SEPTEMBER 1990 ECN-C-90-043
NL91C0215
STATISTICAL EVALUATION OF AN INTERLABORATORY COMPARISON FOR THE DETERMINATION OF URANIUM BY POTENTIOMETRIC TITRATION
DJ. KETEMA RJ.S. HARRY W.L. ZIJP The Netherlands Energy Research Foundation ECN is the leading institute in the Netherlands for energy research. ECN carries out pure and applied research in the fields of nuclear energy, fossil fuels, renewable energy sources, environmental aspects of energy supply, computer science and the development and application of new materials. Energy studies are also a part of the research programme.
ECN employs more than 900 staff. Contracts c;re obtained from the government and Vom national a id foreign organizations and industries. ECN's research results are published in a number of report series, each series serving a different public from contractors to the international scientific world. The C-series is for contract reports that contain the results of contract research. The contractor's name can be found on page 2.
Netherlands Energy Research Foundation ECN Scientific and Technical Information Department P.O. Box 1 NL-1755 ZC Petten The Netherlands Telephone. +31 2246 4323 Fax : +31 2246 3483
This report is available on remittance of Dfl. 20 to: ECN. TWI Department. Petten, The Netherlands. Giro account (postal account) No. 3977703. Please quote the report number.
'':> Netherlands Energy Research Foundation ECN. Petten 1990 - 2 -
Project number: 0979 Contract number: CBNM/ST/85-7O Principal: CBNM Commission of the European Commission, Joint Research Centre, Geel Establishment, Central Bureau for Nuclear Measurements - 3 -
SUMMARY
Upon request of the ESARDA working group on "Low Enriched Uranium Conversion - and Fuel Fabrication Plants" an interlaboratory com parison was organized, to assess the precision and accuracy concer ning the determination of uranium by the potentiometric titration method. This report presents the results of a statistical evaluation on the data of the first phase of this exercise. _ u .
CONTENTS
LIST OF SYMBOLS
1. INTRODUCTION
2. ARRANGEMENT OF THE INTERLABORATORY TEST 2.1 Sample preparation and distribution 2.2 Analyses and submission of results
3. STATISTICAL APPROACH 3-1 Measures of variability 3-2 Mathematical model
4. DATA TREATMENT 4.1 Scanning of original data 4.2 Graphical search for suspected data 4.3 Stragglers and outliers 4.3.1 Cochran's maximum variance test 4.3.2 Dixon's outlier test 4.3.3 Results of outlier tests 4.4 Analysis of variance 4.4.1 Uniform-level approach on solution A 4.4.2 Uniform-level approach on solution B 4.4.3 Split-level approach 4.5 Test of Link-Wallace 4.6 Estimate of expected variances 4.7 Repeatability and reproducibility
4.8 Distribution of differences
5. COMPARISON WITH CERTIFIED VALUES
6. CONCLUSIONS
7- REFERENCES
TABLES FIGURES - > -
LIST OF SYMBOLS
F = variance ratio
F(n.,n_) = critical value for n. and n? degrees of freedom m = average value (of a statistical sample) n = sample size r = repeatability R = reproducibility s = sample standard deviation s1 = within-laboratory variance s* = between-laboratory variance v = coefficient of variation, equal to s/m a = level of significance of a statistical test p = mean value (of a population) o = repeatability standard deviation a. = between-laboratory standard deviation oR = reproducibility standard deviation = indication for the best estimate 1. INTRODUCTION
The ESARDA Working Group on "Low Enriched Uranium Conversion and Fuel Fa brication Plants" (ESARDA-LEU WG) organized an interlaboratory comparison exercise in order to assess the precision and accuracy of the determina tion of uranium by the potentiometric titration method. The special aim of this intercomparison is to give the laboratories the opportunity to inter- compare their results, based on a statistical evaluation. In this way the interlaboratory standard deviation and the individual laboratory devia tions can be assessed, and also their statistical significance.
The interlaboratory comparison exercise is executed in two phases. The first phase dealt with pure uranyl-nitrate solutions the second phase will deal with similar uranium solutions doped with impurities. The Commission of the European Communities, Joint Research Centre, Geel Establishment, Central Bureau for Nuclear Measurements (CBNM) has certi fied and distributed the samples for the first phase. The results have been reported by 14 of the 15 participating laboratories to CBNM. In coded form these results have been transmitted to ECN for the statistical evaluation.
This report deals with the statistical evaluation of the coded results of phase one of the interlaboratory comparison exercise. The whole exercise is a so-called precision experiment. Methods of measurement are character ized by their precision and their accuracy. Precision is used as a general term for the closeness of agreement between replicate results, by applying an experimental procedure under prescibed conditions. Accuracy is defined as the closeness of agreement between the results of a measurement and a reference value. The precision experiment is used as the test method to investigate the type of the variability of the data, and the difference between the results and the certified value(s). Are these differences sig nificant or do they fall in the category of unavoidable random uncertain ties?
The statistical treatment described in this report includes:
- calculation of the within-laboratory variance; - calculation of the between-laboratory variance; - conparison of laboratory means; - 8 -
- calculation of the repeatability; - calculation of the reproducibility; - comparison of the laboratory means with the certified value.
A similar statistical treatment has been made in 1983 for assessing the repeatability and reproducibility of gravimetric uranium determinations in U0-. pellets 111. - 9 -
2. ARRANGEMENT OF THE INTERLABORATORV TEST
2.1 Sample preparation and distribution
For the distribution of the samples to be analysed the CBNM prepared two •other solutions of uranyl-nitrate (called A and B), which are slightly different in concentration. Both solutions had a nominal uranium concen tration of about 7-50 mg.g solution. The uranium contents of both solutions have been certified by CBNM. The certified values have not been communicated to the participating laborato ries before the analyses had been performed and reported. Samples of the solutions A and B have been distributed in sealed glass ampoules, each containing about 60 ml. Each of the fifteen par icipating laboratories received simultaneously three ampoules of solution A and three ampoules of solution B.
2.2 Analyses and submission of results
The participating laboratories were requested to perform the analyses on three different days {here coded with sequence numbers I, II. and III). The first day of analysis should be within two weeks after receipt of the ampoules; the second and third day of analysis should be four and eight weeks after the first analysis. The analysis should be performed immedia tely after the opening of the corresponding ampoule.
For the comparison it was necessary to perform the uranium determinations always in pairs: on each of the three days of the analysis one ampoule of solution A, and one ampoule of solution B should be investigated. For each ampoule the analysis should be performed in duplo (not less or more than two results should be reported). Therefore two aliquots of about 20 g of solution (aliquoting by mass) shruld be taken from each ampoule. The titration method prescribed is the so-called Davies and Gray Method specified in the international standard ISO 7097 U(.
The participating laboratories were asked to give a detailed description of the applied titration method, and also the results of the calibrations performed with mentioning of the reference materials used. The results of the uranium determinations (expressed as mg.g solution) and the descrip tion of the method were sent by fourteen laboratories to CBNM. These re sults have been coded by CBNM, before they have been transmitted to ECN for the statistical treatment. - 10 -
3- STATISTICAL APPROACH
3.1 Measures of variability
Many reasons nay lead to the variability of data. For this experiment in which «any laboratories were involved, factors such as differences in operators, equipment, calibration, environment and sample preparation on different days «ay give a large contribution to the variability. In prac tical situations two numerical measures of variability are recommended to take into account differences in the circumstances of new measurements. These measures are known as repeatability and reproducibility. - Repeatabi 1 ity refers to the variability of results obtained at short intervals of time in one laboratory by one operator using the same apparatus. - Reproducibility refers to the variability of results obtained in different laboratories, which implies different operators, equipment and/or at different time(s).
3-2 Mathematical model
For the interpretation of the results obtained with this type of exercise the analysis of variance technique is a very powerful tool. It gives a good indication of the between-laboratory variance as well as the within- laboratory variance, and it investigates whether the average outcomes of the laboratories are equal, when the experimental uncertainties are taken into account. As mentioned earlier two 'mother solutions' with slightly differenc ura nium concentrations were used. This is known as a split-level experiment. At first, data for both solutions are treated separately as uniform-level experiment, and afterwards the split-level approach is considered. Looking at the approach of the exercise one deals here with a hierarchical (nested) design i.e. the levels of the factor "sample preparation" (day I .. Ill) are nested within the levels of the factor "laboratory", with two replications in each cell. When it is assumed that the laboratories participating were chosen at random out off all laboratories using this method of uranium determina tion, then the next model with random erfects is suitable : - 11 -
y. ., = u • a. • b_ . * e. ., ijk ï ij ijk
where y. .. the result of the k-th repeated measurement on ijk the j-th day for the i-th laboratory; the overall mean for the uranium concentration; a. the laboratory effect; b. the saaple preparation (day sequence) effect; the uncertainty in a Measurement. eijk
The index i denotes the laboratory (i=l..l'l). j the day of analysis by titration (j=I..III) and k the nuaber of replication (k=1..2). In a strict sense, an analysis of variance should only be applied under the following requirements:
- the samples are mutually independent; - the samples originate from a sub-population which follows the normal distribution of Laplace-Gauss; - the sub-populations have the same variance.
Therefore one has first to investigate whether these conditions are ful filled. If the conditions are satisfied or nearly satisfied, one may apply the analysis of variance.
In summary, the evaluation of the data set comprises the following steps:
- tabulation of data; - test on the homogeneity of the laboratory variances; - search on the raw data to trace and eliminate wild results; - analysis of variance; - determination of repeatability and reproducibility; - comparison with certified value; - conclusions. - 12 - k. DATA TREATMENT
4.1 Scanning of original data
In the tables 1 and 2 the raw data are presented for solution A and B, respectively. The supplementary information as supplied by the laboratories (and not included in this report) gave an indication that probably only laboratory 13 has incorporated ^e so-called buoyancy correction in its results. The correction is necessary to take into account the difference in the density of uranium reference materials and its magnitude is depending on the mass values for the analysis. For the determination of the certified uranium concentration of solution A and B this correction was also applied. For this reason the results coming from the other laboratories (except labo ratory 13) were also corrected for this effect (0.1:352). So the values stated in the tables 1 and 2 are all corrected for the buo yancy effect. To get a first impression, some characteristic data were calculated such as mean, standard deviation, variance and coefficient of variation. The average results of the measurements found by each laboratory are illustrated in the graphs of figure 1 and figure 2 for both solutions. A general inspection shows that the data contains several deviating values in each of the solutions.
Some values among the original results (for example those from laboratory 12) deviate so much from comparable entries in the same tables, that a special attention was given to the search of outlying data. k.2. Graphical search for suspected data
Numerical outlier tests were preceded by a first cursory graphical exami nation on the average data from both solutions in each series (I, II, and III). This was performed by means of a so-called Youden diagram |3| , |4|. in which the results of the measurements for two series (here A and B) constitute the co-ordinates of points in a plot. A pair of results yields a single point in the diagram, so there will be as many points as there are reporting laboratories. In the diagrams also horizontal and vertical lines denoting the median values are presented based on mainly all results in a series. - 13 -
In an ideal situation where only random uncertainties of the same size (i.e. belonging to the same population) occur, the points are expected to constitute an elliptic cluster around the point with sedian value co-ordi nates. Because the two solutions are nearly equal in uranius concentration, it is possible to have an indication of the precision of the data in the figures by means of a circle {centered on the intersection of the median lines). A circle of which the radius is about 2.5 to 3.0 times the standard devia tion gives the smallest region that could be expected when approximately a normal distribution applies. When outliers are present, a number of points will clearly lie outside such a circle. The graphical presentations show that at least one labora tory (coded 12) yields obvious deviating results (see figures 3a. 3b. and 3c). Suspected data from other laboratories such as 9 and 11 look also incompatible with other entries. So in these cases where suspected Hota seem to be present, outlier tests have been applied.
4.3 Stragglers and outliers
A calculation of the uranium concentration based on all results of meas urements on identical material done by different laboratories can be influenced strongly by suspicious high or low data. So in consequence of the experiences described in section k.2, a critical examination of the data is important in order to identify and treat out liers or other irregularities. Since it is necessary to handle deviating results with care, and since various outlier tests have different levels of discrimination, a combina tion of several tests is chosen. To facilitate the calculations and to handle conform the recommendations in the international standard ISO 5725 J5| the original data were trans formed to cell ranges and averages for each replication and each solution. In this context one should remember that the range refers to the differ ence between the two replicate measurements on the same day that the cells here refer to matrix positions, when the indications for the labora tories and for the days are used as entries for rows and columns. - Ik -
The data are listed for solution A and B in the tables 3 and 4, respect ively .
In general, one should distinguish between stragglers and statistical out liers . An observation is called a straggler when the test statistic for deviating observations lies between its 5% and 1% critical values. An observation is called a statistical outlier when the test statistic exceeds its 1% critical value. Attention must be paid that for nearly all the outlier tests the assumption is made that the population examined follows a normal distribution. One should realize that an outlier test does not give results clearly dif ferent from those of a test of normality.
*4.^.1 Cochran's •aximua variance test
Since experience shows that between the laboratories large differences exist in the within-laboratory variances (see tables 1 and 2), Cochran's maximum variance test seems very useful |6|. This test investigates the significance of the difference between one large variance value and the other remaining variance values in a set of data. In the case of 2 replicates, the cell ranges w.. can be used instead of the standard deviations. Conform to this rule Cochran's test statistic is given by th:> relation:
2 w r - max IV,2 1
where w = the highest cell range in the set; max w. • the cell ranges for each replication i.
The Cochran test considers only the highest value in a set of ranges and is therefore a one-sided outlier test. Small values of range may be strongly influenced by the degree of rounding of the original results and are for that reason not very reliable. - 15 -
Because of the present lack for recommendations . f statistical tests de signed for a simultaneous examination of the presence of several outliers, the procedure of repeated application of the single outlier test was applied. The results for the solutions A and B are listed in table 5- The ranges which give significant results are also marked in the tables 3a an^ **a. Two asterisks indicate an outlying value, while a single asterisk denotes a straggler.
ty.^.2 Dixon's outlier test
The next step in the analysing of the data set is the application of the Dixon test J7|. This test uses the following statistics :
test criterion
z(2) - z(l) z(p) - z(p-l) 8 ... 12 Q = or whichever is greater Z(p-l) - 2(1) z(p) - z(2)
z(3) - zU) z(p) - z(p-2) 13 or more whichever is greater Q _ . or z(p-2) - z(l) *ip) - z(3) where z(i), i = 1, 2 p denote values in the data set to be tested, when arranged in ascending order.
The data considered here are the averaged cell values. Although some caution must be exercised in drawing conclusions from the results of successive application, this test was repeatedly applied. The tables 6a and 6b present the results. Significant values are also indicated in the tables 3b and 4b.
ty.3.3 Results of outlier tests.
A summary of the most important results of the tests mentioned above is presented below (using laboratory and level codes): I
- 16 -
solution Cochran test Dixon test
outlier straggler outlier straggler
A 9II;9HI;12I;12II; 4lI;9I;9II;12I; 61:611 12III;6H 12II;12III
4iii;9ii;9iH;iii; 61 ^T;ÏHII;9I;9III; 411 B 11III;12III 11I;11III;12I; 12II;12III
Examination of the results in this summarizing list makes it clear that statistical outliers are present for the laboratories coded 9 and 12 at nearly all levels. This is considered as an indication that these laboratories give outlying results for all the measurements. Therefore it was concluded that it was justified to discard all the data coming from laboratories 9 and 12. After this deletion still some statistic outliers remain on the results of solution A and B at single levels for the laboratories coded 6, 4 and 11. Because of the repeated application of the tests and the fact that these outliers occur only at single levels it was not thought to be justified to discard these data at this stage of the analysis. All stragglers were kept in the further analysis in the knowledge that repetition of the tests makes decisions of removal questionable.
4.4 Analysis of variance
The analysis of variance method tests the null-hypothesis that the sta tistical samples originate from populations with the same mean value. In other words, it investigates the significance of differences between the laboratory outcomes. Unless otherwise stated, all further statistical tests are performed at a level of significance a * 0.05. - 17 -
4.4.1 Uniform-level approach on solution A
The analysis of variance (nested design) with random effects was per formed on both c'ata sets (solution A and B) separately. At first the two original data sets (A and B) with 84 observations each were analysed, while afterwards the analysis of variance was performed after removal of the outliers mentioned in section 4.3-3-
The results are presented below :
Analysis of variance for solution A (random effects; all data; N=84)
source of sum of degrees of sample variation squares freedom variance
-44 -4 ('10 ) (•10 )
between labs 5286.8551 14-1=13 406.6812 between days within labs 27H.4578 l4-(3-D=28 96.8378 residual 949.2O6? 84-14*3=42 " 22.6002 total 8947.5194 84-1=83 107-8014
—^—^———^——• • Analysis of variance for solution A (random effects ; laboratories 9 and 12 deleted ; N-72)
source of sum of degrees of sample variation squares freedom variance -4 -4 ('10 4) ('10 H)
between labs 44.5386 12-1=11 4.0490 between days within labs 6.7085 12«(3-1)=24 0.2795 residual 14.6665 72-12*3*36 0.4074 total 65.9136 72-1=71 0-9284
The large differences in the sample variances between the original and cleaned data sets confirm earlier conclusions concerning the presence of the outliers for the laboratories 9 and 12. In consequence of this a further consideration of the last data set was carried out. At first a test is performed on the null-hypothesis th3t thcie are no time effects. - 18 -
The test-statistic is the variance ratio F (Fisher-Snedecor distribution) one-sided for a=0.05-
0.2795-Iff** Because F = r = 0.686. which value is smaller than the critical 0.4074-10 value F [24,36] a 1.82. the null-hypothesis cannot be rejected, in other words: no significant results were found. If also for other reasons there is no need to assume the presence of such a time effect, a pooled variance estimate can be made by combining the be tween days and residual variances. The advantage of this approach is a more reliable estimate because mere degrees of freedom are involved. So a calculation of the pooled within-laboratory variance is allowed:
6.7085*10 + 14.6665*10 _u s* = = 0.3562-10 w 24 • 36 Now the sane kind of test on main effects (labs) was done also:
4.0490*10~'* F ^ = 11.4 0.3562-10
Because F [11,60] • 1.96, one has to conclude that the between-lab va riance and the within-lab variance for case A are significantly different. This means that one has to reject the null-hypothesis that the expected laboratory means are all equal. This implies that at least one laboratory mean value shows a difference from the other laboratory means which .is significant with respect to the observed variability. In a next section 4.5 we return to this subject. - 19 -
4.4.2 Unifor»-level approach on solution B The sume procedure was followed for the data set concerning solution B.
Analysis of variance for solution B (random effects ; all data; N=84)
source of SUB of degrees of sample variation squares freedom variance -4 -4 ('10 ) (•10 4) between labs 3915.7^88 14-1*13 301.2114 between days within labs 2339.7627 14*(3-D=28 83.5630 residual 2590.4781 84-i4«3=42 61.6781 total 8845.9896 84-1=83 106.5782
Analysis of variance for solu tion B (random effects; laboratories 9 and 12 deleted ; N=72) source of sum of degrees of sample variation squares freedom variance -4 -4 (•10 H) (•10 4) between labs 132.35*»2 12-1=11 12.0322 between days within labs 32.5833 12-(3-1)=24 1.3576 residual 111.2097 72-12-3=36 3.0892 total 276.1472 72-1=71 3-8894
The F - test on the time effects in the series I, II, and III leads to the results below. 1.3576-10"11 F * r = 0.439t which value is smaller than the critical 3.0892'10 value F [24,36] - 1.82. The conclusion is that the null-hypothesis (presence of time effects) has to be rejected. In consequence of this it is concluded that no significant time effect is present. Calculation of the poo'ed within-laboratory variance from the between days and residual variances gives :
4 32.5833'10" • Hl.2097'10"'' h s' - 2.3965*10 w 24 • 36 - 20 -
A test on the presence of between-lab effects gives the following value for the test statistic:
12.0322*10 p = __ = 5.02I, which value is greater than the critical 2.3965*10 value F [11,60] = 1.96. So one arrives at the sane conclusions as for solution A: the labora tory mean values are not equal. At least one lab mean value is signifi cantly different from the other lab mean values. fr.fr.3 Split-level approach
In accordance with the approach for chosing the concentrations for this precision experiment, where instead of testing two separate mother solu tions with the same composition, ampoules were prepared with a slightly different uranium concentration, the so-called split-level analysis is suitable. Adoption of this method is useful to avoid influence on the results of other measurements based on experience gained for an earlier analysis while using the same material. Because the measurements on both solutions were repeated each day of analysis, the average values calculated from the first and second analysis of each solution are quantities of interest. So the difference between these average values were considered for each level (day) :
d = m ~ m A-B A B The results are as follows :
..... Analysis of variance for solution A and B (random eTects; laboratories 9 and 12 deleted ; N=72)
source of sum of degrees of sample variation squares freedom variance -4 -4 (•10 4) (•10 4) between labs 68.4480 12-1*11 6.2225 between days within labs 10.3290 12'(3-l)-24 'J. 4 304 residual 31.6738 72-12*3=36 0.8798 total no. 4508 72-1=71 1-5556 - 21 -
It is easily shown that earlier conclusions concerning time and main effects for solution A and B separately, are still valid. The pooled within-laboratory variance as result of rejecting the hypothesis about the existence of time effects is here:
_t 10.3290*10 ' • 31.6738*10"* k s' = = 0.7000*10 w 24 • 36 Also here the conclusion is valid that at least one laboratory mean value deviates significantly from the remaining mean values.
The next step {see section 4.5) can be a further investigation of the laboratory effect that still exists. In other words: one can investigate which laboratory or laboratories show mean results differing significantly from other laboratory means.
4.5 Test of Link-Wallace
As indicated in the former section 4.4, the analysis of variance technique does not answer the question which laboratory or laboratories cause signi ficant deviations. For this purpose the test of Link-Wallace can be applied |8|. The method is useful in a simultaneous investigation of the presence of significant deviations between all mean laboratory results. The requirements for the test are as follows:
- the statistical samples are independent; - the statistical samples originate from a sub-population which follows the normal distribution of Laplace-Gauss; - the sub-populations have the same variance; - the number of observations is equal for all statistical samples.
The test statistic is given by the relation :
Iw.(x)/n - 22 -
where w. (x) = the range between the highest and lowest value in the i-th statistical sample; w(x) = the range between the highest and lowest mean of the samples; n = the number of observations within the statistical samples.
In first instance the Link-Wallace test checks the null-hypothesis that all laboratory means are equal. As mentioned earlier, this situation is not applicable here. The test can also be used to investigate the single differences between the average results of the individual laboratories.
In this case the null-hypothesis reads :
P Q
where u. = the average value for the sub-population i (i=p,q).
Rejection of null hypothesis is necessary when the next relation is true :
Zw (x) D > K pq n
where D « the difference between two laboratory means. pq K * the test statistic.
In the present considerations the laboratories 9 a^cl 12 were excluded, because their results have been marked earlier as statistical outliers. The critical value for K for 12 samples, each with 6 observations, is:
K(6,12) - 0.39 for a =• 0.05 K(6.12) - 0.45 for a - 0.01
Table 7 shows the average results for each laboratory and both solutions as well as the sample ranges defined above. The differences between the average results of the laboratories are pre sented in the tables u« and 8b for the solutions A and B respectively. - 23 -
Calculation of the expression : Iw (x) K — n done For both solutions leads to allowed differences of 0.0095 and O.0165 for a=0.05 and to allowed differences of 0.0109 and 0.01Q0 for a=0.01. Looking at the differences summarized in the tables 8a and 8b the conclu sion is justifiable that for solution A the laboratories 4 and 6 give sig nificantly deviating means, and for solution B the laboratories 4 and 11.
*t.6 Estimate of expected variances
The two-way analysis of variance approach is reduced to an oneway type, caused by the absence of time effects as demonstrated by means of the F-test above (see l'A). Consequently a simplified survey, together with a further evaluation about the expected variances can be made. However, at first some definitions must be mentioned in relation to the simplified model :
y.. * u • a. • e.. Jik ^ 1 ik
The expected variance of a. is called the between-laboratory variance: 2 var(a.) -a. » between-laboratory variance (or shortly: interlab variance)
Within a single laboratory, the random uncertainties are assumed 2 to be approximately normally distributed, with a variance o (e ). This expected variance is the within-laboratory variance:
2 var(e. ) * a * within-laboratory variance
It is assumed that the value for var(e. ) will not vary much between labo ratories, so that it is justified to derive a common value for the within- laboratory variance valid for all laboratories. This coruaon value, which is an average of the variances over the laboratories, is called the re peatability variance
„„„/„I _ ,,-_ƒ_ \ _ „ *• - -„„.»»aki1i'|. > (i&rti Anr«A - 24 -
Now the next sumaary remains :
-4 ' Source of Sample variance (10 ) ! variation t
Unifor»-level Split-level] Expected solution solution both j variance j A B A and B i • ; i between labs 4.0490 12.0322 6.2225 \a * n*o. |
within labs 0.3562 2-3965 0.7000 ;ó 2 ! ! r 0.9284 3-8894 total 1-5556 ! where denotes the best estimate and n=6 (number of replications)
Elementary calculations lead to the following expected variances and stan dard deviations:
- 2 - 2 .. t solution 0 a r r °L °L \ -4 -4 2 (-10"2) i (•10 *) (*io~ ) (•10 H) i uniform A 0.3562 0.5969 0.6155 0.7845 : level B 2.3965 1.5^81 I.6059 1.2673 • split both 0.7000 0.8367 0.9204 0.9594 level A and B
4.7 Repeatability and reproducibility
The variance indicating the total spread between results of measurements, containing the within-lab variance as well as the between-lab variance, can be described as the sum of these two components :
2 2 2
tot w L or, with an equivalent equation :
2 2 2 °R *a r+ °L After determining these variances, and the corresponding standard devia tions one calculates "two sigma values" (corresponding to a 953! confidence level). - 25 -
The repeatability r and the reproducibility R are defined as follows (see |2|J: repeatability r - 2^2 * o = 2.83 * o r r reproducibility R = 2^2. • a = 2.83 " o„ M H
This leads to the following scheme:
Solution R ! - - i i Uniform A 1 0.0169 0.0279 level B | 0.0^38 0.0566 7-7^70 split both | 0.0237 0.0360 7-62G5 level A and B i
4.8 Distribution of differences
The evaluation so far is based on the analysis of variance technique. A possible drawback of the straight-forward application of conventional definitions for averages as well as for standard deviations used is that extreme data can influence the results of these calculations in a serious way. For this reason, and in particular concerning the interlaboratory standard deviation, a comparison with an alternative method of evaluation is of interest. The alternative approach for estimating the between laboratory variance makes use of the distribution of differences between the results of measurements. The term "DoD"-method is used as an abbreviation for this procedure |9|» The method has proved to be suitable to derive meaningful estimates of the standard deviations for data groups comprising outliers without necessity for identification and deletion of these outliers. As already indicated, the absolute differences |D| between all possible combinations of the individual observations representing repetitive analy tical determinations are used as basic elements. These differences are sorted as a function of their absolute value, after which their frequency is determined. By plotting the cumulative frequency Q of their occurrence versus their absolute values, an impression is obtained of the probability of observing a certain discrepancy. Finally, the so-called "DoD" curve is replaced by a smooth curve obtained by a manual fit or preferably by a least-squares fit - 26 - to the frequency curve using all differences. From such a graph a reason able estimate can be derived for the standard deviation of the underlying set of measurement data. Let Q denote the probability that the difference of two basic data is equal to or larger than a certain value |D|. under the assumption of a normal distribution the intersection of the Q=48/K line with the "DoD"-curve estimates the interlaboratory standard deviation (denoted as MsDoD"). The quantity MsDoD" is calculated for the data sets of solution A and B. For reasons of convenience the values are also calculated for the smaller data set with only average laboratory results for each solution. These results are shown in figures 4 and 5- To investigate the effect of the strongly outlying laboratory coded 12 (which had anomalous values in terms of size), on this method, the plots are shown for the two situations with and without this laboratory. The numerical value for S_, _ remains the same DoD for the two situations. The value? of s _. can be compared with values of s, _, which can be de- Don D tot rived from the variance values listed at the bottom of the first scheme in section 4.6.
The comparison of the values for Sn n and S is given in the following scheme:
I Quantity Situation Sol. A J Sol. B calculated
All data involved 0.0112 ' 0.0143 sDoD Average lab data 0.0135 1 0.0134 1 conventional Without outliers 0.0096 | 0.0197 stot
It is noteworthy that for solution B the conventional standard deviation s is appreciably higher than sD0_. This indicates that in case of solu tion B still extreme values are present in the data set. The results show that all standard deviations have more or less the same order of magnitude. - 21 -
5- COMPARISON WITH CERTIFIED VALUES
The certified values for the uranium concentration in both solutions were derived by CBNM Geel. In tables 9 and 10 a comparison is presented between the average values found by each laboratory and the certified values for both solutions (A and B). These tables show that the deviations for a lot of laboratory mean values with respect to the certified values are larger than the standard deviation stated for certification (0.0015. the value as specified by CBNM Geel).
As shown in the tables 9 and 10 the average absolute deviation from the certified value |d| as well as its standard deviation oi ,i were also cal culated with and without the data of the outlying laboratories 9 and 12.
Especially in the case of the clean data sets [dj = OI,I. This is an indication that most of the laboratory results deviate too much from the certified value for both solutions A and B. For calibration purposes of the titration method used, verification by comparison with an appropriate internationally recognized uranium refer ence material is necessary. Either pure uranium metal (for example EC-NRM 101 or NBS 960) or pure U^OQ (for example NBS 950 b), or sintered uranium dioxide of known purity may be used (EC-NRM 110). Fart of the calibration procedure is the dissolution of reference mate rial. Considering a possible dissolution effect we clustered the laboratories according to the type of reference material used (see the data on the next page). - 28 -
1 Material Remarks Mean Stot Solution inclusive outliers 7-4873 0.0195 A U metal exclusive outlier (lab 9) 7.4940 0.0038
inclusive outliers 7.5437 0.122 U oxide exclusive outliers (lab 12) 7.4941 0.0128
Solution inclusive outliers 7-7520 0.0124 B U metal exclusive outlier (lab 11) 7-7483 0.0060
inclusive outliers 7.7927 0.121 U oxide exclusive outlier (lab 12) 7-7389 0.0146
Based on these data the general idea arose that the use of uranium oxide as reference material causes rougly a two or three times higher standard deviation. Also the laboratories using an NBS 960 or an EC standard were investigated.
Standard Deviation from Mean Number of stot certified val. laboratories
Solution EC 110;101 +0.0026 7.4937 0.00410 3 A NBS 960 +0.0038 7.4949 0.00350 6
Solution EC 110;101 +0.0037 7J452 0.00328 3 B NBS 960 +O.O120 7.7535 0.0142 6 NBS 960 excl.lab.ll +0.0064 7J479 0.0042 5
The laboratories using the EC reference materials show for both solutions slightly smaller deviations from the certified values than the laborato ries using the NBS reference material. This difference is not significant enough to justify a conclusion. - 29 -
6. CONCLUSIONS
The original data set consisting of 84 observations originating frcm 14 laboratories comprises for nearly all determinations on solution A as well as solution B statistical outliers. It was found justified to discard all the data coming from laboratories 9 and 12. This conclusion, which is ba sed on the application of outlier tests (significance level Q=0.01) is^ also demonstrated in the summary of results given below. The overall variances for the complete data sets of each solution sepa rately are roughly 10 times larger than those of the clean data sets (without the results from the laboratories coded 9 and 12). The cleaned data sets for both solutions do not show significant devia tions from the normal distribution of Laplace-Gauss. Also it can be stated that concerning the day of analysis by titration. the existence of a time effect is not proved (F-test; one-sided a=0.05)- The numerical data derived by means of the statistical evaluation are sum marized below.
sol. A sol. B split-level 1 original 7-7666 m2 7-5114 data set 0.0108 0.0107 s 0.1038 0.1032 s 1.38 1.33 v (in •/.) cleaned 7.7470 7.6205 "2 7-Wl data set 0.000093 0.000389 0.000156 s (without 0,00964 0.01972 0.01247 labs 9 v (in •/•) 0,129 0.255 0.164 and 12) - 2 o 0.0000356 0.000240 0.0000700 r 5r 0.00597 0.0155 0.00837
vr(in •/.) 0.0796 0.1998 0.1098
-2 °L 0.0000615 0.000161 0.000092 \ 0.00785 0.0127 0.00959
vL(in •/.) 0.105 0.164 0.126
-2 °R 0.000097 0.000400 0.000162 0.00986 0.0200 0.0127 aR
vR(in •/.) 0.132 0.258 C.167 r 0.0169 0.0438 0.0237 R 0.0279 0.0566 0.0360 - 30 -
Looking over the survey above the following statements can be made :
The best estimate for the repeatability standard deviation c is 0.0060 {i.e. relatively 0.080'/t) for solution A. and 0.0155 (i.e. 0.20'/•) for solution B, and for the split-level approach 0.0084 (i.e. 0.110'/.).
The best estimate for the between-laboratory standard deviation a. is 0.0079 (i-e. relatively 0.105'/*) for solution A, 0.0127 {i.e. 0.164'/.) for solution B, and for the splitlevel approach 0.0096 (i.e. 0.126'/,)
The best estimate for the reproducibility standard deviation c_ is 0.0099 (i-e. relatively 0.132'/,) for solution A, 0.0200 (i.e. 0.258'/,) for solution B, and for the split-level approach 0.0127 (i.e. 0.167'/.).
It may be concluded that a relative standard deviation of about 0.25'/, describes very well the total reproducibility variability in the inter- laboratory comparison. An experienced laboratory may have a within-laboratory standard deviation as low as 0.08V» •
There are indications that the use of the uranium-oxide reference material causes roughly a two times higher standard deviation than the metal refer ence material. - 31 -
7. REFERENCES
|l| Verdingh, V., Zijp. W.L.;"Repeatability and reproducibility of gravimetric uranium determinations in U02 pellets". Commission of European Cosmunities , Nuclear science and technology. 1985, Com 4l82.
[2J International Organization for Standardization, International Standard "Determination of uranium in reactor fuel solutions and uranium product solutions - Iron(II) sulfate reduction/potassium dichromate oxidation titrimetric method". Document ISO 7097-1983(E).
131 Youden, W.J.."Graphical Diagnosis of Interlaboratory Test Results". Industrial Quality Control 15 (1959). no 11. p. 24. Reprinted in Journal of Quality Technology 4 (1972) no. 1.
|4| Youden, W.J. and Steiner, E.H. ."Statistical Manual of the AOAC" (Association of Official Analytical Chemists, Arlington, VA, 1975).
151 International Organization for Standardization. International Standard "Precision of test methods - Determination of repeatability and reproducibility". Document ISO 5725" 1981(E), First edition 1981-04-01.
|6| Pearson, E.S., and Hartley, H.O.,"Biometrika tables for statisticians", Cambridge University Press 1970.
|7| Dixon, W.J.;"Processing data for outliers". Biometrics 9 (1973), 74-89.
131 Sachs, L.;"Angewandte Statistik". FUnfte Auflage, (Springer Verlag. Berlin, 1978).
|9l Beyrich, W., Golly, W., Spannagel, G.;"The DoD method : an empirical approach to the treatment of measurement data comprising extreme values", Contribution to Stanchi, L. (editor), Proceedings third Annual Symposium on Safequards and Nuclear Material Management, Karlsruhe, May 1981. Report ESARDA-13 (CEC/JRC, Ispra, 198l). - 32 -
Table 1. Data for solution A
Laboratory Data 1 Lab. Standard Variance Coefficient code sequence (:od e Mean deviation 2 of s S-3 variation I I II III CIO"6) 1 (•10 3) v(X) ; 7.4901 7.5001 7-5021 1 7.4981 4.69 22.00 0.063 i 7.*»991 7.4951 7.5021 i 7.4881 7.4901 7.4911 2 7.4903 1.16 1-350 0.016 ! 7.49OI 7.4911 7.4911 1 7.4956 7.4936 7.4945 1 3 7.4948 0.955 0.912 0.013 7.4959 7.4953 7.4939 7.4662 7.4741 7.4691 4 7.4763 12 "2 154.2 0.166 7.4971 7.4662 7.4851 7.4960 7-4938 7.5001 5 7.4972 3.47 12.05 0.046 7.5028 7.4952 7.4953 7.5211 7-5211 7-5211 6 7.5118 10.5 111.1 0.140 7.5061 7.4981 7.5031 7.4931 7-4942 7.4910 7 7.4926 1-95 3.78 0.026 7.4943 7-4936 7.4395 7.4911 7.4891 7.4891 8 7.4906 1.97 3-89 0.026 7.4911 7.4941 7.4891 7.4115 7.4739 7.4757 9 7.4397 30.5 929.7 0.410 7.4377 7.4026 7.4370 7.5001 7.4911 7.4931 10 7.4949 3.43 11.75 0.046 7.4981 7.4941 7.4931 7.5021 7.4961 7.5051 11 7.4998 4.76 22.65 0.063 7.4921 7.5021 7.5011 7.6919 7.6919 8.2913 12 7.7918 268.1 71856.1 3.44 7.5920 7.5920 7.8917 7-4899 7.4926 7.4829 13 7.4892 4.32 18.65 O.058 7.4911 7-4936 7.4849 7.4801 7.4891 7.4991 14 7-4931 8.07 65.19 0.108 7.4921 7.4951 7.5031
OVÏ;ral l soluitio n A mean - 7.4999 7.5000 7.5345 7.5-L14 46.61 167.1 103.8 s2{#10-6) 47.13 2221.6 2172.1 27919.6 10780.1 v {%) 0.628 0.621 2.22 1.3!3 - 33 -
Table 2. Data for solution B. » .Laboratory Data Lab. Standard Variance Coefficient code sequence code He an deviation 2 of s I 1 s-3 variation II III CIO"6) v(Jt) (•10 5) 7.7449 7-7558 7-7598 1 7.7553 6.79 46.13 0.088 i 7.7648 7-7519 7-7548 I 7-7399 7-7409 7-7419 2 7-74G7 1.16 1-355 0.015 r 7-7389 7-7409 7-7419 7.7494 7-7485 7.7465 1 3 7-7432 1.51 2.29 0.020 t 7-7502 7-7481 7-7465 7.7099 7-7289 7-6989 4 7.7176 13.94 194.3 0.181 7.7169 7.7129 7-7379 7.7377 7-7421 7.7493 5 7-7417 6.41 41.08 0.083 7.7405 7-7324 7-7483 7-7359 7J489 7-7469 6 7-7489 8.71 75.84 0.112 7.7628 7.7519 7-7469 7-7469 7.7466 7.7434 7 7.7457 2.42 5-87 0.031 7-7487 7-7464 7.7422 7.7449 7.7459 7.7469 8 7.7452 I.36 1.85 0.018 j 7-7459 7.7429 7.7449 7.6937 7-7124 7.8807 9 7-7597 79-00 6241.6 1.02 7.6816 7.7686 7.8213 t 7.7^9 7-7419 7.7449 4.00 0.052 ! i0 7-7459 15.99 7.7479 7-7409 7-7509 7.7658 7.7608 7.8438 li 7.7815 49-4 2437-6 0.634 7.8428 7.7439 7.7319 7.7918 7.9916 7.8917 i 12 8.OO83 299.1 89487.4 3.74 7.7918 7.9916 8.5910 7.7491 7-7461 7.7443 !3 7.7458 2.32 5.41 0.030 r 7-7467 7.7422 7.7461 i 7.7479 7.7538 7.7509 1 14 7-7^74 6.94 48.16 0.090 7.7399 7-7379 7-7538
i overall soluti<:> n B ' mean , 7.7474 7.7613 7.7910 7-7666 29-79 66.10 162.7 103.2 ' s^-10 °) 1887-59 4368.8 26475.7 10657.8 : v (%) ! 0-385 O.852 2.09 1.33 - 34 -
Table 3a. Cell ranges (w. .) solution A.
Laboratory Sequence code (j) code (i) I II Ill
1 0.0090 0.0050 0.0000 2 0.0020 0.0010 0.0000 3 0.0003 0.0017 0.0006 4 0.0310 0.0080 0.0160 5 0.0068 0.0014 0.0048 6 0.0150 0.0230** 0.0180 7 0.0012 0.0006 0.0015 8 0.0000 0.0050 0.0000 9 0.0262 0.0713** 0.0387** 10 0.0020 0.0030 0.0000 11 0.0100 0.0060 0.0040 12 0.0999** 0.0999** 0.3996** 13 0.0012 0.0010 0.0020 14 0.0120 0.0060 0.0040 ! outlier (see section 4.3-1)
Table 3b. cell averages (y. .) solution A.
Laboratory Sequence code (j) code (i) I II III
1 7.4946 7.4976 7.5021 2 7.4891 7.4906 7.4911 3 7.49575 7.49445 7.4942 4 7.48165 7.47015** 7.4771 5 7.4994 7.4945 7.4977 6 7.5136* 7.5096* 7.5121 7 7.4937 7.4939 7.49025 8 7.4911 7.4916 7.4891 9 7.4246** 7.43825** 7.45635 10 7.4991 7.4926 7.4931 11 7.4971 7.4991 7.5031 12 7.64195** 7.64195** 8.0915** 13 7.4905 7.4931 7.4839 14 7-4861 7.4921 7.5OH
* straggler ** outlier (see section 4.3-2) - 35 -
Table 4a. Cell ranges (w. .) solution B.
Laboratory Sequence code (j) code (i) I II III
1 0.0199 0.0039 0.0050 2 0.0010 0.0000 0.0000 3 0.0008 0.0004 0.0000 4 0.0070 0.0160 0.0390** 5 0.0028 0.0097 0.0010 6 0.0269* 0.0030 0.0000 7 0.0018 0.0002 0.0012 8 0.0010 0.0030 0.0020 9 0.0121 0.0562** 0.0594** 10 0.0010 0.0010 0.0060 11 0.0770** 0.0169 0.1119** 12 0.0000 0.0000 0.6993** 13 0.0024 0.0039 0.0018 14 0.0080 0.0159 0.0029
* straggler ** outlier {see section 4.3-1)
Table 4b. Cell averages (y. .) solution B.
Laboratory Sequence code (j) code (i) I II III
1 7-75485 7-75385 7-7573 2 7.7394 7-7409 7-7419 3 7.7498 7-7483 7-7465 4 7.7?34** 7.7209* 7.7184*» 5 7.7391 7-73725 7-7488 6 7.74935 7-7504 7-7469 7 7.7478 7-7465 7-7428 8 7.7454 7.7444 7.7459 9 7-68765** 7-7405 7.8510** 10 7.7484 7-7414 7.7479 11 7.8043** 7.75235 7.78785** 12 7.7918** 7.9916** 8.24135** 13 7.7479 7J4415 7.7452 14 7.7439 7.74585 7.75235
* straggler ** outlier (see section 4.3-2) - 36 -
Table 5. Results test of Cochran on cell ranges. — , .—. Sequence Range Cochran *s Critical Probability Conclusion code tested criterion values level C IX 1 5* P {%)
Solution A
I 0.0999 0.816 0-599 0.492 P < 1 outlier I 0.0310 0.426 0.624 0.515 P > 5 II 0.0999 0.632 0.599 0.492 P < 1 outlier II 0.0713 0.874 0.624 0.515 P < 1 outlier II 0.0230 0.723 0.653 0.541 P < 1 outlier II 0.0080 0.317 0.684 0.570 P > 5 III 0.3996 0.987 0.599 0.492 P < 1 outlier III 0.0387 0.700 0.624 0.515 P < 1 outlier III 0.0180 O.505 0.653 0.5**1 P > 5
Solution B
I 0.0770 0.809 0.599 0.492 P < 1 outlier I 0.0269 0.517 0.624 0.515 5 > P > l straggler II 0.0562 0-771 0.599 0.492 P < 1 outlier II 0.0169 0.304 0.624 0.515 P > 5 - III 0.6993 0.965 0.599 0.492 P < 1 outlier III 0.1119 0.709 0.624 0.515 P < 1 outlier III 0.0594 0.688 0.653 0.541 P < 1 outlier III 0.0390 0.951 0.684 0.570 P < 1 outlier .. - 37 -
Table 6a. Results test of Dixon on cell averages for solution A.
Sequence Average , Dixon's test statistics Critical Probability Conclusion code tested values level 1% 5% P (%)
; (7.64195-7-WO 7-6M95 =0.915 0.586 0.670! p < 1 outlier (7-64195-7.4861) i
(7.4861-7-4246) I j 7-4246 =0.826 0.611 0.697 p < 1 outlier (7.4991-7- *»246)
(7.5136-7-^994) I ; 7.5136 =0.516 0.479 0.579 5 > P > l straggler (7.5136-7.4861) 1 i (7.64195-7.499D II 7-64195! =0.944 O.586 0.670 P < 1 outlier (7.64195-7-4906) 1 (7.4906-7.43825) II 7.43825 =0.882 O.611 0.697 outlier (7.4976-7-43825) P < 1 (7.4906-7.47015) II : 7.47015 =0.706 0.479 0.579 outlier (7.4991-7.47015) P < 1 (7-5096-7-4991) II 7 5096 =0.583 0.502 O.605 5 > P > 1 straggler (7.5096-7.4916)
(8.0915-7.503D III 8.0915 =0.968 0.586 0.670 outlier (8.0915-7-4839) (7.4839-7.45635) III 7.45635 • =0.589 0.611 0.697 P > 5 (7.5031-7.45635) - 38 -
Table 6b. Results test of Dixon on cell averages for solution B.
Sequence Average Dixon's test statistics Critical Probability Conclusion i code tested values level 1% 5% P (*> (7.739I-7-68765) 7.68765 =0.765 O.586 0.670 P < 1 outlier (7.7549-7.68765)
(7-8043-7-75485 7-8043 =0.762 0.611 0.697; P < 1 outlier (7.8043-7.739^) (7.7918-7.75485) I 7.7918 =0.701 0.479 0.579. P < 1 ! outlier (7.7918-7.739D (7.7391-7-7134) I 17.713** =0.706 0.502 0.605 P < 1 ! outlier (7.7498-7.7134) (7.75485-7-7498) 7.75485 =0.327 0.530 O.635 P > 5 (7-75485-7.7394)
(7.9916-7.75235) II 7.9916 s0.953 '0-586 O.67O P < 1 outlier (7-9916-7.7405)
(7.7405-7-7209) II 7-7209 -=0.664 JO.611 0.697 5 > P > l straggler (7.7504-7.7209)
(8.24135-7.78785) III 8.24135 •0.910 O.586 0.670 P < 1 outlier (8.24135-7-7428)
(7.8510-7.7573) III 7.8510 '0.866 0.611 0.697 P < 1 outlier (7.8510-7.7428)
(7-78785-7.7573) III 7-78785 '0.665 0.479 0.579; P < 1 outlier (7.78785-7.7419) (7.7419-7.7184) III 7.7184 'O.692 0.502 0.605: P < 1 outlier (7-75235-7.7184)
(7J573-7.75235 III 7.7573 =0.341 '0-530 0.635 P > 5 ' " (7.7573-7-7428) - 39 -
Table 7. Mean values and sample ranges between highest and lowest value in each sample.
Laboratory Soluti on f code ;k B Mean Mean w J - 1 7-4981 0.0120 7-7553 0.0199 j 2 7-4903 0.0030 7-7407 0.0030 j 3 7-4948 0.0023 7-7482 0.0037 4 7-4763 0.0309 7.7176 0.0280 ! 5 7.'•972 0.0090 7-7417 0.0169 i 6 7-5118 0.0230 7-7489 0.0269 1 7 7-4926 0.0048 7.7457 0.0065 J 8 7-4906 0-0050 7.7452 0.0040 ! 10 7.49*19 0.0090 7.7459 0.0100 | 11 7-^998 0-0130 7.7815 0.1119 | 13 7-4892 0.0107 ( 7.7458 0.0069 14 7-4931 0.0230 7.7474 0.0159 Talie 8a. Differences between average values of individual laboratories for solution .. laboratory 1 2 3 ^ 5 6 7 8 10 11 13 14 -4 code (•10 4) 1 - 78 33 218 9 137 55 75 32 17 89 50 2 - 45 140 69 215 23 3 46 95 n 28 3 - 185 24 170 22 42 l 50 56 17 4 - 209 355 163 143 186 235 129 168 5 - 146 46 66 23 26 80 41 6 - 192 212 169 120 226 187 7 - 20 23 72 34 5 8 - 43 92 14 25 10 - 49 57 18 11 - 106 67 13 - 39 14 -
Table 8b. Differences between average values of individual laboratories for solution B.
Laboratory 1 2 3 4 5 6 7 8 10 11 13 14 code (•10- 4 ) 1 - 146 71 377 136 64 96 101 94 262 95 79 2 - 75 231 10 82 50 45 52 408 51 67 3 - 306 65 7 25 30 23 333 24 8 4 - 241 313 281 276 283 639 282 298 5 - 72 40 35 42 398 41 57 6 - 32 37 30 326 31 15 7 - 5 2 358 1 17 8 - 7 363 6 22 10 - 356 1 15 11 - 357 3**l 13 - 16 14 - - 41 -
Table 9- Comparison with certified value for solution A.
Laboratory standard mean d. d. code (in mg U*g ) deviation from 1 code material certified val. » (7.4911) _j_ 5R (in mg U-g )
1 NBS 960 ü metal 7-4981 +0.0070 0.710 2 U02 7.4903 -0.0008 -O.O812 3 EC 110 U02 7-4948 +0.0037 0.375 4 NBL 97 U308 7.4763 -0.0148 -I.50 5 EC 110 U02 7.4972 +0.0061 0.619 6 NBS 950a U308 7.5118 +0.0207 2.10 7 NBS 960 U metal 7.4926 •0.0015 0.152 8 NBS 960 U metal 7.4906 -0.0005 -O.O507 9 CEA Fl. Anemone U metal 7-4397" -5-21 U metal -0.0514" 10 NBS 960 7.4949 +0.0038 0.385 11 U metal 7.4998 NBS 960 •0.0087 0.883 12 U308 7-7918" NBS 950b + 0.3007" 30-5 EC 101 U metal 7.4892 13 -O.OOI9 -0.193 14 U metal 0.203 NBS 960 7.4931 +0.0020
|d| all laboratories 0.03026 ML „ 0 ,8, 303 oiji all laboratories 0.07898 o(d| °-
|d| without labs 9 and 12 0.005958 ]dT m 6 u,yw oi ,i without labs 9 and 12 0.006196 o.dj
* 5R • 0.009857 ** outliers - 42 -
Table 10. Comparison with certified value for solution B.
Laboratory standard mean _1 d. d. code (in mg U*g ) deviation from 1 code material certified val. (7-7415) _! aR # (in mg U*g )
1 NBS 960 U metal 7.7553 +0.0138 0.690 2 U02 7.7407 -0.0008 -0.0400 3 EC 110 U02 7-7482 +0.0067 0.335 4 NBL 97 U308 7.7176 -O.O239 -1.20 5 EC 101 U metal 7.7417 +0.0002 0.010 6 NBS 950a U308 7-7489 •0.0074 0.370 7 NBS 960 U metal 7.7457 +0.0042 0.210 8 NBS 960 U metal 7.7452 +0.0037 0.185 9 CEA Fl. Anemone U metal 7.7597** +0.0182*» 0.910 10 NBS 960 U metal 7.7459 +0.0044 0.220 11 NBS 960 U metal 7.7815** +0.0400** 2.00 12 NBS 950b 1308 8.OO83** +0.2668** 13-3 13 EC 101 U metal 7-7458 +0.0043 0.215 14 NBS 960 U metal 7.7474 +0.0059 0.295
]d| all laboratories 0.02859 jdL . 0 kl2 u 41 oi .i all laboratories 0.06941 o.d. * ^
jdj without labs 9 and 12 0.009608 |d| . ft v oi,i without labs 9 and 12 0.001151 o,d| " '°^
oR = 0.0200 outliers SOLUTION A -t>-
7.5 2H
7.51 -
7.5?n;0n -
T
- .8- 9 ;3 li 3 10 12 Lab. n" Fig. I. Data for solution A (buoyancy corrected). SOLUTION B mcU. c -t- 7.78- 7.77- 7.76- 4> -4- IJ 7 10 U ;9 11 Lab. n* i Fig. 2. Data for solution B (buoyancy corrected) -e- - 45 - 7.8500 : u 12 7.7600 oe a 7.6700 "o « 7.5800 Series I 7.4900 7.4000 *— l *----»»--*« * M • 7.4000 7.4900 7.5800 7.6700 7.7600 7.8500 mg U-g-1 solution A 8.0100 r + 12 7.8880 a o s 7.7660 o 9 + 7.6440 Series II e 7.5220 7.4000 1.1* I.,...... !.. .„.,,.,«. 7.4000 7.5220 7.6440 7.7660 7.8880 8.0100 mg U.g-1 solution A Fig. 3o and 3b : Youden diagram solution A versus B. (average dota laboratories series I ond II) - 46 - 8.30OO 8.1200 en c o » 7.9400 o J* 7.7600 7.5800 7.4000 ^—"*• "• •—^^ 7.4000 7.5800 7.7600 7.9400 8.1200 8.3000 mg U.g-1 solution A Fig. 3c : Youden diogrom solution A versus B. (overage data laboratories series til) - 47 - 91 All laboratories 82 il A 73 ii A 03 to CO O 64 4) C O 91 55 c V «J 45 «O - 27 4> c *> - 18 6 u lab 12 sDoD=0.0H2 9 0 0.000 0.072 0.144 0.216 0.288 0 360 Absolute difference [D] (in mg U.g-1) II A CO to vG (3 4» V CL. 0.000 0.016 0.032 0.048 0.064 0.080 Absolute difference [D] (in mg U.g-1) Fig. 4 : Distribution of differences with fit using overage data for solution A. (48 % value indicates the estimated sDoD of the interlab standard deviation) - 48 - 91 82 it All laboratories A 73 it m A *> O! o 64 V c C w 18 E 3 b V sDoD=0.0U3 9 O. 0 0.000 0.060 0.120 0.180 0.24u 0.300 Absolute difference [D] (in mg U.g-t) 78 Lab 12 deleted 70 S" II A 62 ii a> A 4> O 55 - C o 47 g 4) Q=m 39 s •o 31 o o M 0 w 23 u e 16 E « tDoD=O.OI04 a> 8 z 0.000 0,014 0.028 0.042 0.056 0.070 Absolute difference [D] (in mg U.g-1) Fig. 5 *. Distribution of differences with fit using averoge dota for solution B. (48% volue indicotes the estimoted sDoD of the interlob standord deviation)