEcs thc cmpirical rulc rtatcd io
Chaptcr rl, that in multiplication or division. the 'numbcr of significant
figurcs rctaincd in thc answcr should bc that in thc lcast accurate,qrrantity
to thc total
cntcdng thc calculations.
EXERC'SES
. .1. A studcnt dctcrmincs thc dcnsity of a liquid by wcighing e samplc
from a 20 ml pipcttc, bclicvcd to bc accuratc.to *0.01 ml. Thc mass, found
by taking thc diffcrcncc bctwccn lwo wcighings, is 16.820 g. Thc cstirnatcd
crrorofaach wcighing is *0.0029. '
.
'a- Estimatc thc crror in thc dcnsity, taking thc crrors in zr'and t'
to bc indcterminatc.
. b- If thc crrors in cach wcighing wcrc kaoun to bc 0.002 g,.and that in
'
Grror in thc dcnsity?
-. l/ wzs kaoam to bc 0.01 ml, what would bc thc(Sccti6n
9.5) to dcrivc
L Use the rclationships ofdiffarcn(iat calculus
thcnrlcsgivcnfordctcrminatccrrors in Tiblc ll.6 (scc Examplc 9-l0 and
thc ircrciscs at thc cnd ofscction i.5).
I l.i Astudcnt dctcrmincs thc pcrccntagc o[chtorinc in a compound by
drrating a wcighcd samplc with AgNOl. His mcasurcincnrs arc:
oassbcakcr
+ iamplc'
mass bcakcr
vohlmcAgNOl f
A
B
pcrccntagcofCl = l00
conc-
AgNOI *{
x &'x .U x l5-5
(8
-
.{)
II
ERROR ANALYSIS
Wc saw in Chaptcr 4'that thc crrors associatcd with cxpcrimcntal
mcasurcmcnts or with guantirics calculatcd from cxpcrimcntal data can bc
cxprcsscd quitc simply in tcrms of significant 6gurcs. Thc rulcs govcrning
thc usc of signiFcant figurcs amount to an clcmcntary lorm of crror.analysis, which is sufficicnt for most of our purposcs in gcncral chcmistry. Ordinarily, rhc quantitics that wc mcasure in thc gencral chemistry laboratory
are bascd upon a singlc trial or, at mostr upon duplicatc detcrminations.
Such cxperimcnts hardly justify a more sophisticatcd rypc of crror analysis
than that dcscribcd in Chaptcr 4.
Wc may, howcvcr. havc thc opporrunity to carry out morc cxact cxperiments whcrc it is imporunr thar r.,c bc ablc to cstimatc quitc accuratell, thc unccrtaintics associatcd with individual mcasuremcnts or with
rcsults dcrivcd from scvcral successivc mcasurcmcnts. Lct us suppose, for
cxamplc, that wc arc askcd to dctcrminc rhe pcrccntagc of chlorinc in a
sample of sodium chloridc, givcn thc cquipmcnt neccssary ro dctcrminc
thc pcrccntagc to four significant figurcs and rhc timc to makc fivc succcssivc dctcrminations. Wc might obtain thc dara listcd in Tablc I I .l .
Thcrc arc scvcral qucstions wc mighr raisc conccrning rhis data.
I . Since wc wcrc ablc, iri principlc, to obtain thc pcrccntagc of chlo'rine
to +0.01 pcr ccnt, why didn't wc gct thc samc vatuc in cach
trial
?
2. Should we rcjcct any o[ thcsc rcsults? (Trial 2 looks a littlc
dubious.)
3. 'What should wc rcport for lhc pcrccnragc of chlorinc? Thc avcragc
o[ all fivc trials? Thc avcrage omitting trial 2? Somc othcr
4.
.
answcr?
Should wc rcpcat thc cxpcrimcnt a fcw morc timcs to gct a
lcr" value lor the percentage ofchlorinc?
"bct'
ERNOfl ANAI,TS'S
5- How confidcnt can wc bc of thc valuc thar wc rcpon lor thc pcr_
ccnragcof chlorinc? What are (hc chaoces thar it wiil bc
-ithio
*0.01 olthe rruc valuc? *0.05?
Thcsc arc rypical of rhc kinds of qucstions that wc will arrcmpr to nna
.*_*:o for in rhis chaptcr. Bcforc procccding funhcr. ir will bc ,,.."r=rry
to defioc ccnain tcrrns that are uscd rcpcatcdly in crror analysis.
II.I
ACCURACY AND PRECISION
. . T-hc aicurecy of a mcasurcd quanrity indicatcs rhc rxrcnt ro
egrccs wirh ihc truc valuc. It is dcscribcd in terms o[thc crror:.
Error
Thcimaller
th-c
-
lrror, ihc
Obscrvcd valuc
-
(r
Truc valuc
more accirratc a mcasurcmcnt
which it
is.
r.l)
f
- Thc prccision of a mcasurcmcnt altor..s us ro estimarc its icproduci_
- b-di,y- It is dcrcribcd by thc dcviutioa, which is thc diffcrcn..
L.,r...r,
rhc cibscrircd valuc and thc averagc vatue. obtaincd from a ,..i.,
of .rr.rlltrcmcnis:
Dcviation
-
Obscrvcd value
-
Averagc vatuc
(
r 1.2)
Thc smatlcr thc dciiation, thc morc prccisc rhc mcasurcmcnr isTo illustratc rhc distinction bctwccn prccision and accuracy, considcr
ttrc data givcn in Tablc t l.l for rhc p.r..r,,"g. of chlorinc ln
sodium
dloridc- Thc avcrage valuc,
r4rqsr which
wtllgll t5
is olIe
oftcn rcfcircd ro as thc ariihmetic
is rcadily found ro bc 60.50 pcr ccnr.
Arithmcricmcan
r
60.50
+
60.41
+
60.5J
+
60.5a
+
60-52
5
-ry-6o.sb
Thc true valuc for thc pcrccnragc of chlorine in sodium chloride is,
ro
significant figurcs,'60-66 perccnr. (This numbcr is bascd upon
ih.
wcights of sodium and chtorinc, which havc bccn vcry
carctully
164
CHAPIER
dctcrmincd roy'uasignificanr 6gures.) Conscqucntly, tor
.t
l.l.
Equarions I l.t and I t.2 bccome:
Error =
valuc
Obscrvccl
'Deviation = Obscrvcd valuc
-
I'
thi dau otTablc
60.66
66.50
Thcse cquations can bc used to cxpress thc prccision and accuracy of tach
mcasurcmcnt o[ thc pcrccnragc of chlorinc (Tablc I t.2).
:
fobtc t
1.2
P*cirion ond Acrurocy.of hcorured perccntogci of Chtorinc
Trirt o-t::**
Eluc
I
60.50
2
60.4 I
:3
'4
l'
60.53
.
16
-0.2t
-0.13
-O-12
.
-0.
:
-
60.54',
S.
Err.or i.a.iio;:
60-52.'
-0-t4
ii
NoCl.
Pcrccnlr''
1Pc1c91t i.ir::
'.. dcviation
crrori i.- 1:
0.00
-0.09 ,"."
*0.03;':'
*0.04
. -:'
+0.02. ',.'
Clcarly, the prccision o[ thcsc mcasurcmcnrs is bcttcr than their accuta.cy. In evcr,v rrial, thc crror is gr.catcr in magnitudc than thc deviation.
ln gcncral, wc can always cxpcct thc prccision of mcasurcmcnts to surpass
thcir.accuracy. Onlv if the arirhmctic mcan coincidcs cxactly with rhc truc
va[uc will our mcasurcmcnts bc as accuratc as they are pricisc.
'
Frcgucntly, r,c rcfcr ro.,'h..para.nt..ror, o, thc perccnt.dcviation.
Thesc quanrirics arc dcfincd as lollows:
Pcrccntcrror==4"
value
I ruc
perccnt dcvia.tion
'
1gg
(11.3)
x too
= Antnmettc
: ,,:Pc'i1,tion
mcan
Ttrchumbcrs in ihc lasr rwo cotumns ar rhe right of Tablc
culatcd in this manncr.
(t 1.4)
tt.2 wcrc
Exomple I l.l Four diffcrcnr srudcnts mcasurc thc boiling point
o[ a ccrtain organic liquid at 760 mm Hg pressurc. Their rcsults
54.9'C. 54.4'C, 54.I'C, and 54.2'C
cal-
.
ERROn
'
ANAIYSIS
I6s
Thctruc boiling point, to-rhrcc slgnificanr 6gures, is 54.0'C. Dctermine. lorcach mcasurcmcnrr thc crror, dcviation pcrccnt crror.
and pcrcent dcviation.
Solulion Thc cmor can bc calculated dircctly from Equation
l.l.
I
In ordcr to usc Equatiori 1.t.2 ro calcularc rhc dcviarion, rze
must nnrt obtain thc arirhmcric.mcan. Wc could do rhis by adding
thc lour numbcrs and dividing thc sum by
amount of arithmaig wc might wrirc:
Arithmctic mcan
-
5.i.0 +
.
54.0
+
0.9
[our. To reducc rhc
+ 0.4 + 0.1 +
!9 -
0.2
i+.1
_4
For thc 6rsr measurcmcnr,
J. nur.,
Error = 5't-9 - 5{-0
..
Pcrccnt
:
".ro. . .il
Dcviarion
-#
Pcrccnt dcviation
Procccdir.rg
.
' 5.r.9
x fio - a!/o
-'54.1
-
,. lN =
0-j;
ag.|yo
Pcrccnt
:. Error
's.r.+
54.t
54.2
0-9;
similarly for cach mcasurcmcnr:
Obscrvcd
value
54.9
=
-
crror
0.9
0.4
0.1
0.2
Pcrccnt
Dcviation
dcviation
+2
0.5
+0.7
+0.2
+0.4
0-0
0.0
-0.1
-0.2
-0.6
-0.4
+0.9
EXERC'SES
l.
in
('K - 'C + 271-2')-
Rcpcat thc calculations
prcsscdin'K
Examplc
ll.l
for .rcmpcraiurcs
e:(-
- 2- In 1894, I-ord Raylcigh dcrcrmincd thc mass of nirrogcn gar filling
a ccrtain containcr a( a known prcssurc and rcmpcrarurc. Thc rcsults o?
succcssive wcighings wcrc as follows:
2.31029, 2.30999,
2.3103
2.3101
g,
2.31
g,
l6
2.3t0Og. 2.3102g, 2.)l0t
g, and
2.3096 g
g.
r66
CHAPIER TI
Thc nitrogcn hc uscd was prcparcd from air by rcmoving oxygcn, *atcr
vapor, and carbon dioxidc. Whcn hc pcrforincd similar cxpcrimcnts wirh
chcmically purc nilrogcn, he obtaincd a mass of 2.2997 g. Taking this to
bc thc "rruc" mass, .calculatc rhc dcviations and crrors o[ cach of thc
masscs listcd abovc.
I
I.2
TYPES OF ERRORS
'
Thc most scrious crrors that studcnts makc in thc chemistry labora-'
tory (or in lilc, for that mattcr) arc oncs which could cithcr bc avoidcd or
corrcctcd [or- Thcsc arc callcd dctcrminatc crrors. As an cxamplc, consider a studcnt.who is attempting to analyzc a mctal oxidc by hcating it
in a srrcam of hydrogcn to form rhepurc mctal.
Mo(s) + H,(s).- MG) + HrO(g)
lf he spills pan o[ his samplc, his rcsult is likcly to show a rathcr largc
crror. This crror could bc avoidcd by bcing morc carcful; thc only way to
"corrcct " for it would bc to rcpcat thc cxpcrimcnt.
As an. cxamplc of a dctcrminate crror which could bc corrccicd for
quirc rcadily, considcr thc dilcmma o[ a studcnt who determincs tirc dcnsity o[ bcnzcnc at 20'C by wcighing samplcs of thc liquid issuing lrom a
l0 ml pipcttc. Hc obtains thc following masscs:
8.681 g.8.678 g,8.683 g, and 8.678
S
Finding thc arithmctic mcan o[ thcsc masscs to bc 8.680 g, hc dividcs by
t0.00 ml and conhdcntly rcpons a dcnsity of 0.8680 g/ml, rusting that his
exccllent precision insures high iccuracy. Unfortunatcly, thc truc valuc is
0.8790 g/ml; thc srudcnt has madc an crror of morc than t pcr ccnt. Hc
failcd to rcatize that a l0 ml pipcttc docs not dctivcr cxactly tcn ml of
liquid. Had he takcn thc trouble to calibrate his pipctte with distillcd
warcr, he would havc lound that it dclivcrcd about 9.90 ml. Hc worjld thcn
havc rcponcd:
.
dcnsity bcnzcne
=
8.680 9/9.90
mI =
0.977
t7.,
with an crror of about 0.2 pcr ccnt.
Eiomplc
I1.2
A studcnt dctcrmincs thc gram cquivalcnt wcight
of a mcral by rcducing ir whh hydrogcn. . Hc makes rhrcc wcighinss:
"
tcst lubc
mass = I
tcst tubc
tcst tube
4 mctal oxidc
+ mctal
maSS
: I
mass = C
EnnoR.ANAtySrs
...
167
. and calculates thc gram cquivalcnt wcighr using rhc cquation:
c-E.w.
,
-
g.0oogo
I
massmctal
(c - -{)
= 8.oooso
-. x (g _ C)
f,assoxygen-
How will cach o[ thc following dctcrminarc crrors affccr thc acwill thcy makc ir largcr or smallcc rhan
curacy of his rcsutt (i.c.,
thc truc valuc) ?
a. ln wcighing thc tcs( tubc. hc rccords a masi ooc gram
.
grtarcr than rhc truc niiss-
b. After weighing rhc mcrat oxidc
of the oxidc.
c.'Hc fails ro conyen all
plus tcsr tubc, hc spiils pa rr
o[thc mcral oxidc (o mcral.
Solurion
a- Mass-l is too targc;
8 and C erc prcsumably
corrccr.
From thc cquaiion for G.E.W., wc scc rhar rhc catculatcd
mars of mctal, C
- A, will
bc too smatt. Hcncc. thc
- .{). witt bc
G.E-W., which is dircctly proponional to (C
'
too small-
b. Hcrc,
I
and 8 will bc corrcct. Miss C will bc too srhall.
rhird rvcighings. Looking ar ihc cquarion for G.E.W.. rvc scc lhar
(C -..1)will bc roo small, (B - C) will bc too largc, and
thc calculatcd G-E.W. wilt bc too smalt.
c. Again, A aad B arc corrc(; C is roo largc bccausc not all
of rhc samplc is rcduccd. Following thc rcasoning of parr
(b), rvc dcducc thar thc calcularcd G.E.W. will bc too
bccausc samplc is losr bcrwccn thc sccond and
largc.
Dctcrminatc crnors account hr thc facr that thc accuracy o[ cxpcriments rarcly cquals their prccision. Wc 6nd, howcvcr. rhat cvcn.whcn a[
dctcrm-rnatc crront arc ctiminatcd, a scries o[ mcasurcmcnts shows a ccr-
tain amounr of scarrcr, reflcctcd in dcviations lrom thc mcan. Thcsc dcviations arc duc to what arc callcd indctcrminate crrorr. Thc a-djcctivc
l'indctcrtninarc" implics rhat thcsc crro6 cannor bc corrccrcd for. In orhcr
words. wc cannot in any rational way adjusr our rcsutts to compcosatc for
or to ctiminatc crrors of this typc.
Indctcrminatc crrors rcsutt from inhcrcnt impcrfccrions in thc insrrumcnts and tcchniques uscd to carry ou( mcasurcmcnrs. To i[ustratc this
point, considcr thc studcnt who dctcrmincs rhc density of bcnzcnc using a
carcfu-Ifycalibrarcd pipctrc and a high quality analytical balancc. In four
trials, hc might obtain dcnsitics o[:
0-8782
g/ml,
0.8794
g/ml,
0.8785
g/ml, and
0.8779
g/ml
r6E
CHAPTER
I'
Thc indcrcrminatc crrors thar arc responsibtc
for dcviations in this scries
(a) slight
flucr-uations in tcmpcraiure;
with tcmpcraturc.
thc dcnsity o[ a riguid varics
(b) Evaporation ola smail amounr of bcnzcnc bcforc
(c) Failurc ro makc rhc benzcnc rcvcl in rhc pipctte it is weighcd.
.oin"ial .r"",ty
with thc graduation
mark.
(d)???
Evcn though indctcrminarc
crrors cannot bc corrcctcd [or, thcy can bc
rrcarcdstaristically to reil us how thlf
1c likcly to affcct thc rclijbili,y;;
our mcasurcmcnts. This rypc of analysis
is bascd on the so-callcd normat
crror curve shown in Figurc I l.l. This curvc tclts us
rhc rctativc
t .qu.n"y
o[ thc various dcviarions rhat wc can cxpcct ro 6nd
if wc makc a t"rgc num._
bcr of mcasurcmcnts. Thc 6grrc allo*i us to make
somc gcncral observations about the magnirudc of indcterminatc crrors.
. l.
Sincc rhc curvcs arc symmctric about thc midpoint. rcprescnting
rhc arithmcric mcan, positivc and ncgatirc dcriations
.q;;ii;
likcly.
";.
2. Sinct thc curvcs risc to a maximum ar rhc midpoinr,
sma[ dcvia-
.
rions occur morc frcqucntly than large dcviarions.
shapc of rhc curvc is dctcrmincd by rhe inhcrcnt
prccision o[
the measurcmcnt. l[ the instrumcnd or techniqucs thai *c
usc arc
3. Tlre
z
o
tr
.o u.t
o
E
tr
o
o
z
ul
f,
o
t
u,
o
lr
o
o.2
u,
E
C)
tr.
ut
f
o
u,
tc
lr
-2-1
0 1
-2 -1
2
DEVIANON
(a)
FIGUIE tf
.l.
fhccrrcrcsrc:
0
1
DEVIATION
(b)
(c,r.
lor prccirioq
(6L higlr prccirion.
2
A^{ALYS'S
t
69
rdativcly crudc or "sloppy," wc can cxpcct to havc an crror curvc
of thc type shown at thc lcft o[ Figurc I l.l, wirh a rclativcty high
fr,cquencT of largc dcviatioos. As rvc rcfine our mcasurcmcnrs to
hrprovc their precision, wc would cxpc€r ri, appmach ahc crror
disribudon shown ar rhc right of Figurc I l.l, whcrc large d6vi3tions arc highly improbablc.
l.
A studcnr dctcrmining thc G.E.!V. of a metal makcs thc following
cruciblc
crucibtc
-
-
+ mctal
sulEdc
n'"" t!l'l
l6.0ss
- * mass sulfur= l6.0ss
" ,.!4
(C -
cruciblc
c.E.w.
-I
mass - I
mass - C
mass
+
mctat
J.lA)
cach o[ rhe following dctcrminarc errors affccr rhc accuracy o[ h is
rG.E.W.?
a- Afrcr wcighing thc cruciblc plus mcral, he spills somc of thc
metal.
b. Somc of thc mctal sulfidc (MS) is oxidizcd to sulfatc (lvtsO. )c- Thc mass of thc crucibtc plus mcral sulfidc is incorrcctty rc.
cordcd as 16-900 g; it should hasc bccn 16.100 g.
Using Figurc I l.l, cstimatc rhc rclarive frcqucncy o[ obtaining dcof0-5 vs 1.0 for mcasurcmcnrs associarcd wl1[ inhcrcnrly
a. low prccision (Figurc I l.la).
b- high prccision (Figurc I l.lb).
A dass of 20 studcnts mcasurcs thc lcngth of a ccnain spcctrat linc
lollowing rcsults:
zo5o.i
zo5o
A
2054
A
2052
i
2049 205t
2041 20.t9
2o5t 2048 2050
2043
2053 2052 205t
20{5
2o5o A
2049
205t
2050
a- Find thc arirhmctic mcan and rhc.dcviation o[ cach mcasurement-
b. Consrruct
'
an "crror curvcl' by plorring thc lrcqucncy (numbcr)
o[cach dcviation vs thc magnirude ofthc deviation.
I7O '
I
CHAPTER
,,
I.3
MEASNES OF PTECISION
Tro q-c &ril qrrffs ert uscd ro dcscribc
rhc acartcr o[ cx-
er:lc &rilh t rtzdilyEcutarcd by takiig rhc sum of
dcvirirc, drrn !Edtn &uo rhc arithmetic mcan, and dividing
it by thcrotd dcrotdeaarins-
viation.-Thi
thc
a-whcrc a is thc
erege ddatioa,
dividual dcviedoo,
D
ldl
,
lll
(r r.s)
r,cprtscnts rhc magnirudc of an inof triats.
nt3lqi'g is-:fun, and a is thc numbcr
Exomple I l3 Frc sudcrrts rcpon thc fotlowing pcrcentagcs of
chlorinc in a lamplc:
t9-8a 1957, t9.68, l9-7t, and t9.75
Calculatc thc arithrnctic mcan o[ thcsc rcsults and thc avcr:rgc
deviation.
:'
Solution To obtaio rhe arithmctic mcan, it
I
9.50
as.
is convcnicnt to takc
our.basc numbcr and writc:
Arithmcric mcan = 19-50
+
0'32
+ 0'07 +
0'-18
)
+ 0'21 +
0'25
-le.5o*+_te.zl
(Somc pcoplc prcfcr to c:lrry onc cxtra digit in thc mian; in this
case, they would writc t9.706 for thc mean. Wc shall round off ro
t9.71 to simplify thc arithmctic-) Wc procccd ro calculatc rhe individual dcviations, d aid the magnitude o[ cach dcviadon, I / | .
1234
PcrccntagcolCl 19.82 19.57 19.68
d
+0.1r -0.r4 -0.03
0.1 I
0.14 0.03
t
I I
Trial
a-
o.il + 0.14 4
0.03
5
19.7t
0.00
0.00
+ 0.00 +' 0.04
--- : ooir.
t9.75
+0.04
0.04
ERROR
ANA(rstS
Thc arrnsc
I7I
dcrhtln girts
ofour data. Unfortuner*n
stendard dcvietioa
t
us e quetiterir 6.imarc of thc prccision
hes ao iirccr stadstical significancc- -Thc
i
a oooc signiEczor quantity in that ir dcrcrmincs
thc shapc of thc crnor crrrE q,
bc .!ci.t a *itt, s..i., o[ mcasurcme
nrs.
rhc.vcragc dcviilirq ir caorpt bc
"
calcularcd
.Unlikc
cxacdy from a limitcd arnounc of crpcriocanl daaIt
i;;;;.;
the approximatc
-r,
rderbn:
bc cstimatcd from
(t
At
1.6)
whcrc c is rhc srandard dcyiarion
and, as bcforc. d is ao individual dcwia_
tion and n is thc numbcr of triats-
Exomple I 1.4 Esrimate
givcn in Examplc t l.J-
Sotution Wc rccall rhat thc arithmctic
mcan is 19.71. Calcutating dcviations
from rhc rncan, wc havc:
Trial
ofCl 19.82 tg.5t t9.68 19.71 19.75
+0. t I
{
-0. t4 -0.0i +o.OO +0.04
dz
o.0r2r 0.0re6 0.000e 'o.o6oo -J.*,u
Pcrccntagc
xd2
-
9.9121+ 0.0t96 + 0.0009
+
O.O0O0
+ 0.00t6 _
0.0342
-l-5-t-4
Dd211n
- t) -
o.o34zl.r
-
0.0086
Comparingrhe answcrs to Examptcs
ll.l and 1t.4, wc scc thar thc
standard dcviarion. 0.092,
is grcalcr'rhan',L.
"r.r"g. dcviation, 0.066.
*:+".::'u'u|*::';:*ii;*iyi:,:xi;i,rf
prccision of rhcir rcsutrs! r,
.rn
of trials, thc standard dcviation
,ri".1,..n*rs.r.
i.'r-r,'oJn.ii:;';.":'r.1ii[iffi::
approachcs ,/. of rhe avcragc
dcviation:
o
-1
o,as n
+ .o
(n
.-
numbcrof rriels)
(r r.7)
t72
CHAPIER
'I
Thc imponancc thar wc attach to thc standard dcviation rcflccts thc
'influgncc
ir has on thc shapc of thc crror curvc- It can bc shown (sec
Excrcisc 4 ar rhc cnd of rhis sccrion) that a smail vatuc bf a corresponds to
a sharp, stccply rising curvc, on which thc vasr majority o[ dcviaiions arc
vcry closc to zcFo-' Convcrscly, a largc o tcads to a broad, squat crror
curvc, on which largc dcviations havc a rclarivcly high probabiliry.
In Figurc ll-2, wc havc shown a gcrciar lorm of the crror curvc similar ro thosc shown in Figrrrc I t.l, cxcept that thc divisions along thc horizonral axis arc cxprcsscd as mulriplcs o[ rhc standard deviatio-n, c
_
lx
-3a, -2o, -o,O,o,2o,3al. To intcrprct this curvc, lct us took at
rhc
shadcd arca, which is boundcd on thc lcft and righr rcspectively by rhc
r - -o
-
vcnical lincs
and x i.'rThis arca is proporrional to Uc jnbiOtt;E
ol obrcruing o dcoiatioa within ont unii o! o of thc arirhmctic mcan, llc"tcd ar
thc midpoinr o[the curvc. Thc shadcd arca compriscs a littlc more than
of thc total arca undcr thc crror curyc. This mcans thar if wc werc'to
makc a largc numbcr of trials, wc would cxpcct about ,[ of them to fa[
I
within thc rangc:
U:otoM.4o,orM*.o
z
o
tr
!r.l
o
lt
o
o
lr,
)o
lrl
E
u-
-3c -2a -o
FIGURE
0
o
DEVIATION
11.2. Normolcrror(vrc.
2t
la3
ERflOR .AI{AI,Ys's
whcrc rl{ is rhc arithmcric mcan and a is thc standard dcviation. slightly
would fall
lcss than f. of thc rriats would rhow larger dcviations and hcncc
outsideo[rhis range.
lcr us
As wc movc finhcr e*ay [rom thc midpoinr o[ thc cfror curvc.
rcgion
Thc
arca'
rotal
o[
thc
ponion
greatcr
a
still
s y io *2n,wc cnclosc
b"t*l"or - -2aand i - +2a comprises about 91'pcr ccnt o[ thc total
arca. Wcintcrprct this to mcan that i[wc madc a largc numbcr o[mcasurcmcntl wc would cxPcct about 95 pcr ccnt of rhcm-to lall in thc rangc :
M-2a
to M+2a, or lt-2o
wotrld
Orily about 5 pcr ccnr, or I in 20. would fall outsidc this rangc; i'c''
show a.dcriation grcatcr in magnitudcthan 2c.
Onc featurc of Tablc ll.3 is panicularly notcwonhy' It can bc sccn
that hatfofrhc dcviations can bc cxpcctcd to fall in thc rangc:
M-
O-674a
to M +
O-674a
('t/ *
0'674o)
Thc quandty 0-6744 is somctimcs rcfcrred lo as thc probablc dcviation:
(l l-8)
p - 0-614a
with rhc undcrsranding that rhcrc is a
choscn
5G.50 chancc
that a dcviation,
et random, -o,rtd occcd thiptobablc deviation, p'
Exomph I
..
l5
Rcfcrring to Examplc I1.4 and Tablc
l-l'l:
Ass,r-ing rhat a largc numbcr o[ studcnts dctcrminc thc
pcncclntagc
of chtorine in a samplc, within what rangc
woutd you expcct S) Pcr ccnt of thc rcsults to fall?
174
CHAPIER
b. What
pcrcenragc o[ thc class woutd you
rcsults grcatcr rhan t9.80 pcr ccnt?
II
.*p.., ro ,.pon
t_
Solution
'l
a. 90 pcr ccnr of thc rcsulrs shoutd falt in thc rangc (Tlblc
r
r.3):
M - l.64ia to ll + 1.64'>o (M *
1.645a)
In Examplc 11.4, wc found rW = 19.71, o
rangc musr rhcn bc:
19.7t.
-
,
t.645 (0.092)
=
0.092.
riir
= r9.7t + 0.t5
In other words, 90 per ccnt o[ thc studcnts *ould bc cxpcctcd to submit rcsults bctwccn 19.56 and 19.86.
b. t9.80 falls ibout onc standard dcviation aboua thc mcan.
Wc would cxpcct about 32 pcr ccnt of thc studcnts te !c:
porl results dcviating lrom'ihc"incan.by morc than bnc
' sta.ndard dcviation. Of thcsc, half would bc on thc liigh
sidc ( > 19.80) and half on thc low sidc ( < I 9.62). Wc dcducc that about onc halfot32 pcrcenr, or 16 pcr cenr o[
the studcnts would rcport'results highcr rhan t9.80 pcr
ccnt.
fil
E
A word ofcaution is in ordcr conccrning thc kind of prcdictions made
in Examplc 11.5. Likc any predicrions madc on thc basis o[thc lar^s of
probability, rtq con fu crpcclid to bc ualid on$ whca uc arc dcaling with a largc
aumbcr oJ obscntarionr. With only a fcw rcsults to work with, wc can cxpccti
at bcst. an crror pattcrn which crudcly approximates that shown in Figure
I 1.2 or Tablc I I .3. Indccd, if wc havc only a lcw obscrvations,.we cannot
expcct ro obtain thc rruc a?irhmctic mcan, M, on which thc 6gure and
tablc arc based. .h is highly likcly that if morc rcsults wcrc availablc. thc
apparcnr mcan would changc to a ncw valuc, hopcfully approaching morc
and morc closcly thc rruc mean of an infinitc numbcr of trials.
EXERC'SES
.
Calcutarc a and s for thc data in Table I t .l .
czrclrol cxpcrimcntcr obtains thc following vatucs for thc atomic
wcight of cadmium: t t 2.25, I I 2.36,-t 12.32, ll2.Zl, t t2.30, and I t2.36.'
a. What are thc arithmeric mcan and rhc standard deviation?
I
2. A
l7s
ERNOI ANAI,YS's
.' b. lf tha
c.
oak6 onc monE mcasuncmcna, what is thc
litclihood thrt it wilt fall within +0.06 of t'hc mcan?
If many morc trials err madc, within whar rangc would you
cxpcct 50 Pcr
ccil
of thcm to fall
?
3. Rcfcrring to Eamplc ll-5:
a. Vfithin what rangc would you Gxp€ct 99 pcrccnt o[ thc rcsults
.
to tal!?
b. Wlietpcnctntagcof eclass would you cxpcct to rcporr rcsults
.
lcss than 19.53 pcr ccnt?
4- Thc cquation of thc crror curvi is:
r-G
Plota curvco[;r vs: in thc rcgionr
e-c-t
b-u-0.5
lli4
= -2 tox - *2
whcn:
REtlABltlTY OF THE MEAN
WhcB. rir€ carry out a quantitativc cxpcrirncnt. wc arc always rcstrictcd to a li-mitcd numLcr o[ rriats, oftcn as fcw as two, scldom rnorc
6vc dr six.. Thc valuc which wc rcpirn is ordinarily thc arirhmctic
mcanofthc rcrcrat trials. The gucstion ariscs as to how much conEdencc
we can placc'in rhis valuc. It wc could makc an infinitc numbcr o[ dctcrmiaations, irow ctosc could wc rcasooably cxpect tic truc mean to comc
thin
to thc mcan calculatcd on thc basis o[a fcw rrials?
Qucstions such as this can bc answcrcd statistically in tcrms o[ what
are known as confidcncc lcvcls. Considcr. for cxamplc, Excrcisc 2 at rhc
qnd of Scction t lJ, in which wc showcd that thc mcan valuc o[ rhc atomic.
wcight o[ cadmium, bascd on six riats. was 112.30. It can bc shown statisticelly (scc Excreisc I at thc cnd of this scction) that thc truc mcan wilt
fall within +0.02 of t t2-30 at thc "50 pcr ccnt ionfidcncc lcvcl," which wc
intcrpret to mcan that thcrc is a 5G,50 chancc that thc truc mcan would
fall in rhc rangc lt2-28-ll2-32- Again. wc can show thet thc truc mean
wilt hll within *0.05 of t 12-10 at thc "90 pcr ccnt conGdcncc lcvcl." I;
othcrwords..thc chanccs arc 9 in t0 rhat rhc truc mcan, which coutd bc
dctcrmincd only by making an infinitc numbcr of triats. would fatl bctwccn
ll2-25 atd l l2.l5.
:
Rctiability timits cao bc assigncd to e olcula(cd mcan by using the
so-catlcd "t-tesql! rvhich makcs usc o[thc cquation:
.
n{ -
Calcutatcd mcan
+
talrf
,t
(l1.9)
176
CHAPIER
II
nbcrc M is thi rruc mcan, , is.thc standard dcviarion, n is.rhc numbcr of
trials, and I is a statisrical paramctcr which can be obtained from tablcs
such as Tablc t t.4.
.90pcrccat
99 pcr.ccn1.'-r'
63.657 r .i; ..i
2
9-9?5 .,.: ..
3
4
5.841
',';::.:;
. l.ool i;i.i
5
''
6.
,
8
l0'
4.032.
;r,i;
..3.5s9
.;ri..:a'i
2.576
':,!'i
. .-3.250 .:!,:!ii
To il.lustrarc how Eguation li.9 is used. lcr us considcr a.spccific
cxarnpic.' Supposc a studcnt, askcd to dercrminc the molccular weight of
an organic solutc, mikcs thrcc trialslvith the folloil,ing.rcsijlii:
t59.0, l6l.O, and 160.0
.Clcarly, rhc calculated mcan is 160.0. The standard dcviation (Equation
I
t.6) is:
o_tr_y@@=,0
Subslituting in Eguatibn I l:9, wc obtain i
ru
-
160.0
+ t(t.0)l\/1-
Reading across Tablc t t.4 ar n
160.0
- 3:
+ t.otlt.1 = 160.0 +
0.59,
.
l! = 160.0 + O.AfefOjS)
- 160.0 + 0.5; r:rnge - 159.5 90 pcrc-cnt lcvcl: ,.:
,90-O * 2.920(0.59)
t60.0 * 1.7;
= t58.3 50pcrcdnt lcvcl:,
t60.5
.
nranSc
99 pcr ccnr
lcvcl: M =
160.0
ll|n
*
.
9.925(0.59)
a a
t6l-7
ENSOR ANATYs'S
t77
This calculadoa tclls us th:r il an infinitc numbcr of trials wcrc rnadc,
thcrc is e 50-!0 charrc rhet thc rruc mean woutd tatt within *0.5 of I 60.0.
9 chancc in t0 rh:t k rould fatt within * l-7 of 160.0, and 99 chanccs
in
Ifi) that it woutd bc yirhi; *5.8. Purring,ir anothcr rvay. thcrc is onc
chance in nvo drat rk mrc mcan witl tall ouuidc thc rangc 159.5:t 60.5;
t chancc in t0 th:t it ritl htl outsidc thc rangc t58.3-t61.7, and onty ooc
chancc in tfil thr ir sill bc lcss rhan 154.2 orgrcarcr than 165.8.
As rhis cnmplc tluintcs, thc highcr thc confidcncc lcvcl wc dcrnand.
the lcss prcciscly wc an
thc value if thc ruc mcan. To choose rwo
cf,tnemc crses: wc can bc 0 pcr cent conEdcnt that r.hc truc mcan coincidcs
pify
cxacdy with rhc calculatcd mcan (,U - 160.0 + 0.0). bul wc can
pcr ccnt confidcnt that the truc mcan falls bctwccn * - and
-*!
be
Exomple I 1.5 Thcobscrvcil mcanofa scrics of dctcrminarions
thc dcnsity of a liquid is t-69.g/rnl. Calculatc rcliability limirs
thc 90 pcr ccnt confidcncc livcl if:
.;2-'. " . - 2
cl,. i:0.05ia - 5
a. o:=
b. c -
0.10, a
0.05; a
...
o[
at
:
l
- .:'
S'olution'
a. FromTablc ll-4. wcfind rhat for a
confidcncc lcvct,
.
b.
100
M
:
-
t-
- Z irthe
90
pcrccnt
6.314
1.69*6'i(oj.lo) = r.69*0.45
\/2
- r.6e+ryiggt- r.6e*0.22
c.t-z-rrz- M -t.6s*2'l(Los) -
t.69+0.05
Wc scc from Examplc ll.6 (and from Equarion ll.9) rhat rhc rcliability or thc mcan dcpcnds upon thc magnirudc of rhc srandard dcviarion.
Laric valucs o[o imply poor prccision and lcad to a rcrativcly unrctiabrc
mcan. Thc reliability of thc catculatcd mcan is also a function of the numbcr'of obscrvations upon which it is bascd. Thc morc trials wc make. rhc
more coofidcncc wc havc that thc mean wc calculate is a good approximation to thc truc mcan. ln Examptc 11.6, wc found that whcn r incrcascd
STIAPIEh IT
178
.
lrom 2 to 5, thc limia of M , at thc 90 pcr ctnt conndcne lcvcl, rharpcncd
from *0.22 ro +0.05It should be notcd, howcvcr, that bcyond a ccnain point.thcrc is littlc
to bc gaincd by increasing the numbcr of obscnrations. Rdcrring again
whcn c - 0'05'
to ExJrnplc t t.'6, lct us see what happcns ifwc makc n - 6
Hcncc
lcvcL
Gent
ar
rhe
90
I
2-Ol5
wc
6nd
Pcr
From Tablc I l.{,
M
- r.oc *
zo(ojls)
=
a/6
r.6e +.0.04
Comparing this rcsult with that calculatcd in Examplc I l'6, part J"l'-I"
,". ,i r. in-...rsing n from 5 ro 5, had very littlc cffccl on thc rcliability
timits of M (+0.04 instcad of *0.05).
wc
Strictllspcaking, Equation tl.g tctli ds only thc dcviatibn which
wc
can
If
mcan'
truc
and
obscncd
the
..rror,"bly c*pict bctwccn
-"y
should
assumc thal thcrc arc no detcrminatc crrors invotvcd, thc trilc mcan
this
coincidc with rhc truc vatuc o[ tirc quantity wc arc mcasuring' In
with
associatcd
to
bc
crror
crr.,, Eqrurion I 1.9 should tctl us thc cxpcctcd
thc n1can. Thqs io Examplc ti-6, iran (c), if th.e prccision and accgfircy
dt*tg,::
of thi dcnsity -."rr*-.nt, arc thp samc, wc could rcport tt
x'oulo lrc
bc 1.69 * 0.05 with 90 pcr ccnt conhdcncc that the rue density
within thc indicarcd timits.
is known
Equation 11.9 can bc uscd in this manncr to catculatc what
mcan'
of
thc
crror
probablc
as thc
P.E.=
+
!n
(50 pc, ccnl confidcncc lcvcl)
(r.10)
thar thc crror of thc mcan will bc lcss
arc conthan thai catcutarcd by Equadon I l-t0. In Pnicticc' thc chanccs
indtcatco
thc
within
will
fall
valuc
truc
sidcrably tcss than cvcn that thc
ccrp.irrrr.ily bccausc thc accuracy of thc mcasurcmcnt is almost
ln principlc. rhcrc
is a 50'50 ctrancc
.rrrg.,
tain ro bc poorcr than thc piccisionby
Thc cxprcssior, fo. ti. probabtc crror is oftcn furthcr simplificd
Equarcwriting
and
(Tablc
ll'4)
n
rhat
ukin! thc t valuc ro bc
tion
ll.t0as:
",
l,-L- =
0-67o
(I.ll)
{n
-
mcan
givcs us an cstimatc of thc probablc crror of-thc
for:
I
lincc
which is cvcn monc optimisric than'that in Equadon lt't0'
0'67'
finitc numbcr olobscrvations will bc grcatcr than
Equation
lt.lt
t
j
a
ERROn
17,9
ANAI'YS,S
EXERC'SES
t- comidcr rhc data of Ercrci; 2, Sccrion 11.3. What arc thc rcliand 99 per ccnt
ability limits of rhc mcan rt the 50 PGr ccnt, 90 pcr ccnt'
confidcncc limits?
for thc pcrA student, in two trials, finds thc vatucs 16'6 and l7'2
sqr-c o[ his
Gcnt
99
to
bc
wishcs
Pcr
ccntageotchlorinc in e'samptc- If hc
rcpons?
hc
valuc
on
thc
put
hc
what limits rhould
wcight
"n.rvJ.,
3. 1 studcnt fin& thc following valucs for thc gram.cquivalcnt
Z
o[ao acid:
202-{, 199.8,201'7. and 2fl)'8
Gcil confidcncc
Calculate thc rcliability timir of thc mcan at the 50 Pcr
II ' tl'
Equation
from
calculatcd
crror
thc
io
lcrcl and comparc
Probablc
I
I.5
REJECTION OF
A
RESULT
that a studcnt in the gcncral chcmistry laboratory'
makcs 6ve dctcr'
askcd to nnd tirc molcctlar wcight of an organic liquid'
Lct us
suPPosc
mir.rations with thc [ollowir-rg rcsults:
.,.-.
:
.
$4.2,t+g-0, tsr.:.'ts2.9. and
154'5
run was carricd out in rhc samc manner- Thcrc
numbcr
was no obvious dctcrminatc crror in any of rhc trials' .Yct' thc
thc
calculating
in
rcjcctcd
it
bc
shouid
linc;
o[
out
to
bc
148.0 appcars
So far as hc knows, cach
mcan?
Thc qucstion wc arc rcally asking is"'Docs thc valuc 148'O rcflcct
i d.t..*in"t" crror of which thc studcnt was unawarc?" lf this is thc casc.
What
it should bc rcjcctcd. To answcr this qucstion, we must isk anothcr'
up-normally
turn
weuld
arc thc chanccs that a dcviation of this magnitudc
in carrying out a scries of expcrimcnts? Unfortunatcly' rhcrc ari no hard
.nd f.rt irlcs which will givc us dcfinitivc answcn to thcsc gucstions'
of thrce
Chcmists and othcr scicntiits frcqucntly cmploy onc or anothcr
cmpirical rulcs to dccidc whcn to rcjcct a doubtful rcsult'
l.Ilrc2Jorulc.Thcvatucisrcjcctedi[itsdcviationfromthcrrial
2'5 timcs
mcan, catculatcd by ignoring thc doubtful valuc, is grcatcr than
wcight
data
thc
molccular
lo
rutc
this
aAfplying
dcviation.
thc avcragc
discusscdlaborc, wc 6rst rakc thc mcaq ofthe four"rcliablc" valucs:
154.2+ 153.5+ 152.9+ t54.5
4
-
l5i.B
C}'APTER I
0-4+OJ+0.9+0.7
Os
-
'
0-6
Thc dcviation of rhc suspcacd valug 148.0, from thc mcan, 153.g is 5.g.
Sincc
I
.
5.8 > 2.5(0.6)
-
t.5
this rulc would clcarly rcjcct thc valuc 148.0.
2. The 1o rule. This rulc is simitar io the rulc givcn in (t), cxccpt
thar rhc critcrion for rrjcction is 4a rathcr than 2.5a. Ctcarln if we usc rtis
rulc insrcad of (t), wc arc lcss likcly io rcjca doubtful vatucs. If thc dcviation o[thcsuspccred value wcrc J rimes. the avcrage deviarion, it would bc
rctaincd ifwe uscd the 4a rulc and rcjccrcd ifwc used the 2.5a rulc.
In thc casc o[thc molccular wcighr data. thc 4a rule would advisc us to
rcjcct thc numbcr t48.0.
s.8:4(0.6) -2.4
3. Ihe Q test. This rcst, which is uscd morc frcguently today than
cither of the prcceding rulcs. is bascd on thc following proccdure:
(a)- Arrangc thc rcsults in asccnding order:
'
Exarnptc: ilg.o, t52.9.153.5,154.2,154-5
(b). Obtain
rhc diffcrcncc bctwccn thc suspcctcd valuc and thc vatuc
nearcst to it:
152.9-148.0=4.9
(c). Calcularc.rhc rangc, i.e., rhc diffcrcncc bcrwecn the highcst and
lowcst valuc, including the suspcct:
tic.s-r4g.o=6.5
(d)- Obtain a quoticnr, Q by dividing the answcrobtaincd in stcp (b)
by thc answcr obuincd in (c):
Q-4.el6.s=0.75
(c). Comparc to rhc valuc
o[ Q found in Tablc t 1.5, or similar rablcs.
Q cxcccds rhat givcn in rhc rablc, rcjccr thc suspccted
valuc. Looking arTiblc t t.5, wc find that eae6 forn - 5 {5 obscrvations)
is 0.64- Sincc 0.75 is grcatcr rhan 0.64, wc would rcjecr the numbcr 14g.0.
If thc calLulatcd
Etfoi
ANA{.fSrS
This rulc rcnds to
the rcnsc that
ir is
bc. morc
lcss
,rringinr thao cirhcr rulc ( l) or rulc (2) in
ritcty to
Examplc I1.7).
advi-sc us
to.rcjcct a suspcctcd vatuc (scc
Each ofthc rulcs that wc bavc srarcd is bascd
upon sratisricat consid_
.T!i1la
of onc rypc or :norhcc
q rcr,, i"
can bc applicd
3,.rcjc-ctcd |-"ni"ut"r,
with 90 pcr ccnt con'dcn'e rhat rhc
varuc is rigniica".ry diF;;;;
from thc othcrs. Thar iq g0 pcr ccnt if rhe ,atu.s
rcjcctcd on rhc basis of
thc Q tcst rcflcct dctcrmin ti..ro- rathcr than
normar rtarisricat ffucruations' Each of rhc rurcrsuffco r.o- o* trnJ"-..,"r
dcfccc with a
rirn-,
ited numbcr of obscrvations, wc arc nor in a position
to accuratcly csrima tc
thc truc mcan, and hcncc thc ruc dcviation,tf,h.
,urp..,.d value. Li thc
care of the molctular wcight dacrminations,
for cxample, it is quitc possiblc that if wc wcrc to czrry out tcn dctcrminations
rathcr rhan 6vc. urg
might obain thc following rcsults:
t,19-6, t52.0,
l{8.{. t54.8, 15t.2
inwhichcasc the trial mean would shift trom
l5l.g to t52.i, and onc can
rcadily show (sec Excrcisc I ar thc cnd
of this sccrion) rhat each of thc
thrcc iulcs would advisc us to ralain thc
numbcr 14g.0.
Exomptc tI.7 A srudcnt obtains rhc
folowing rcsuks lor thc
molariry of an NaOH sotution: 0.504,
0-510, O]:i.t, ana o.Sfa.
Applythc {c rulc and thc tcst ro at.ia. *t.ri.r
Q
rhc
numbcr
0.538 should bc rcjcctcd.
Solulion
'4a rulc:
-
triatmcan
- zltl
+ 0.5t0 + 0.514
l
0.005+0.00t+0.00s
g--
a
0'504
-
ffi-0.00a
Dcviarion ofsuspccrcd valuc
0.029
> i(0.004)
-
-
0.029
0.016. R.ict
-
0.509
Q rcst:
cHAfiee
tt
0.02{
-o-50{ o-5ro o-5t1-t.518
0.034
Q-0-024/0.034-O-Zr
From Tablc I t.5, wc find Qu.. - 0.ZO for a 4.
Sincc Q < Q.., wc shoutd rakiz rhc valuc 0.53g.
Not infrcqucntly' wc find.rhaq wi(h a rimircd numbcr of riais, rhc
e
rcst adviscs us to rc..in a valut which the 2.5a rule or thc 4d rutei..,odi
rcject- Whai do u,e do in such a ca;i? The obvious answdr is to carry out
morc dcrcrminatioos, in which casc the thrcc rutcs become
-o.. ,"ii"bt"
and niorc ncarly consiitcnt. If wc cannot do this, wc havc to accept thc
lact rhar sratistical considcrations arc no( vcry hcrpful when wc arc limited
ro a fcw trials- wc ma1 wcll find oursclvcs wirh the dilcnima of choosing
betwccir thc Qrist, irhich is likcly ttr retain invalid rcsuts, aird the 4a anj
2-5o rulcs. which art succcssiyely morc likely to rcjccr vatid rcs,itts.
EXERC'SES
1.. 4pply tfrq !.5a.ru|c, thg 4a rutc, and thc Q resr to rhc tcn molccular
wcight ualucs givcn on p. l8l to scc whcthcr l48.0should bc rcjcctcd.
. 2- A srudcnr dcrcrmincs thc pcrccntage bf iron.iri an'oic, obtaining
six valucs:
41
.O, 55-2,49.4, 5Q.l , 49.6, and 50.5
Apply,rhc Qtcsrsucccssivcty ro
any of thcm should bc rcjictedncccssary. 50.5, and so on.
I
I.6
ERROR OF
A
it. tigh.st and lowcst numbcrs to scc if
That is, test 552 6rst. thcn-47.0, thcn. if
CALCUTATED RESULT
Most of thc quantitics that we dctcrminc in thc raborarory arc bascd
upon calculations involving morc than onc mcasured quantity. Associatcd
with cach individual mcasurcmcnt. thcrc witt bc a ccnain crror which may
bc csrimatcd, pcrhaps by a formula such as that givcn by Equation I t.t 1..
Hovr should thcsc errors bc combincd to cstimatc rhe totat crror ro bc associated with a calculatcd rcsult?
In chaptcr 4, wc dcscribcd horv this qucstion courd bc ans*crci in an
approximatc way by using thc rulcs o[ significant figurcs. A.morc cxact
trcatmcnt lcads to lhc rulcs givcn in Tablc t 1.6.
ERROR /ANAIYSrS
, Thccquations listcd in column I of Tablc tl.6 givc lhc crrorr to bc
8. I E l, and
expcctcd in a calculatcd rcsult whcn thc crrors in '{ and
with. the
wc
arc
conccrncd
(Notc
that
I A I arc dctcrminatc in origin.
addirion
[n
signs.)
with
thcir
not
and
crrors
of
thcsc
absolutc magnitudc
and subtraction, lhc total crror is t\c sum of thc crrors in thc individull
quantitics- For multiplication and division, thc fractiooal (or pcrccnt)
crror ih rhc surir or producr is cqual ro the sum of thc lrcctioaal (or pcr-
ccnt) crrors in
I
and
I
(Examplc I l-8)-
Eromptc lI.8 Thc dcnsity of a liguid is dctcrmincd by 6nding
thc mass and votumc of a samplc and applying thc dcfining
cquation:
p=m/Y
I[thc
mass is known tp bc
in crror by 0-l pcr ccnt and thc volumc
by'1.0 pcr ccnt, *hat is thc pcrccnt crror in p?
Solrrlion I[thcpcrccnterrorin
Similarty:
?E
m
is0.l, thcn:
!: -
O.OO!
= o.oro
Hcncc: E,/p
- 0.001 + 0.010 = 0.011
Pcrccnt crror
- l0o$r/p -
l.l7o
Whcn thc crrors associatcd with mcasurcd quantitics arc indctcr'
minate in origin, wc usc Column 4 of Tablc I1.6 to cstimatc ihc crror in a
deiivcd rcsult.
CHAPIER IT
Exoraple IIJ A studcnt dctcrmincs thc gram cquivalcnt wcight
of iron by rcducing a wcighcd sample o[ an gxidc of iron to tbc
mctal and using thc cquation:
.
G.E.w.Fc
-
8.fi)0gO x
-I!lL
wr. oxygcn
His wcighings, along with thc cstimatcd iadctamiaabcrror, arc
as
follows:
cmpty tcst
rcsr tube
tcst
tubc:
+
tube'*
iron
oiidc:
iron:
*
*
t4.076 *
12.602
0.002 g
l{.709
0.002g
0.0029
Hc calculates thc grarn cquivalcnt wcight to bc:
8.ooogx#H=18.63s
Estimatc thc crror in:
. i. fnc wcight of.iron" .
b. Thc wcight of oxygen.
c. ThcG.E.W.ofiron:
Solution
a.
Using rhc last column in Tablc I 1.6:
'
*
x l0-')r+ (2 x l0-')2
- 4 x l0-'+ 4 x l0-'- 8 x l0-'
E =,(8 x lO-r;ur = 2.8 x t0-rg
Ez
(2
b. Hcrc again, two wcighings. cach with an crror of 2 x
l0-'g, arc involvcd:
Hcncc: E=2.8 x l0-tg
c. Thc gram cquivalcnt wcight is catculatcd by taking thc
ratio o[ rwo weights, cach of which is in crror by 2
/ r \' =\/z.a x ro-'\' /2.8 x ro-'\2
,ar-l "\-t=r/.
\'*r/
= 3.6 x
t0-'+
20
x l0-'
= 24
x l0-'g:
x l0{
Ennoa.lfl
ttTn
:
185
E -izt x l0-r;rn - l.s r
tE-63
E
- :It'will
-
(o.m1e)ir8.6l)
bc mtcd-from Examplc
-
ll.9
to-'
o.oe
thar thc'fractional crror in thc
cntircly to tha(
gram cquivalcnt ucighr (E/18.61) can bc attributcd almost
x t0-'/0.633); thc smattcr lractional crmr in
x l0-!/t.{74) madc a rctativcly small cootribution
inrhewciglrtotort3cn(2-g
rhc v,cight of iron (2"8
crtor. It is gencratly truc that whcrc indctcrminatc crrorr arc
involtc4 it is thc crror in thc lcast accuratc quantity'rhat dctcrmincr thc
megnitudc of thc ovrrall ermr. This justiEcs thc cmpirical rulc rtatcd io
Chaptcr rl, that in multiplication or division. the 'numbcr of significant
figurcs rctaincd in thc answcr should bc that in thc lcast accurate,qrrantity
to thc total
cntcdng thc calculations.
EXERC'SES
. .1. A studcnt dctcrmincs thc dcnsity of a liquid by wcighing e samplc
from a 20 ml pipcttc, bclicvcd to bc accuratc.to *0.01 ml. Thc mass, found
by taking thc diffcrcncc bctwccn lwo wcighings, is 16.820 g. Thc cstirnatcd
crrorofaach wcighing is *0.0029. '
.
'a- Estimatc thc crror in thc dcnsity, taking thc crrors in zr'and t'
to bc indcterminatc.
. b- If thc crrors in cach wcighing wcrc kaoun to bc 0.002 g,.and that in
'
Grror in thc dcnsity?
-. l/ wzs kaoam to bc 0.01 ml, what would bc thc(Sccti6n
9.5) to dcrivc
L Use the rclationships ofdiffarcn(iat calculus
thcnrlcsgivcnfordctcrminatccrrors in Tiblc ll.6 (scc Examplc 9-l0 and
thc ircrciscs at thc cnd ofscction i.5).
I l.i Astudcnt dctcrmincs thc pcrccntagc o[chtorinc in a compound by
drrating a wcighcd samplc with AgNOl. His mcasurcincnrs arc:
oassbcakcr
+ iamplc'
mass bcakcr
vohlmcAgNOl f
A
B
pcrccntagcofCl = l00
conc-
AgNOI *{
x &'x .U x l5-5
(8
-
.{)