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The reporting of chemical analyses of silicate rocks
Geeehimieaet CosmochImScs d&s, 1857,Vol. lt:pp. 247 to 251. PergamonPress Ltd., London
The
reporting of chemical anROBEBY
A.
CHALMERS* (Received
of silioa~
l!odss
and E. S. PAUE t
7 August 1956)
&ptppeeI is made to tXlldySt8 to report aZls&?%S in Such & way that the degree Of pm&ion of the am-+& is included in the quoted results. Examples are given of t,he caloulmtion of the statistical errors in en analysis, and of a method of quoting results.
Ab&sct-_An
INTRODUCTION THE wide spread of results obtained in two recent investigations of precision and accuracy in silicate rock analysis (FAXRBAIRN et al., 1951; FAIBBAIRN and SGHAIRER, 1952) has caused some concern amon~t chemists and ~trolo~sts. Ia the first of these papers it is stated that “The retention of two figures beyond the decimal point, except for very minor constituents, becomes ridiculous for most petrological purposes. The chemist is justified in presenting his results in this way since addition to 100 per cent is a test of his work.” The first of these statements is probably true, although CHAYES (1953) has advocated retention of the second decimal, on the ground that if it is rounded off it cannot be reconstructed by the user of the analysis, who may wish to know it. CHAYES’S argument is open to the same objection as is the second of the statements quoted above. First, the analyst quotes his results in terms of various oxides, assuming that the elements he has determined are accompanied in the rock by stoiohiometric proportions of oxygen, although this may not be the case. Secondly, as has been pointed out by SCHLECHT (1953), the errors in chemical separations are likely to be greater than those associated with the instruments used, and are certainly harder to control. Thirdly, even if no non-Daltonian compounds are present, and no chemical or manipulative errors have been made, there will still remain an uncertainty in the results, arising from the statistical error inherent in all observed numerical data. In other words, the errors in even the most painstaking work may be sufficiently large to make the reporting of four significant figures unjustifiable. SCHLECRT (1953) has criticized the estimation of errors by calculation of the observational errors in the instruments used, on the grounds that the chemical errors may be greater in magnitude than the observational errors and will not usually be susceptible to calculation. Although this is true, it is rather a negative contribution to the problem of providing a criterion for the number of significant figures to be quoted. It is usually fairly easy to estimate the standard deviation of the observational errors and from this to give limits, which would be exceeded only rarely, for the observational error. If then it is desired, the number of significant figures quoted can be adjusted so that the observational errors can be reasonably expected to have no effect on the figures given. It will in general l Chemistry Department, University of Aberdeen, Old Aberdeen. t Department of Mathematios, University of Durham, South Road, Durhnm.
ROBERT
A. CHALMERS snd E. S. PAOE
be preferable to state the data as observed and the estimated standard deviation of the observational error, whether or not the former are truncated in a final or summary statement of the results. In such a summary statement it will usually be found that not more than three si~i~cant figures can be justified. Two examples will confirm this and illustrate the method. STATISTICAL ERRORS
Consider an analysis conducted by the classical method, and suppose that the analytical balance used has a standard deviation of O-1 mg at the loads used. That is to say, in about 99 per cent of weighings the error‘ of a single weighing will not exceed 0.26 mg. About a gram of sample is weighed and silica determined in the usual way by fusion with sodium carbonate, dehydration in the presence of hydrochloric acid, filtration, recovery of dissolved silica from the filtrate and recovery of any further dissolved silica by coprecipitation with the ammonia precipitate and further treatment of the mixed oxides. The weight of silica will be determined by measuring the loss in weight when the crude silica ignition products are treated with hydrofluoric and sulphurio acids, and reignited. The possible error arising from statistical fluctuations can now be calculated. Suppose that in weighing the sample there is an error of x mg in the first weighing and y mg in the second weighing. Then we are implicitly supposing that both z and y are independent random variables following the normal or Gaussian distribution with zero mean and standard deviation 0.1 mg. Then the error in the weight of the sample is 2 - y mg. This error is again a normally distributed random variable with zero mean, but now it has standard deviation [(O*1)2+ (0*1)2]- = 0.14 mg. [In statistical language we have Variance (x - zf) = Variance x + Variance y Standard deviation = (Variance)‘.] If the silica has been initially separated by dehydration, and the silica that has escaped the initial separations has been recovered from the mixed oxides, as described above, then four weighings will have been made in determining the total weight of silica in the sample. Consequently the standard deviation of this weight will be twice the standard deviation of a single weighing. The percentage of silica in the rock is then calculated from the ratio of the weight of silica to the weight of sample. Let these weights be X and Y respectively (we are supposing that X and Y are normally distributed random variables with means the true weight X of- silica.and the true weight Y of the sample, and standard deviations 20 and a%!2 respectively, where u is the standard deviation of a single weighing). The observed percentage of silica is then 100 . X/Y. To the first order (Variance X .P Variance of 100. X/Y = IO4 ---p2-j- -?%. Variance Y , 1 ‘i Taking the case where (I = O-1 mg and a sample of about 1 g is taken, and where the sample contains about 60 per cent of silica, we obtain standard deviation 248
The reporting of chemical analyses of silicate rocks
of observed percentage of silica = 0,022 per cent. This is, in 99 per cent of all determinations of silica in this rock by the method described, the observational error will not exceed about 0.00 per cent of the rock total. It is, however, necessary to modify slightly the reasoning given above, to take account of the fact that in practice ignitibn products are weighed “to conThat is, heating8 and weighings are repeated until two consecutive stant weight.” observations agree within some previously decided limit, say k mg., the observed Obviously k should weight being taken as the mean of the last two observations. not be less than the error reasonably to be expected in a single weighing. If indeed true constant weight has been reached, the last two weighings are independent observations of the same quantity and the observed mean has, therefore,
standard
deviation
a/2/2,
and mean
the true
weight.
This
means
This refinement in calculation, however, that the standard deviation of X is ads. makes only a negligible difference to the 99 per cent limits of observational error, O-06 per cent, already quoted. If, however, the true weight has not been reached when both weighings are made, the recorded value will be distributed about a mean greater than the true weight. This is clearly a chemical error. In the case of volumetric determinations it is possible to estimate the observational error in the burette readings, and the error involved in the control of addition of reagent, which determines the error involved in-observing the end-point. If the smallest amount of reagent that can conveniently be added is 0.03 ml, then an end-point that occurs within such an addition may be in error by almost the whole of the increment. If the total iron is determined in the mixed oxides from 1 g of rock, by reduction to the ferrous state and titration with ceric sulphate, the observational error can be calculated as follows. In order to estimate the standard deviation of the volume of reagent added from the burette we need to consider the errors in reading the burette at the beginning and the end of the titration, and the error arising from the actual position of the end-point within the last increment of titrant. Let us suppose that each reading of the burette is accurate to &O*Ol ml, and that the smallest drop of titrant that can be conveniently added is O-03 ml. The error in the amount of t&rant needed lies between -0.02 and +O*OS ml. The standard deviation will, then, be of the order of O-02 ml .* We are not considering the error arising from drainage in the burette during the titration, an error that may amount to a few hundredths of a millilitre (KOLTHOFF and SANDELL, 1952). Apart from errors arising from the use of the burette, there is likely to be an error of about 1 part per 1000 in the standardization of the titrant. In the case of 0*0500 N ceric sulphate, 1 ml is equivalent to 3.993 mg of ferric oxide, and the error in standardization is therefore equivalent to an error of about 0.004 mg of ferric oxide per til of titrant; the standard deviation is then about 0.002 mg/ml. Let x ml be the amount of titrant used, and y mg/ml be the ferricoxide equivalent of ‘the titrant. Then the observed amount of ferric oxide is * For these readings there is a good case for supposing the distributions to be rectangular, in which however, the standard deviation will not be greatly affected. An analogous example in quite a different context is given by BARNARD (1951). case,
249
~{l,lW;liT
2-y
1np,
and
we have \‘nriancc
\\‘IIVII .T ix about, ahout x per cent
to the (~j)
.4. (‘tl.\I.wcl~s cant1k:. s.
first
order,
using
= y-2 . Variance
bars
L’.\(:E
to denote
x + f2 . Variance
20 ml and y is about, 4 mg/ml (i.e. a gram of total ferric oxide ix being analysed):
\‘ariailcc
in amount
of ferric
means
as before,
y.
of a sample
containing
oxide = 8 x 10e3 (mg)2,
i.c, I Iw standard deviation of the amount of ferric oxide is OWI mg, and the standard tl(*viat ioII of t’hc pcr(*entagc of ferric: oxide is approximately 0.01 per cent. That~ is. t 11th !)!I-]wr wnt, limit,s of error are about &0.026 per cent. The relative error in t I)(%cl(~tc~rmillatioi~ of total iron could in this case clearly be about 3 parts per 1000. analysis by the classical method some constituents will be III il complatc For such a determination the variance will tlvtc~rni~lcvl illdirectly by difference. I)(h the SIIM of t,he variances of the contributing factors, just as was the case for Thus for an indirect determination t Iw tlrt,c~rnliilatioii of the total weight of silica. of alumina. soInc five or six factors are involved, and the standard deviation is likrly t,o bc about twice t,hnt given above for the total iron, even neglecting the (brror arising iI1 weighing any alumina recovered from the filtrate from the ;lI11IllOlliiL lmbcipitatioli. ‘I’hc mc%hotl of analysis of errors given above may be applied to determine t Iw st,ailtf;irtl tlcviation of a determination calculated from observed results llsillg 0111y mii~t~iphation ad divis!on. Approximate results can be obtained wllc%Ilot~ll(~rfutlctions of t,he observations are used, from formulae such as 1’ariance ,f(z) = [J’(X)]” Variance (k\o, I!).i-‘: J)av1es, I,&, IIX r($iiril to
z
l!Hi). tllv ])rc%cntnt,ioJi of results: it is seen that the g!+per cent Iitllit, of error ~ttllolltlt,s. ill cases of major constituents, to several units in the sc~c~)~itl tlcainial place of percentage. Therefore it, is best that the results be prescntcbtl itI full. i.e. giving t,wo decimal places koget’her with the estimated standard 111 this way the accuracy of the results is given, limits on the error (I(~viatioil. (-;~II I)(b ckly cnlculatctl. mid the results can be compared with others. It is sotrlc$iJnc~s necessary in a snrnmar~ t’o express each result by one number only. ( ‘i(~ill*l~ t.his can only be achieved with some loss of information, and previously th(b itlforlnntion on the accuracy has bre~l sacrificed as only the result itself, to tllch fllll Ilunlbcr of decimals, has been quoted. This method of reporting can impression of t’he accuracy attained, and so we suggest, ii1ij);lrt il Inislentling I II;L~ it’ t,llis met,hocl of summary presentation is unavoidable, the result should 1~ clllot,cbtl to such a number of decinial places that variation within the W-per f~c~llt limits of (‘rror causes at, most1 a variation of one unit in the last figure quoted. I Lot 1, nlt4 Ilods of presentation are exemplified in Table 1. The obser$ed differences i II it .tllll)licntc~ alIa1.ysis arc quoted for comparison with t,he estimated statScal (~ITOI*S. It, is seen t,hat, in 110 case of the summary reporting are we justified in (iriot illg t,o more thall the first decimal place. Indeed, for four of t’he constituent’s, I~‘c>().(‘~0. H,O( -+), and l’,O,, the statistical error could cause t)he first decimal t 0 Iw w17ided tlifferenhly. (‘hrnlical errors. which we have not considered here. would add further to 2,X)
The reporting of chenric&l &nnlyse:Sof silie&te rocks
the doubt felt about the accuracy of the determinations. In general it may be said that inadequate reporting of analytical results limits the value of the infornlatiol~ contained in them, and may make them misleading. In particular. it must be stressed that in those cases, such as in petrology, where the results arc made the basis for comparisons, it is essential that an estimate of the precision Table II
I
SiO, x12f-b
b-C,O:, Fe0 ‘I’itl, t’&( f .\I&$ )
K,O xa20
H,O (-) H,O (+I MI10 I’,(), _-.__- ___~
___
.- _.~__
OJJlO 0.017 0.013 0.012 0.005 0.007 0.008 0.014 0.020 0*008 0.016 0~005 O.OM
III
0.02 0.20 0.07 0*03 nil nil
0.07 0.04 0.03 0.03 0.01 0.03 ~.
1.00.24
Total I. 11. III. IV.
52.82 13.40 1.03 6.96 0.42 IO*25 9.70 0.20 0.99 0.61 3.64 0.17 0.05 -I--
1
nil -._I__
I\‘
52.8 13.4
l.(J 7.0 6.4 10.3 9.7 0.2 1.0 0.6 3,6 0‘2 0.1 100.3
Mean of duplicate analyses. Estimated st&nd&rd deviation. Observed difference between duplicete lleter~nin&tio~ls. Resuhs rounded off to imply est,imated error.
and accuracy of the results should be available in order that the significance of differences may be properly assessed. This conclusion has been repeatedly stressed in the literature but ~lnfo~unately has been more often ignored than heeded.
References (1951) St&tistics applied to assembly processes. Statistical Metlwd in Ilntlustriul I’mhtction p. 55. Royal Statistical Society, London. ('H_AYESF. (1953) In defence of the second decimal. Amer. Mineral, 38, 784. D.I~IFZ0. L. (Ed.) (1947) Stat&id Methods irt h?esearch and ~r~(~~~t~o~ (2nd edition) 1’. 37. Oliver and Hoyd, London. FAIRB~IRN II.W. and SCHAIRER ;i. F. (1952) A test of the accuracy of chemical analysis of silicate rocks. Amber. Mineral. 37, 744. F~IRBAIRN H. W.,SCHLECHT TV. G.,STEVEN~I~.E.,DENNENM*.H.,AHRENSL.H., and CHAYES F. (1951) A co-operative invcstigat.ion of precision and accuracy in chemical, spcetrochemical. &nd modal analysis of silicate rocks. United States Gedogical Survey Bulletin So. 980. krocmom 1. %f. and SANDELI, E. B. (1952) Textbook of ~~u?~~~~t~~~ ~~~Tg~~n~~~ Alta.lysi.s (3rd edition), p. 425, X&cmill&n, Xew York. Ii 40 C. R. (1952) Advanced Statistid Methods in Bionletric Eesearclrr p. 209, WiIey, Xew York. SPHLECHT IV. U. (1953)The probable error of & chemical analysis. I’nited States CeoZogicnl bVwre2/ Bulletin So. 992, p. 57. Ua~mam
Ii.
ii.
231
The
reporting of chemical anROBEBY
A.
CHALMERS* (Received
of silioa~
l!odss
and E. S. PAUE t
7 August 1956)
&ptppeeI is made to tXlldySt8 to report aZls&?%S in Such & way that the degree Of pm&ion of the am-+& is included in the quoted results. Examples are given of t,he caloulmtion of the statistical errors in en analysis, and of a method of quoting results.
Ab&sct-_An
INTRODUCTION THE wide spread of results obtained in two recent investigations of precision and accuracy in silicate rock analysis (FAXRBAIRN et al., 1951; FAIBBAIRN and SGHAIRER, 1952) has caused some concern amon~t chemists and ~trolo~sts. Ia the first of these papers it is stated that “The retention of two figures beyond the decimal point, except for very minor constituents, becomes ridiculous for most petrological purposes. The chemist is justified in presenting his results in this way since addition to 100 per cent is a test of his work.” The first of these statements is probably true, although CHAYES (1953) has advocated retention of the second decimal, on the ground that if it is rounded off it cannot be reconstructed by the user of the analysis, who may wish to know it. CHAYES’S argument is open to the same objection as is the second of the statements quoted above. First, the analyst quotes his results in terms of various oxides, assuming that the elements he has determined are accompanied in the rock by stoiohiometric proportions of oxygen, although this may not be the case. Secondly, as has been pointed out by SCHLECHT (1953), the errors in chemical separations are likely to be greater than those associated with the instruments used, and are certainly harder to control. Thirdly, even if no non-Daltonian compounds are present, and no chemical or manipulative errors have been made, there will still remain an uncertainty in the results, arising from the statistical error inherent in all observed numerical data. In other words, the errors in even the most painstaking work may be sufficiently large to make the reporting of four significant figures unjustifiable. SCHLECRT (1953) has criticized the estimation of errors by calculation of the observational errors in the instruments used, on the grounds that the chemical errors may be greater in magnitude than the observational errors and will not usually be susceptible to calculation. Although this is true, it is rather a negative contribution to the problem of providing a criterion for the number of significant figures to be quoted. It is usually fairly easy to estimate the standard deviation of the observational errors and from this to give limits, which would be exceeded only rarely, for the observational error. If then it is desired, the number of significant figures quoted can be adjusted so that the observational errors can be reasonably expected to have no effect on the figures given. It will in general l Chemistry Department, University of Aberdeen, Old Aberdeen. t Department of Mathematios, University of Durham, South Road, Durhnm.
ROBERT
A. CHALMERS snd E. S. PAOE
be preferable to state the data as observed and the estimated standard deviation of the observational error, whether or not the former are truncated in a final or summary statement of the results. In such a summary statement it will usually be found that not more than three si~i~cant figures can be justified. Two examples will confirm this and illustrate the method. STATISTICAL ERRORS
Consider an analysis conducted by the classical method, and suppose that the analytical balance used has a standard deviation of O-1 mg at the loads used. That is to say, in about 99 per cent of weighings the error‘ of a single weighing will not exceed 0.26 mg. About a gram of sample is weighed and silica determined in the usual way by fusion with sodium carbonate, dehydration in the presence of hydrochloric acid, filtration, recovery of dissolved silica from the filtrate and recovery of any further dissolved silica by coprecipitation with the ammonia precipitate and further treatment of the mixed oxides. The weight of silica will be determined by measuring the loss in weight when the crude silica ignition products are treated with hydrofluoric and sulphurio acids, and reignited. The possible error arising from statistical fluctuations can now be calculated. Suppose that in weighing the sample there is an error of x mg in the first weighing and y mg in the second weighing. Then we are implicitly supposing that both z and y are independent random variables following the normal or Gaussian distribution with zero mean and standard deviation 0.1 mg. Then the error in the weight of the sample is 2 - y mg. This error is again a normally distributed random variable with zero mean, but now it has standard deviation [(O*1)2+ (0*1)2]- = 0.14 mg. [In statistical language we have Variance (x - zf) = Variance x + Variance y Standard deviation = (Variance)‘.] If the silica has been initially separated by dehydration, and the silica that has escaped the initial separations has been recovered from the mixed oxides, as described above, then four weighings will have been made in determining the total weight of silica in the sample. Consequently the standard deviation of this weight will be twice the standard deviation of a single weighing. The percentage of silica in the rock is then calculated from the ratio of the weight of silica to the weight of sample. Let these weights be X and Y respectively (we are supposing that X and Y are normally distributed random variables with means the true weight X of- silica.and the true weight Y of the sample, and standard deviations 20 and a%!2 respectively, where u is the standard deviation of a single weighing). The observed percentage of silica is then 100 . X/Y. To the first order (Variance X .P Variance of 100. X/Y = IO4 ---p2-j- -?%. Variance Y , 1 ‘i Taking the case where (I = O-1 mg and a sample of about 1 g is taken, and where the sample contains about 60 per cent of silica, we obtain standard deviation 248
The reporting of chemical analyses of silicate rocks
of observed percentage of silica = 0,022 per cent. This is, in 99 per cent of all determinations of silica in this rock by the method described, the observational error will not exceed about 0.00 per cent of the rock total. It is, however, necessary to modify slightly the reasoning given above, to take account of the fact that in practice ignitibn products are weighed “to conThat is, heating8 and weighings are repeated until two consecutive stant weight.” observations agree within some previously decided limit, say k mg., the observed Obviously k should weight being taken as the mean of the last two observations. not be less than the error reasonably to be expected in a single weighing. If indeed true constant weight has been reached, the last two weighings are independent observations of the same quantity and the observed mean has, therefore,
standard
deviation
a/2/2,
and mean
the true
weight.
This
means
This refinement in calculation, however, that the standard deviation of X is ads. makes only a negligible difference to the 99 per cent limits of observational error, O-06 per cent, already quoted. If, however, the true weight has not been reached when both weighings are made, the recorded value will be distributed about a mean greater than the true weight. This is clearly a chemical error. In the case of volumetric determinations it is possible to estimate the observational error in the burette readings, and the error involved in the control of addition of reagent, which determines the error involved in-observing the end-point. If the smallest amount of reagent that can conveniently be added is 0.03 ml, then an end-point that occurs within such an addition may be in error by almost the whole of the increment. If the total iron is determined in the mixed oxides from 1 g of rock, by reduction to the ferrous state and titration with ceric sulphate, the observational error can be calculated as follows. In order to estimate the standard deviation of the volume of reagent added from the burette we need to consider the errors in reading the burette at the beginning and the end of the titration, and the error arising from the actual position of the end-point within the last increment of titrant. Let us suppose that each reading of the burette is accurate to &O*Ol ml, and that the smallest drop of titrant that can be conveniently added is O-03 ml. The error in the amount of t&rant needed lies between -0.02 and +O*OS ml. The standard deviation will, then, be of the order of O-02 ml .* We are not considering the error arising from drainage in the burette during the titration, an error that may amount to a few hundredths of a millilitre (KOLTHOFF and SANDELL, 1952). Apart from errors arising from the use of the burette, there is likely to be an error of about 1 part per 1000 in the standardization of the titrant. In the case of 0*0500 N ceric sulphate, 1 ml is equivalent to 3.993 mg of ferric oxide, and the error in standardization is therefore equivalent to an error of about 0.004 mg of ferric oxide per til of titrant; the standard deviation is then about 0.002 mg/ml. Let x ml be the amount of titrant used, and y mg/ml be the ferricoxide equivalent of ‘the titrant. Then the observed amount of ferric oxide is * For these readings there is a good case for supposing the distributions to be rectangular, in which however, the standard deviation will not be greatly affected. An analogous example in quite a different context is given by BARNARD (1951). case,
249
~{l,lW;liT
2-y
1np,
and
we have \‘nriancc
\\‘IIVII .T ix about, ahout x per cent
to the (~j)
.4. (‘tl.\I.wcl~s cant1k:. s.
first
order,
using
= y-2 . Variance
bars
L’.\(:E
to denote
x + f2 . Variance
20 ml and y is about, 4 mg/ml (i.e. a gram of total ferric oxide ix being analysed):
\‘ariailcc
in amount
of ferric
means
as before,
y.
of a sample
containing
oxide = 8 x 10e3 (mg)2,
i.c, I Iw standard deviation of the amount of ferric oxide is OWI mg, and the standard tl(*viat ioII of t’hc pcr(*entagc of ferric: oxide is approximately 0.01 per cent. That~ is. t 11th !)!I-]wr wnt, limit,s of error are about &0.026 per cent. The relative error in t I)(%cl(~tc~rmillatioi~ of total iron could in this case clearly be about 3 parts per 1000. analysis by the classical method some constituents will be III il complatc For such a determination the variance will tlvtc~rni~lcvl illdirectly by difference. I)(h the SIIM of t,he variances of the contributing factors, just as was the case for Thus for an indirect determination t Iw tlrt,c~rnliilatioii of the total weight of silica. of alumina. soInc five or six factors are involved, and the standard deviation is likrly t,o bc about twice t,hnt given above for the total iron, even neglecting the (brror arising iI1 weighing any alumina recovered from the filtrate from the ;lI11IllOlliiL lmbcipitatioli. ‘I’hc mc%hotl of analysis of errors given above may be applied to determine t Iw st,ailtf;irtl tlcviation of a determination calculated from observed results llsillg 0111y mii~t~iphation ad divis!on. Approximate results can be obtained wllc%Ilot~ll(~rfutlctions of t,he observations are used, from formulae such as 1’ariance ,f(z) = [J’(X)]” Variance (k\o, I!).i-‘: J)av1es, I,&, IIX r($iiril to
z
l!Hi). tllv ])rc%cntnt,ioJi of results: it is seen that the g!+per cent Iitllit, of error ~ttllolltlt,s. ill cases of major constituents, to several units in the sc~c~)~itl tlcainial place of percentage. Therefore it, is best that the results be prescntcbtl itI full. i.e. giving t,wo decimal places koget’her with the estimated standard 111 this way the accuracy of the results is given, limits on the error (I(~viatioil. (-;~II I)(b ckly cnlculatctl. mid the results can be compared with others. It is sotrlc$iJnc~s necessary in a snrnmar~ t’o express each result by one number only. ( ‘i(~ill*l~ t.his can only be achieved with some loss of information, and previously th(b itlforlnntion on the accuracy has bre~l sacrificed as only the result itself, to tllch fllll Ilunlbcr of decimals, has been quoted. This method of reporting can impression of t’he accuracy attained, and so we suggest, ii1ij);lrt il Inislentling I II;L~ it’ t,llis met,hocl of summary presentation is unavoidable, the result should 1~ clllot,cbtl to such a number of decinial places that variation within the W-per f~c~llt limits of (‘rror causes at, most1 a variation of one unit in the last figure quoted. I Lot 1, nlt4 Ilods of presentation are exemplified in Table 1. The obser$ed differences i II it .tllll)licntc~ alIa1.ysis arc quoted for comparison with t,he estimated statScal (~ITOI*S. It, is seen t,hat, in 110 case of the summary reporting are we justified in (iriot illg t,o more thall the first decimal place. Indeed, for four of t’he constituent’s, I~‘c>().(‘~0. H,O( -+), and l’,O,, the statistical error could cause t)he first decimal t 0 Iw w17ided tlifferenhly. (‘hrnlical errors. which we have not considered here. would add further to 2,X)
The reporting of chenric&l &nnlyse:Sof silie&te rocks
the doubt felt about the accuracy of the determinations. In general it may be said that inadequate reporting of analytical results limits the value of the infornlatiol~ contained in them, and may make them misleading. In particular. it must be stressed that in those cases, such as in petrology, where the results arc made the basis for comparisons, it is essential that an estimate of the precision Table II
I
SiO, x12f-b
b-C,O:, Fe0 ‘I’itl, t’&( f .\I&$ )
K,O xa20
H,O (-) H,O (+I MI10 I’,(), _-.__- ___~
___
.- _.~__
OJJlO 0.017 0.013 0.012 0.005 0.007 0.008 0.014 0.020 0*008 0.016 0~005 O.OM
III
0.02 0.20 0.07 0*03 nil nil
0.07 0.04 0.03 0.03 0.01 0.03 ~.
1.00.24
Total I. 11. III. IV.
52.82 13.40 1.03 6.96 0.42 IO*25 9.70 0.20 0.99 0.61 3.64 0.17 0.05 -I--
1
nil -._I__
I\‘
52.8 13.4
l.(J 7.0 6.4 10.3 9.7 0.2 1.0 0.6 3,6 0‘2 0.1 100.3
Mean of duplicate analyses. Estimated st&nd&rd deviation. Observed difference between duplicete lleter~nin&tio~ls. Resuhs rounded off to imply est,imated error.
and accuracy of the results should be available in order that the significance of differences may be properly assessed. This conclusion has been repeatedly stressed in the literature but ~lnfo~unately has been more often ignored than heeded.
References (1951) St&tistics applied to assembly processes. Statistical Metlwd in Ilntlustriul I’mhtction p. 55. Royal Statistical Society, London. ('H_AYESF. (1953) In defence of the second decimal. Amer. Mineral, 38, 784. D.I~IFZ0. L. (Ed.) (1947) Stat&id Methods irt h?esearch and ~r~(~~~t~o~ (2nd edition) 1’. 37. Oliver and Hoyd, London. FAIRB~IRN II.W. and SCHAIRER ;i. F. (1952) A test of the accuracy of chemical analysis of silicate rocks. Amber. Mineral. 37, 744. F~IRBAIRN H. W.,SCHLECHT TV. G.,STEVEN~I~.E.,DENNENM*.H.,AHRENSL.H., and CHAYES F. (1951) A co-operative invcstigat.ion of precision and accuracy in chemical, spcetrochemical. &nd modal analysis of silicate rocks. United States Gedogical Survey Bulletin So. 980. krocmom 1. %f. and SANDELI, E. B. (1952) Textbook of ~~u?~~~~t~~~ ~~~Tg~~n~~~ Alta.lysi.s (3rd edition), p. 425, X&cmill&n, Xew York. Ii 40 C. R. (1952) Advanced Statistid Methods in Bionletric Eesearclrr p. 209, WiIey, Xew York. SPHLECHT IV. U. (1953)The probable error of & chemical analysis. I’nited States CeoZogicnl bVwre2/ Bulletin So. 992, p. 57. Ua~mam
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