Lognormal-type distributions—III

Geoehimiea et Cosmochimica Acta, 1057, Vol. 11, pp. 205 to 212. Perga~~nPress Ltd., London

LognomaMype distributions--III L. H. AHRENS Depart,ment of Chemistry, University of Cape Town (Receive&2 August 1956) Ab&&--Aspects of recent papers by MILLER and GOLDBERG, and by AWREY, am discussed, and further examples of lognormal-type distributions are given. There is nom’ a large body of evidence to show that lognormal-type distributions are very common indeed. Attention is drawn to extreme positive skewness in a histogram of grain-size distribution of pyroxene from the Merensky Reef, Bushveld Igneous Complex.

Two recent papers (MILLER and GOLDBERG, 1955; AUBREY, 1956) discuss frequency distributions of elements in specific geological materials. Both criticize the general conclusion of lognormality arrived at by AHRENS (1954a) in his first but neither discusses the second (AHRENS, paper on lognormal distributions, 1954b)-apart from reference-which was written partly in a response to a criticism by CHAYES (1954) so as to set out more clearly the meaning and purpose The second paper also gave further examples of the so-called lognormal “law.” of distributions approaching lognormality and discussed other aspects of the The present paper is written partly application of statistics to geochemistry. as a reply to some of the criticisms in the papers of MILLER and GOLDBERG, and AUBREY, and partly to provide further examples of lognormal-type distributions. In the discussions which follow we consider first aspects of EDWARDS and GOLDBERG’S paper and, after examining some new examples of frequency distribution discuss aspects of AUBREY’S paper. D1scuss10w

(1) Aspects of the Puper by &filler and ~~l~ber~ and GOLDBERG tested the analytical data used by AHRENS (1954a) and also some on meteorites and sediments. Concerning AHRENS’ general observation about lognormality, they conclude that, though some of the observations could be lognormal, others could not (their Table 1). At the same time (p. -ES), but unfortunately as a footnote only, they make an important statement: “In fact, it should be made clear that great difficulties are involved in showing that any set of empirical data is precisely described by a specific function. Thus the search for specific functions to describe universes in nature seems fruitless. On the other hand, ifauniverse is approximately described byapartitular function, statements of prediction and comparison can be made. Herein lies the usefulness of such investigations!~’ The italics and exclamation mark are MILLER and GOLDBERG’S and give correct emphasis; their statement may be compared with that given by AH[RESS (1954b, MILLER

205

L. H.

AHKENS

after &IILLEK alid (kbLT)BERC Ilad Writtell 124.-I 25) IjIlt wbicll their paper: “We seek to GIKI a distribution which fits the various sets of observat,ion (several c~lenlt~nts) reasonably well and which may therefore be used for the geochemical purpose of comparison and prediction. The lognormal dist,ribution fits all sets of observations examined thus far reasonably well. and this conclusion has been expressed in the form of a geochemical law: but ill so doing, it is not meant to imply t,hat exact fit is to be expected in any particular case. that some other distribution might sometimes show slightly better fit, or that on very rare occasions a really serious exception will be found.” \Vith very few slight, exceptions, all observations considered so far (AHRENS. 1!)~,4a and b), together with those to be considered, are approximately described and accordingly, for the purpose of comparison by a lognormal distribution, and prediction, overall lognormality may be assumed. The same certainly could not be said of an overall assumption of normality, even if we stretch the t’erm “approximate”. We may now consider further d&a on the distribution of the concentration The fresh data will, where possible. of elements in specific geological materials. be presented in the form of frequency-distribution diagrams. As before (AHRENS. 1!)~4a, b), their shape will be used as the main guide as to whether t,he observations approximate closely or not to either a normal or a lognormal distribution. A more rigorous test of whether t,he observations conform precisely or not to one of these distributions will not, be attempted, for reasons given before (AHRENS. 1954b): see also the first’ t’wo sentences in MILLER and GOLDBERG’S statement above.

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(2) E’wther Examples of lhsfritmtions

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( ~OL~WHZIID’~ (1954, pp. 45i-4%) refers to the frequency distribution of pllosphorous ill different igneous rocks: basalt,. andesite. rhyolite, gabbro. diorite. syelrite, and granite. The number of observations ranges from S3 for rhyolite to 342 in basalt. All hist,ograms are reported as very asymmetric and positively skc~cd, and accordingly the arithmetic mean (abundance) in each of the seven rock types is usually found t,o be far greater than the mode, the most frequent c~oncc~ntration. The rat,io of arithmetic mean to mode in six of the seven types is from about, ~2-3; in one (granite) it is 1.2. (For discussion of the relationship between the ratio arithmetic mean/geometric mean and dispersion in lognormal tlistribubions. see AHREKS, 1954a, b). The original data are unfortunately not a\-ailable for detailed examinat,ion, but, provided analytical error (see below) is not) excessive, they lead to the firm conclusion of +ve skewness in each example, and distributions presumably approach lognormalitjy. 111 the ~MRU~IRN et ab. study on the accuracy of conventional methods of chemical analysis. t’he relative deviation for phosphorus is given as 39% and 33 0 (8 respectiyel,v for granite (:-I and diabase iv-1 (FAIRBNRS. 1%2). This is

Lognormal-typedistribut,ions-111 qaite an appreciable error and may in fact amount Do a third or so-occasionally possibly more-of the spread reported by GOLDSCHMIDT(see his Table LXXIV), where relative deviations usually lie between 70% and 105%. Nevertheless, as the observed dispersion is quite considerably greater than what seems to be a fair estimate of analytical error, we may assume that t,he histograms do reflect reasonably well the actual distribution of phosphorus in these igneous-rock types. It is of interest to note that GOLDSCIIM~DT(1954, p. 458) raises the question as to which quant,ity should be used, arithmetic mean or median, when comparing t,he phosphorus content of various rock types (see also AHRENS, 1954c). On the same page it is also stated that the high phosphorus values in the asymmetric this is not so, as these high values histogram may be regarded as anomalies; are found in almost every example of large dispersion examined so far, and are merely confirming evidence that distribution is not approximately normal but of a positively skewed lognormal or related type. (b) lMolyb&nurn in basalt + diabase, and in yabbros and ubtramafics The distribution of MO in granite (KURODA and SANDELL, 1954) has been discussed (AHRENS, 1954a) and is considered a good example of a lognormal-type distribution (see Fig. 16 of that paper). Here we consider the distribution of MO in other igneous rocks and again use the data of KURODA and SANDELL. Eightyfour specimens of basalt -+ diabase have been analysed, and Fig.l(a) shows a distrib~ltion curve similar to that shown by KURODA and SANDELL. Extreme positive skewness is again evident. The distribution of MO becomes fairly symmetrical and bell-shaped when a geometric interval is used (Fig. 2a). A fitted curve has been drawn. We note that the distribution is not perfect, and parGcularly a deficiency in the O-Z-0*4-p.p.m. interval and an excess iu the 0*4O.S-p.p.m. interval is apparent. Such irregularity is not, regarded as serious when taking int,o consideration the problem of sampling and the possibility of significant analytical error (values given by KURODA and SAXDELL are to within 0.1 p.p.m. MO, which at, say, #+1--0.4 p.p.m. is a large tolerance); fit is quit,e good and, for the geochemical purposes outlined before, a lognormal distribution may be assumed. A much smaller number of observations is available for gabbros and ultramafics, but the data are quite sufficient to indicate that distribution is markedly positively skew in both rock types (see Fig. 1 of KVRODA and SANDELL).

()?;ISHI and SAXDELL (1955) have determined As in a large number of igneous rocks. I~~speetioi~ of their frequency-distribution diagrams (see their Fig. I) shows that decided positive skewness is present in each type except volcanic glass -t felsite and rhyolitc, which were considered as one group. Ry far the largest number of obser~~ations were made on granite and basalt + diabase and the distribution of As in these two types will be considered further. Part of their Fig. I (granitic rocks and basalts + diabase) is redrawn here as Figs. l(b) and l(c). together with two histograms using a geometric interval (Figs. 2b and 2~). In each of t,hese two examples, extreme positive skewness has been strongly 207

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Lognormal-type

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A slight posit’ive skewness remains, reduced by the use of a geometric interval. in As in granite, a comparison of the fitted curve with the observed however; distribution shows this tendency as very slight indeed, but it is somewhat greater in the distribution curve of As in basalt + diabase. In this example a fitted curve has not been drawn. Although the one example in particular shows some distinct departure from lognormality, prediction and comparison of dispersion on the basis of assumed lognormality would not be seriously in error. Gross distortion would, however, result if normality were to be assumed, leading in these two examples, as in many others, to predictions of negative concentrations of elements in granite and in basalt + diabase. ONISHI and SANDELL (1955) have discussed the occasional strong enrichment of arsenic in igneous rocks. They draw attention to the presence of secondary sulphide,* as in the specimen from Cook County (their Table 5), and suggest secondary mineralization as a cause for high As enrichment. This may be the reason for the presence of some skewness in Figs. 2 and 3 and illustrates a typical difficulty which is encountered when examining the distribution of an element in a specific geological material. (d) Beryllium

and boron in Japanese granites

HAMAGUCHI, KURODA, and GORAI (1955) have determined Be in forty-six Japanese rocks which they describe as intrusive-type granitic rocks. Their histogram, copied here as Fig. l(d) shows decided positive skewness; t,his may, be compared with Fig. 2(d), in which a geometric interval has been used. The curve as drawn fits the observed distribution reasonably well and, for the purpose of predict’ion and comparison, a lognormal distribution may be assumed. The distribution of boron in Japanese intrusive granitic rocks is also positively skewed (see HAMAGUCHI, KURO~A, and GORIA, 1955). The boron concentration is, however. below the detection limit in eight specimens, and this example will not be examined in detail. (e)

Fllborine in Japanese igneous rocks

Olle example of a positively skewed lognormal-type distribution given in the first paper (AHRENS, 1954a) was that of fluorine in granite (Canadian and New England). KOKUBU (1956) has determined fluorine in Japanese igneous rocks and draws attention to the asymmetry (positively skewed) of the distribution of fluorine in ninety-nine rocks grouped under ‘(volcanic rocks” (his Fig. 1). The arithmetic mean is almost twice the most frequent concentration. (3) Aspects of the Paper by Aubrey AUBREY (1956) raises the question of a negatively skewed distribution. He shows theoretically, considering first a two-component system and then a system with more components, that if the distribution of concentration of one or more of t,he major components is positively skewed, the distribution of the others will be negatively skewed. Further, that there will be minor elements associated ~

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wit,11 the negatively skewed major elements and hence departure from a positively skewed lognormal-type distribution will not be rare. Distinct evidence of this interesting predicted possibility has yet t’o be found. Slight tendencies of negative skewness have been pointed out by AUBRET (I 956) in the distribution curves of SC (d), K (gr), Rb (gr), and Co (d) given by AHRENS (1954a). Dispersion in these examples is very small and, unfortunately therefore,

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Fig. 3. Frequency distribution diagrams of grain size of orthopyroxenes from three localities in the Merensky Reef (Bushveld Igneous Complex). After SCHMIDT (1952).

analytical error makes a fairly significant contribution to the total spread of (This possibility was not discussed earlier (AHRESS 1954a and b, observations. hecause main emphasis was on the elements with moderate to large dispersion: for t2hose with small dispersion, it ma,tters little as a rule, for t,he purpose of comparison and predi&ion, whether distribution is assumed to be normal or logIlormal.) This does not, however. rule out the possibility that t!he distributions of SC. K, Rb, and Co are slightly negatively skewed, as pointed out by AUBREY, l)ut significant analytical error introduces a complicating factor; compare with t’he exa,mple of phosphorus above. If the theoretical considerations of AI-BREY do apply t,o natural environments. it is not clear why distinct examples of negative skewness have not yet been found. The possible examples of SC and C’o may provide a clue as to where negatix-ely skewed examples might be found. These t,wo elements, together wit.h Ni, are usually associated with Fe and Mg in ferromagnesian minerals, arid t,he dist)ribntion of Fe, Mg, arid Ni iu a selected suit,e of dia,bascs and related rocks mav be instruct’i\-cl. For further discllssiou ill this rf?Sp?Ct. see L~lyl~REY (r!k%i). ‘i’hc fresh data given in t’he present, paper, together with that given before, ljrovides a large body of evidence that positively skewed lognormal-t#ypc distributions are very common indeed, aud leads t,o the collclusiou t,hat, if examples of distinct ncgati\-e slzwness are found. their proportion is likely to be slnall. (4) Tote o’n Grain Nrrmduy

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Kelat,ecl to the general problem of the statistical nature of the dist’ribution of concentration of au element in specific igneous rocks aud minerals, is that of the statistical nature of crystal grain-size distribution of minerals iu igneous 211

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