A Plan for Studying the Accuracy and Precision of an Analytical Procedure FREDERIC J. LINNIG, JOHN MANDEL, and JEAN M. PETERSON National Bureau of Standards, Washington,
D. C.
This work was undertaken to clarify the concepts of accuracy and precision and to show how statistical methods available in the literature may be used in a flexible plan for studying the accuracy and precision of a n analytical procedure. The plan, illustrated by application to a particular analytical method, involves the segregation of relative-type and constant-type systematic errors. The effect of relative-type errors is generally more important in the determination of major components, while constant-type errors are more serious in the determination of trace components of material. Methods are presented for detecting the sources of variability in the results which reduce the precision of an analytical procedure. The relationship between the concepts of accuracy and precision of a method is shown to depend on the particular application involved. Methods having no satisfactory theoretical basis may be studied using these techniques and maximum efficiency may be achieved in the study by designing the experimental work to fit the particular problem in hand. Segregation of errors and detection of sources of variability facilitate their elimination or control and lead to a more realistic estimate of the limitations of the method. The segregated relativetype and constant-type errors may be helpful in the quantitative study of the chemical factors involved in some equilibrium reactions.
T
H I S paper deals with the investigation of a chemical problem in which the statistical approach has been used from the start-i.e., in the actual planning or design of the ecperiment-as well as in the interpretation of the resulting data The problem involves the study of the accuracy and precision of a method for titrating fatty acid in solutions of GR-S synthetic rubber. I n the preparation of GR-S the soap used as an emulsifier partly reacts during the coagulation with sodium chloride and sulfuric acid. Because the organic acid produced as well as the remaining soap affects the rate of vulcanization and therefore the properties of the vulcanized stocks, it becomes necessary to control within specified limits the quantity of these materials present in the coagulated rubber. The method studied here for the determination of one of these constituents, organic acid as fatty acid, is illustrated in Table I as the procedure used in studying precision. The work described here including the statistical planning was
preceded, of course, by the development of a procedure which was satisfactory from the operational standpoint, including the choice of adequate solvents and an indicator giving a detectable color change ( 7 ) . The following considerations are pertinent in titrations of this type: The organic character of the solvents used does not readily permit the application of the usual methods for determining the accuracy of the end point. The determination of fatty acid must be made in the presence of varying quantities of soap-i.e., from 3 to 6% of fatty acid must be titrated in the presence of from 0.00 to 0.76% soap from the same acid. The indicator must give accurate resultq in the presence of each of a number of differently colored suhstances used to stabilize the rubber. Inorganic salts present in GR-S are insolutJle in the organic solvents used and, therefore, should not afl ert the titration. These considerations involve different phases of a study of svstematic errors-Le., the study of accuracy. I n the absence of sufficient information concerning acid-base titrations in these solvents the usual a priori choice of an accurate indicator could not be made. The use of instrumental methods, such as potentiometric or conductometric titrations t o determine the concurrence of the color change and the equivalence point in these solutions, would in itself involve considerable study. A simpler approach is the one ordinanly employed in which known added quantities of fatty acid are titrated t o determine the material “found” in the presence of the substances that might be most likely t o interfere, in this caw soap and stabilizer. It is shown that with this approach the statistical theory of a straight line is invaluable not only in clearly defining the practical limits of the accuracy of the procedure but also in assigning chemical interpretations to the sources of error which the study discloqes. .4nalysis of variance techniques is shown to be useful in determining the reproducibility or precision of the method as a control procedure. The study as a whole affords some insight into the relationship between the concepts of accuracy and pi ecision as applied to the study of the errors involved in a given analytical procedure. Eisenhart (6) has discussed these concepts from the standpoint of the selection of a method of knomm accuracy and precision best suited t o making a particular type of meaqurement, while Mandel and Stlehler ( 9 ) have introduced the concept of sensitivity as a measure of the suitahility of a method. A concept related to sensitivity and referred t o as the “Cute” or “goodness” of a method, was introduced somewhat earlier in the German literature by Reichel (IO) -
S t e p s in Procedure Sample
Dissolved in Aliquot used Titrated xitli Indicator used Result Blank correction Calculation
Table I.
Procedures Used in Studying Precision and Accuracy Precision
Production, 5
g.
250 ml. 100 ml. 0.1N KaOH (alcoholic) m-Cresol purple MI. reagent .M1. reagent reiiuired for sol\-ent and indicator Fatt?l acid, ci - nil. of reagent (corrected) X S X m e. 0 R-t. of sainple
1102
Accuracy [ Purified rubber Simulated production 1 gtabilizer IF%y acid (j g,) 250 ml. 100 ml. 0.l.Y S a O H (alcoholic) m-Cresol purple 511. reagent See Table VI and text (See Figure 1, Tables 111 t o V I , and t e s t )
i
wt.
X
2 . n X 100
V O L U M E 2 6 , NO. 7, J U L Y 1 9 5 4
1103 The effect of soap was studied only in the presence of stabilizer
Table 11. Stabilizer
series of Experiments
B C
soap Present, G.
-4,the least opaque of t'he three and consequently the one least
Quantity of F a t t y iicid Added, G. 0.02 0.05 0.15 0.25 0.50 KO.of determinations -
0 0054 0 0270 0 0540
..
2 3 4
0 0230
2
5
0.0270
2
1
.I
Specific Plan of Experiment
2
.
2 2 2 2 2
2 ,,
2 2
2 2 2
2 2 2
2
2
2
2
ACCURACY
Experimental Plan. The study of accuracy involved t,he titration of a number of solutions made up as shown in Table I. Each solution simulated a 5-gram sample of production rubber. The purified rubber was prepared by extracting production rubber with ethyl alcohol-toluene azeotrope, in order t o remove all but the smallest traces of fatty acid, soap, and stabilizer. The known quantity of st abilizer added was equivalent to that usually present in 5 grams of production rubber. The three types -4, B, and C (Stalite, phenyl 2-naphthglamine, and BLE) are named in the order of increasing opacity in solution. Soap was added in three different quantities: a medium amount close to the usual quantity present in 5 grams of rubber and t'wo other quantities close to the upper and lower limits of the expected range. Fatty acid, the constituent to be analyzed, was also added in a peries of carefully selected known quantities as explained in the discussion of the analysis of the data by the statistical approach. Table I1 shows more specifically how the entire experiment was planned and the exact, quantities of materials that were used. The two determinations indicated for each quantity of added acid in each of' the five series xere made on aliquots of two separate solutionp, each of which was prepared as described in Table I.
likely to interfere in the detection of the end point. The effect of soap was expected to be negligibly small. However, if contrary to expectation the effect proved to be large enough to be of practical importance, the three sets of duplicate deterniinations made in the presence of the smallest and largest quantiLies of soap should be sufficient for its detection. As it was known that the 5 to 1 mixture of toluene and alcohol used as solvent' requires a certain quantity of reagent to produce the color change of the indicator, four rubber-solvent blank determinations were made in the presence of e:tch stabilizer, and four solvent blank titrations were made to find whether this second more readily determined blank might, be an adequate substitute for the first. The reason for making this particular number of blank determinations i p explained in the discussion of t,he analysis of the data. In all cases, the errors involved in weighing out the materials and in t,aking aliquots are negligible with respect to the test errors. Also the variability of the test, measurement is assumed to be independent of the quantity of material titrated. Under these conditions the application of the usual least squares formulas for linear regression is legitimate. To avoid introducing day-to-day variability, as discussed later, into the study of the straight-line relationship, all the determinations and blanks made in the presence of a given stabilizer were completed on the same daHowever, the order of weighing the sample, the order of pos ion on the hot plate, and the order in which the titrations were made within the day were entirely random. Randomization is necessary to neutralize the effects of possible gradual systematic changes in techniques during the course of the day. Such systematic changes are often encountered and aredue to a variety of causes such as fatigueof theoperator or in this case the gradual darkening of the rubber solution caused by the effect of sunlight on the stabilizer. A random
Standard Procedure
Investigated Procedure
Fatty acid I
Calculation :
I
I I
I I
I
I
I
Figure 1. Determination of Accuracy Customary verBu8 statistical approach
.
I
I
ANALYTICAL CHEMISTRY
1104
order is easily determined from tables of random numbers given in standard textbooks on statistics (2, 6, IS). Criterion of Accuracy. A judgment of accuracy is facilitated by comparison with a known standard of reference. Since the organic acid used in GR-S is a mixture of fatty acids of unknown composition, t'he equivalent weight could not be calculated for use in determining the quantit,y of material found in the titration. The ASTM procedure for the determination of acid number ( 1 ) was therefore chosen as the best available crkerion of accuracy. The diagram in Figure 1 illustrates how the equivalent weight of the fatty acid determined by the ASTM procedure is used to determine the quantity of acid found in the procedure under investigation. The equivalent weight of the acid used in this study was 275.1. This criterion of accuracy is not absolute. Sliewhart (18) has given an interesting discussion of the essential operat,ional nature of all criteria of accuracy. Analysis of the Data. It is advisable to plan quantitative esperiments that have progressed beyond the exploratory stages in such a manner that some accepted statistical technique may readily be used to analyze the resulting data. The present data are most satisfactorily treated by a statktical procedure known as regression analysis. Nevertheless, a more customary intuitive approach may ah0 be used. To demonstrate the advantages of the statistical procedure, two series of data, those numbered 2 and 5 in Table 11, are analyzed by both methods. This comparison is followed by a general presentation of the results relating to all five series of data. CI.STOMARY APPROACH. The resultj for the two series chosen are given in Tables I11 and IV. They represent determinations of fat,t,y ecid made in the presence of the same (medium) quantity of soap but of different statiilizers: the lightest colored, .4, and the most opaque, C. For rach titration, the milliliters of O.lO5ON alcoholic sodium hydroxide corresponding to the prescribed 100-ml. aliquot (see the section on statistical procedures for milliliters used for seriea 5 ) were converted according to the
_______
--
~.-
method indicated diagrammatically in Figure 1 into milligrams of fatty acid found. These converted quantities are given i n the third columns in Tables I11 and IF'. It is immediately apparent that the fatty acid found exceeds that added in both sets of data. Column 4 is included to show that the ratio of the quantity found to that added decreases, rapidly a t first, then more s l o ~ l y ,as the amount of material titrated increases. This suggests the presence of a constant, systematic error. Therefore, the values 111 column 3 are corrected by the rubbersolvent blanks noted in Tables 111 and IV. A study of column 5 of these tables shows that in the case of stabilizer A the corrected ratios are distributed about a value close to unity, while in the case of stabilizer C these quantities are, nine times out of ten, distinctly greater than unity. Thus, a correction for the blank seems to remove all systematic errors from the first set of data (stabilizer A ) but leaves a residual error in the second set (stabilizer C). This rejidual systematic error, in contrast to that caused by the presence of a blank, is not constant, but proportional to the amount of material titrated. For example, found - blank if the ratio added were equal to 1.10, then in titrating 20 mg. of fatty acid an error of 2.0 mg. would result, while in titrating 500 mg. an error of 50 mg. would be involved This type of error may be called a relative error.
Y
P
z
Y = rnX+b
Titration 1 2 3 4 5 6 7 8 9 10
Ratio, Founded/Added 1.358 1.269 1.094 1.103 1.050
(Found - Blanha) Added0.998 0.957 0.960 0.971 1.004
1.032 1.021 1.005 1.017
1.004 0.993 0,991 1.003
...
a Rubber-s( ent blank equivalent to 6 92 mg. of equivalent t o 5.96 mg. of f a t t y acid
Table IV.
Titration 1 2 3
4 5 6 7 8 9 10
/
/
a W
INTERCEPT b FOR COMPARISON WlTH BLANK TITRATION
Soap present 27 0 mg.)
F a t t y Acid, Mg. Added Found 19.2 26.1 22.2 28.2 51.5 56.3 52.4 57.8 149.6 157.1 150.4 . .. 249.1 257.1 245.2 250.4 480.0 482.5 498.5 507.0
~
rn FOR COMPARISON WITH "STANDARD" PROCEDURE
Table 111. Titration of Fatty Acid i n Presence of Stabilizer A (Series 2
- BLANK X
k
I
GRAMS OF MATERIAL TITRATED-X
Figure 2.
Youden's Method for Studying Accuracv
...
t t y acid; solvent blank
Titration of Fatty Acid in Presence of Stabilizer C (Series 5. Soap present 27.0 mg.) F a t t y Acid, Mg. Ratio (Found - Blank=) Added Found Founded/Added Added 20.0 28 0 1.398 1.064 20.0 24.5 1.224 0.890 50.0 58.5 1.170 1.036 50.0 57.8 1.156 1.022 150.0 157.8 1.052 1.008 1.062 1.018 153.7 163.2 250.0 257.8 1.031 1.005 250.0 259.3 1.037 1.010 500.0 512.4 1.025 1.012 1.006 500.0 509.2 1.018
a Rubber-aolvent blank equivalent to 6.68rng. of f a t t y acid; solvent blank equivalent to 7.40 m g . of f a t t y acid.
While the customary approach has shown qualitatively the need for a blank correction in both sets of titrations and the probable presence of a relative error in the second set, it fails in a t least the following respects: The ratios in the last column show considerable variability which increases as the amount of fatty acid decreases. As a result therr is no definite criterion for judging whether the blank id an adequate correction for the constant type of error in the data in Table I11 or for estimating the relative type of error affecting the data in Table IV. If the experimental blank is inadequate as a correction for the constant type of error, the segregation of the two types of error8 may be accomplished only by a tedious trial and error procedure. STATISTICAL APPROACH.Linear regression analysis, the statistical procedure best suited to the analysis of this type of data, was first applied to the study of accuracy by Youden (16). In addition to giving the desired estimates for constant and relative errors, this method simultaneously permits the evaluation of their reliability and furthermore yields a measure of the random error affecting the titration. This method is indicated graphically in Figure 2. The ratio Y - blank indicated in this figure is analogous to the ratio
V O L U M E 26, NO. 7, J U L Y 1 9 5 4 found - blank given in column 5 of Tables 111 and IF’, except added that in order to simplify the statistleal calculations, the values Y and blank in the numerator of the ratio are not converted but left in the units in which they were obtained-milliliters of reagent. The simple algebraic transformation indicated in the figure shows that if values of I’ are plotted as a function of the corresponding values of X, a straight-line relationship is obtained. This geometric interpretation leads a t once to important conclusions. The geometric and statistical aspects of a straightline fit are related to the chemical aspects of accuracy and precision as follows: The slope of line, m , is the volume of titrant required per gram of material. This experimental value may be compared to a stoichiometric value predicted from the known normality of the reagent and the equivalent weight of the material. This comparison constitutes a test for one aspect of accuracy. Since both values are ratios, disagreement between these values constitutes a relative type of error. I n the present case the equivalent weight was determined by the ASTM procedure as described in Figure 1 and the criterion for this aspect of accuracy conutitutes essentially a comparison of the values obtained by the standard procedure and the investigated procedure. The intercept of line, b, is the volume of titrant required for a solution containing X = 0 gram of material. This should, of course, be identical within test error to the experimental blank titration. A comparison of these values constitutes a second aspect of accuracy. Disagreement between the value of intercept b and the experimental blank indicates a constant type of error. The ability to detect this disagreement depends, of course, on the precision of the value3 obtained for the intercept and the experimental blank. T o each quantity of material, X , there now correspond two values Y , the experimental volume of titrant and the ordinate inferred from the fitted straight line. The discrepancy between these two values constitutes a measure of experimental error that can be expressed as a standard deviation, u. It is customary in regression studie8 to refer to this standard deviation as the standard error of estimate The exact knowledge of u requires a large (theoretically infinite) number of experimental values. With a limited number of data, it is possible to estimate the value of u with a precision depending on the number of determinations. To avoid confusion, an estimate of u thus obtained is denoted by the letter s. The calculation of s is given in the section on statistical procedures. In regard to the selection of quantities of added matexial, the precision with which the slope and intercept are determined depends on two factors: the experimental error of the method expressed by u and estimated by s and the number and relative spacing of the weights, X , of added material. Since, as mentioned previously, the slope and intercept of the line are important factors in measuring accuracy, the number and spacing of the values of S become an important part of any plan designed for the study of accuracy. Specifically, the following steps are involved in the selection of the plan, before any experimental work is started A tentative set of X-values is selected, mainly on the basis of experimental feasibility. I t will be assumed that the selection was ab indicated in Table I and in the section on statistical procedures. The precision of the slope and intercept are calculated on the basis of this set by means of Equations 1 and 2 given in the section on statistical procedures. In this case urn = 1.83
U, in
Ub
ml. of reagent per gram of material
= 0.475 u, in ml. of reagent
1105
I t was known from previous work in developing the procedure that u is of the order of 0.03 ml. of reagent and that the number of milliliters of reagent per gram of material is approximately 35. Consequently, the relative precision of the slope can be espressed as: Standard deviation of slope Slope
-
1.83 X 0.03 35
which is about 1part in 500. Since GR-S contains approsiniately 5% of fatty acid, the precision of the slope will be of the order of 0.01% fatty acid. This was considered adequate for the purpose of this investigation. Precision of the correction for the constant-type error is caald a t e d as follows: The standard deviation of the difference between the experimental blank and the extrapolated intervept b ia given by U ( b - blank1
=
\’UL?
$-
U&,,L
In the present case ubl =
On the other hand,
(O.-175)?u?= 0 . 2 2 6 ~ ~
the variance of the average of the blank titrations, depends on the precision of a single blank titration and on the number of titrations made Assuming that the precision of a single blank titration is the same as that of a sample titration and letting n be the number of blank titrations
ug
uElank,
It is desirable to make the two components of u(b-blank)-i.e., and &lsnR-of the same order of magnitude. I n the present
caJe, this leads to U2
~
&
~
-~ 11
k
0.226 U’
which is satisfied approximately for n = 4. Therefore, it was decided to make four blank titrations for each series, giving U(b-l,lank)
= 40.226 U2
f 0.25 U z
= 0.69 u
Incidentally, the use of sets of four blank titrations allows for an evaluation of the precision of blank titrations and its comparison with the precision of sample titrations. The quantity ( b - blank) represents the constant type of error remaining after correction by means of the blank titration; it is therefore a correction to the blank. Since, in actual practice, the blank is used as a correction for a single sample titration having a standard deviation u, it seems reasonable to take this second-order correction into account only if its magnitude is appreciable with regard to u. In the present plan this secondorder correction is determined with a precision measured by 0 7 u. .2s a result of experimental fluctuations, the observed discrepancy between the blank and intercept will in general differ from zero, even if no real difference exists. Adoption of a 5 % level of significance ensures that such chance effects will not be mistaken for real differences more than 5 times in a 100. On the other hand, should a real difference exist, then the chance of its detection increases with its magnitude. I n this case, a real difference of 1.5 u has a chance of being detected of approximately 50%, and a real difference of 3 u has a cahance of being detected of approximately 95%. Thus, a real difference would be suspected if the estimated difference is several times the value of U, and if of importance additional experiments would be required to evaluate the discrepancy more precisely. I n general the degree of assurance for detecting true differences (“power” of a test) depends on the precision of measurement and the number of measurements (5). The number and spacing of the tests and the number of blank determinations used in the additional ex-
1106
ANALYTICAL CHEMISTRY
periments would therefore be planned in such a manner as to yield sufficient power in the detection and estimation of important discrepancies. Predetermining the precision of the slope and intercrpt in this manner as a function of the precision of a single determination and on the basis of a tentatively adopted plan for the e-iperiment, permits the selection of that plan which gives adequate degrees of assurance that the unknown effects will be detected and, if necessary, estimated with a preassigned precision. I n this case it also allows for a similar detection of the effect of the different stabilizers and of the presence of different quarititiep of soap on these t n o types of segregated errors. It is possible that the experimental points instead of scattering a t random about a straight line show evidence of curvature. Chemically, this means that the volume of titrant per gram of material depends on thr amount of material initially present in the solution. The presence of curvature if appreciable n ould constitute a serious limitation t o the usefulness of the analytiral procedure. When it is possible t o make the duplicate determinations on exactly the same quantity of added material, the inc:ease in the standard deviation attributable t o curvature may be e-timated by an analysis of variance ( 6 , 8).
Table V. Accuracy of Fatty Acid Titration
Source of Value Standard procedure Slope m from regression study
0
(Relative-type errors) Slope m (for Comparison u i t h Eitandard Procedureia. MI ‘G Soap __ Stabilizer Piesent, C G A B
,
34.61 34.61 None 34.61 ....... 0.0054 34.71 10 . 1 5 ...... 0.0260 , . ..,, 35.04 i0 . 2 2 ....... 0.0270 34.60 1 0 . 1 0 ...... 34.88 i O . O g 5 0.0540 34.50 1 0 . 1 2 ...... .......
Values following 1 sign are standard errors of measuremcnt.
Table VI.
Source of Value Rubber-solvent blank Solvent blank Intercept b from regression study 5
Accuracy of Fatty Acid Titration
Soap Present, G. h-one None 0.0054 0.0250 0.0270 0.0540
(Constant-type errors) Intercept b (for Comparison vitli Blank Titration)a, MI Stabilizer A B C 0.096 1 0 . 0 0 7 0.086 10.002
0.078 i0.021
.
,,,,
..
0 . 0 8 6 f0 . 0 1 1 0.088 =tO.O16
0.082 1 0 . 0 0 2 0.092 1 0 . 0 1 0 0 . 0 7 0 + 0,001 0 . 1 0 2 i0 . 0 0 2
....... ...,, .. .......
0.070 10.023
., .... . . . . . .. . 0,097
+ 0.010
.. . . .. .
Values following i sign are standard errors of measureinent.
Tables V and VI list the values of the slopes and intercepts and their standard errors for each of the five series of titrations calculated from the least squares theory for linear regression ( 4 , 16). The pertinent calculations used in this method are outlined in the section on statistical procedures. I n the case of the three series made in the presence of stabilizer A there would be little reason t o expect a variation in precision. Indeed, Bartlett’s test (2, 4, 13, 16) for the homogeneity of variances showed no significant differences among these variances. They were therefore pooled as indicated in the section on statistical procedures and the standard errors of the slopes and intercepts were calculated from the pooled value using the relationships given in the section on statistical procedures. For purposes of comparison there are given the values of the slopes denoted “standard.” These values were calculated from the normality of the reagent and the equivalent weight of the acid, as determined by the ASTM procedure. Mean values obtained for the blank titration of the rubber solution and the solvent alone are also included. COMPARISON OF STATISTICAL AND CUSTOMARY APPROACHES. Application of the statistical method t o series 2 and 5 leads t o
the observations that follow. For series 2 (stabilizer .4),the slope found by the regression method, 34.60, is in excellent agreement with the standard value. When judged on the basis of the standard error, 0.10, this agreement indicates the absence of a relative-type error, a t least within the precision of the data. Similarly, there is very satiPfactory agreement between both blanks (solvent and rubber-solvent) and the intercept of the fitted regrrspion line. On the other hand, the slope for series 5 (stabilizer C ) is higher than the standard value by an amount ml. of reagent to 0’2T grams of fatty acid‘ The intercept for this series is in agreement with both blanks. I n the present case these conclusions are in good agreement with those obtained by the customary approach. However, the objections t o the customary approach are overcome in the statistical approach as follows: the standard deviation of intercept b allows one to judge whether the blank titration is an adequate correction for the constant type of error; similarly, comparison of slope rri with the standard value is facilitated by the standard deviation of the slope; and simultaneous segregation of the two types of errors circumvents the necessity of using trial and error methods. I n addition to these advantages, the statistical method evaluates the standard error of estimate u which in the absence of curvature is a measure of the precision of the method. hlthough curvature may be deterted either visually or by its tendrncy t o increase u, its presence if most efficiently found by the method discussed later. EFFECTOF STABILIZER.Examination of Table V indicates a tendency for the slope to be higher for the more opaque stabilizers, B and C, than for the lighter stabilizer, A. The value for stabilizer B, of medium opacity, is unexpectedly high. This is probably due to the poor reproducibility in this series as reflected by the high standard deviation of the slope given in Table V. T o determine conclusively the relationship between the value of the slope and the visual opacity of the solution aould require additional experimental work to ascertain the values with greater precision according to the method outlined previously. It would also be helpful, in such a study, to nullify the effect of any slight consistent variation attributable to making the series of tests for diffelent stabilizers on different days (see the discussion of precision). This may be accomplished by randomizing the determinations among days as well as the other factors mentioned previously. Such an approach might tend to increase slightly the value of u and require a modification of the number and spacing of the quantities of added material for the comparison of the slopes and the detection of curvature either visually or by its effect on u. It may also introduce biaq into the comparison of the intercept with the experimental blank which may change from day t o day A more satisfactorj approach involves the use of incomplete block designs described by Youden ( 5 , 16). This design would also require a modification of the number and spacing of the quantities of added material to yield the required power (see the discussion of the analysis of data) in making statistical tests of significance in the comparison of the slopes. The analysis of the resulting data would be somewhat more involved than in the present case, but it would ehminate the effects of day-to-day variability from the comparison of stabilizers. Examination of Table VI shows no effect of stabilizer on the value of the intercept. In each case the solvent blank is in excellent agreement with the rubber-solvent blank and the intercept. Inspection of the first three EFFECTOF SOAP(CURVATCRE). series of data in Tables V and VI shows that the slope decreases with increaEing quantities of soap and that this decrease in slope is associated with a corresponding increase in intercept. Though h e present data do not establish conclusively the existence of these trends, their presence here suggests a method of ~~~
V O L U M E 26, NO. 7, J U L Y 1 9 5 4 studying curvature in relationships involving chemical reactions. I n the present case, the product of the titration of fatty acid with sodium hydroxide is soap. If the titration is performed on a larger quantity of fatty acid, the amount, of soap a t the end of the titration will be larger. Consequently, the effect of adding soap to the solution that is t o be titrated can be expected t,o be similar to that of titrating a larger quantity of fatty acid without addition of soap. From the observed trend of the slope with incrmsing amount,s of soap it could be espected that if any of the sets of data had been extended over a large range of fatty acid addcd, it would have shown a gradual decrease in slope-Le., curvature-as the amount of fatty acid added (and consequently the amount of soap produced a t the end of the titration) increased. In Figure 3 it is shown that the observed increasing trend in the intercept ip entirely consistent with this hypothesis. This figure illustrates a relationship hetween material found and addcd, which exhibits marked curvature. If the plot is assumed to cover a large range of values, the slopes of the tangents A , B, and C' approximate the values for 171 obtained when a series of andyses are made in the region very close t o D, E, and F , respectively. These decreasing slopes determined in the presence of inrreasing quantities of end product a , b, and c, give rise to incrrsasing values for the corresponding intercepts a',b', and c'.
1107
ence of stabilizer C and t o the trends in the slopes and intercepts obtained in the presence of different quantities of end product are not so obvious. Considering the curve for the change of p H during the titration of a weak acid with a strong base in an aqueous medium aa shown in Figure 4, it is reasonable t o assume that in the case of solutions in organic solvents the property measured by the indicator follows somewhat the same type of curve as this relationship for p H in aqueous media.
,
E.!.
I
I
IOC
PH=pKa f l o g
[Residual Acid]
I Q E
I
I
pH=pK,tlog
[Base]
I
I I
I I'I
IO
I 20
I 30
I
'40
I
I
I
I
50
60
70
80
I I I I I 1 I 90 100 I 1 0 120 130 I 4 0
PERCENT NEUTRALIZED
Figlire 4.
Chemical Interpretation
D
z 3
,
zc 1
4 cc w
G
B b'
a
MATERIAL ADDED
Figure 3.
Effect of End Product on Slope and Intercept
Again, a modification of the experimental plan to achieve a better comparison of the slope8 and intercepts would not only enable the esktence of the trends in the present study t o be estahlished with certainty but would afford a quantitative estimate of them as well, From these rstimates the degree of curvature and its effect on the analytiral results could be inferred. In conclusion, it is seen that from the over-all standpoint, the results of the test are probably accurate within 1% relative, equivalent t o ahout 0.05% fatty acid in normal GR-S which contains about 5 % fatty acid. The use of the solvent blank in place of the rubber-solvent blank will introduce an error no greater than 0.01 ml. of reagent, which is equivalent to about 0.01% of fatty acid. The accuracy of the method is satisfactory from a practical st andpoint. Chemical Interpretation. The fact that relative and constaut types of errors as well a8 combinations of these may be present in this procedure indicat'es that perhaps three chemical factors w e involved. The first is obviously t.he quantity of reagent required to bring the solvent to the point a t which the indicator changes color. This is the constant type of error that is adequately correct,ed for by the solvent blank. However, the other chemical factors leading t o the relative type of error in the pres-
For the titration of a given acid the pII to the left of the equivalence point depends on the ratio of concentration of salt to residual acid as shown in this section of the figure. At the equivalence point the relat,ion t o the right of the broken line begins t o take effert, hut only gradually If the equivalence point occurs at a p H of 9.8 as shown and the color change of the indicator ocrurs at a p H of, say 8.0, only 90% of the original quant,ity of acid present xi11 have been titrated a t the end point. This leads to a relative t!-pe of error, the abPolute value of which depends on the quantity of acid titrated. Thus, the use of an unsatisfactory indicator changing a little before or a little beyond the equivalence point will give rise t o a relative type of error. The reason for the relative type of error in the presenre of one stabilizer and not the other now becomes apparent. I n using an indicator in a dark solution the color change is observed a little later than in a light solution. This is the same a i using an indicator t h a t changes slightly beyond the equivalence point and consequently leads to a positive relative type of error. Refcrying once more t o the equation on the left-hand side of Figure 4 it is apparent, that the addition of the salt formed at t,he end of the reaction will tend to raise the p H of the equivalence point. If the p H a t which the indicator changes remains the same, the addition of salt formed a t the end of the titration (in this case soap) reduces the quantit.y of reagent required t o reach the end point below that required for the equivalence point. This is equivalent to a decrease in the value of the slope m. I n view of the previously discussed relationship between the addition of end product and curvature indicated in Figure 3, there should also he an associated increase in the intercept. Thus, the possible trends in the values of the slopes and intercepts obtained in the presence of different quantities of added soap, if proved t o be real, could be a result of this effect. This is of course a measure of buffer action, the third chemical factor. The introduction of salts other than soap would, of course, affect the reaction in a similar manner but only in so far as they affected the activity coefficients of the substances involved in the reaction. Inorganic salts present in rubber were eliminated from this study because they are insoluble in the medium used. Soluble organic
ANALYTICAL CHEMISTRY
1108
salts might have an effect. The apptoac-h einployed here might be useful in the quantitative study of other chemical equilibria. Aspects of Accuracy. Table VI1 presents a summary of the way in which the importance of factors related to accuracy depend 011 the objective t o be attained as suggested by this study. For applications involving large quantities of thc substance to be analvzed, the relative type of error will be of niajor importance and hence the precision of tlie estimated slope will therefore be the governing factor in calculating the number and spacing of the determinations to be made. Equation 1 in thc section on statistical procedures shows that the precision of the slope is greater if equal numbers of determinations are made at a point near the Y a\is and a t a point far out along the X axis than it is if tlic deterniinations are cquallj- spared along this axis.
T a b l e VIJ. Objective
.T : h l e ~11, IV, V, and V I ) are given here for illustration.
ANALYTICAL CHEMISTRY
1110 y, I ,G . / 2 5 0
All.
0.0200 0.0200 0,0500 0.0500 0.1500 0.1537 0.2500 0,2500 0.5000 0.5000 .v = 10 rx = 1 . 9 4 3 7 Z y = 28.086
ar1./1oo
hll. Aliquot 0.387 0.339 0.810 0.800 29185 2.260 3.570 3.590 7.095 7.050
Consequently
From these equations, si and S d are readily computed. Number of Determinations Required to Evaluate Precision. I n the situation described in the preceding section the value of s t is based on A’ pairs of duplicates. The precision of such an estimate is given as follows: u(sz) =
2 2 : = 0.67692369 Zzy = 9.632632
-
si)
hlultiplying by 2.5 to convert t o the basis of 250 nil. of solution and rounding off, gives a result of 34.88. b =
28.086
- (13.95246859)( 1.9437) 10
b
=
0.096658681 = 0.097
The intercept may be left on the basis of a 100-ml. aliquot since the blank titrations are also made on this quantit’y of rubber solution or solvent,. T o obtain s, the following table is constructed: X
0,0200 0.0200 0.0500 0.0500 0,1500 0.1537 0,2500 0.2500 0.5000 0.5000
yr =
Y 0.387 0.339 0.810 0.800 2.185 2.260 3.570 3.590 7.095 7.050
=
mz
+b
d = Z , - y c
0.3757 0.3757 0.7943 0.7943 2.1895 2.2412 3,5848 3.5848 7.0729 7.0729
4Zd2 10
-2
10,0113 -0,0367 +0.0157 + O . 0057 - 0.0045 + O . 0188 - 0.0148 +0.0052 +o. 0221 -0,0229
= 40.00338615
s = 0.0205735
8 =
0.020
Equivalent weight of acid = 275.1 Normality of alcoholic sodium hydroxide = 0. I050 Average solvent blank (ml.), 0.102 Average rubber-solvent blank (ml.), 0.092 Reliability of Slope and Intercept. The quantities 7th and b, when calculated on the basis of a limited number, say AV,of 5, 1~ pairs obtained experimentally, are subject to fluctuations, the magnitude of which decreases as the number of experimental values increases. Under the usual assumptions of least squares calculations the fluctuations in m and b can he expressed as functions of those in y-Le., as functions of s. These relations are as follows: in milliliters of reagents per gram of
(I)
It is seen from these equations that the precisions of m and b depend also on the quantity Z ( x - f ) ’ , which is related to the spacing of the z values chosen for the experiment. Substituting the values employed in studving the organic acid test: X - 0.02; 0.02; 0.05: 0.05; 0.15: 0.i5; 0.25; 0.25; 0.50; and 0.50; then S = 10, Z(z)’ = 0.6758, Z(s - Z ) 2 = 0.2994, and thus: sn = 1.835, in milliliters of alkali per gram of organic acid and Sb = 0.4755, in milliliters of alkali. Calculation of Components of Variability. If duplicate measurements are made on each of N days, the analysis of variance will yield “mean-square” values both for da to day and n-ithinday variations. These mean squares arehere ¬ed, respectively, by MSd and MS,. These values will vary somewhat from experiment to ex eriment because of the inter lay of chance effects. They will, {owever, fluctuate about fixex values a-hich can be shown ( 2 , 4, IS, 16) t o be approximately equal t o (s:
+ 2s;)
and s:, respectively
5s -
(approx. )
4%
If S is made equal to 8 this evpression becomes
-
1 . 9 4 3 7(zs.oss) 10 (1.943712 (0.67692369) - -~ 10 m = 13.95246859 (9.632632)
ni =
+
MSd = s: 2s: (approx.) and JIS, = s: (approx.)
Si
- 0.25 si
diG ~
Thus, while in such a situation the precision of si is rather poor (about 25y0 coefficient of variation) i t is nevertheless sufficient to locate definitely its order of magnitude, since the range of variability of a quantity is approximately equal to plus or minus twice its coefficient of variabilit,y. Moreover, the test for dayto-day variability is based on an F value ( 2 , 4, 5 , 13, 1 6 ) with 8 - 1 and 8 degrees of freedom, capable of detecting daily variations that would be sufficiently large to cause the over-all standard deviation, sr, to be appreciably larger than si. SUMMARY
Two types of systematic errors limiting the accuracy of a n analytical procedure have been considered and a statistical method has been presented for segregating these errors simult,aneously. This segregation becomes especially meaningful for equilibrium reactions in which the segregated errors appear to he related t o t,he chemical factors involved in the equilibrium. Yet, the method itself is particularly useful in studying analytical procedures for which no chemical theory is available. The intimate relation between the c0ncept.s of accuracy and precision has been shown and it appears possible, theoretically at least, t o ferret out many of the sources of error and correct for them as systematic errors, thus including them as phases of accuracy. As a matter of practical convenience, however, only t,he gross errors or fundamentally important errors may be so treated; the others must necessarily be considered as random errors related t o the precision of the test. LITERATURE CITED
(1) Am. SOC.Testing Xaterials, Philadelphia, “1949 Standards,”
Part 5, p. 500, Designation D 460-46, 1949. (2) Brownlee, K. A , , “Industrial Experimentation,” 4th ed., Brook-
lyn, N.Y., Chemical Publishing Co., 1952. (3) Daniel, C., and Heerema, S . , a. Am. Stat. Assoc., 45, 546 (1950). (4) Davies, 0. L., “Statistical Methods in Research and Production,” London, Oliver and Boyd, 1947. (5) Dixon, W. J., and Massey, F. J., “Introduction to Statistical .4nalysis,” New York, McGraw-Hill Book Co., 1951. (6) Eisenhart, Churchill, Photogrammetric Eng., 18,542 (1952). (7) Linnig, F. J.,Peterson, J. lI.,Edwards, D. AI., and dcherman, w.L., A4NAL. CHEM., 25, 1511 (1953). (8) Ifandel, J., J . Chem. Educ., 26, 534 (1949). (9) Mandel, J., and Stiehler, K. D., J . Research SatZ. Bur. Standards, to be published. (10) Reichel, E., 2 . anal. Chem., 109,385 (1937). (11) Reita, L. K., O’Brien, A. S., and Davis, T. L., A 4 s . CHEM., ~ ~ . 22, 1470 (1950). (12) Shewhart, W. h.,“Statistical Method from the Viewpoint of Quality Control,” Kashington, D. C.. Department of -4griculture, 1939. (13) Snedecor, G. W., “Statistical Methods,” 4th ed., Ames, Ia., Collegiate Press, Inc., 1946. (14) l\‘ernimont, Grant, L4N.4L. C H E M . , 23, 1572 (1951). (15) Youden, Vi-.J., Ibid., 19,946 (1947). (16) Youden, W. J., “Statistical hIethods for Chemists,” Sew York, John Wiley &- Sons, 1951. RECEIVED for review October 2 5 , 1953. hecepted May 5 , 1951. Presented before the Division of Analytical Chemistry a t the 122nd meeting of the AMERICAI CHElrcicAL SOCIETY,.4tlantic City, K. J., September 1952. Work performed as a part of the research project sponsored by the Reconstruction Finance Corp., Office of Synthetic Rubber in connection with the Government Synthetic Rubber Program.
