GEOLOGIJA 28 29, 255—269 (1985 86), Ljubljana
UDK 550.4.01:553 = 20
Theorems and laws defining the distribution of the geological
random variable
Teoreme i zakoni koji definišu raspodelu geološke slučajne promenljive
Veljko Omaljev
Geoinstitut, Rovinjska 12, 11000 Beograd
Abstract
This paper treats the theoretical basis of distribution of population of
chemical elements (atoms) in a geological body according to 6 theorems
and 3 Laws of distribution. The mathematical background of these theo-
rems and laws of distributions of geological random variables are well
beyond the framework of this paper; this material has been presented
in a paper at the 27*'< International Geological Congress, Moscow, on
August 6, 1984.
By the dual nature of a sample as a unit magnitude the existence of
two geological populations is introduced within a statistical sample that
represents the elementary population of the investigated chemical element
within a geological body. This finding represents an important discovery
that modifies the methodology of statistical investigations in geology
(geochemistry). This author has directed his efforts to further develop-
ment of this new methodology of statistical study of distribution of che-
mical elements in geological bodies.
Kratak izvod
U ovom radu je data teoretska osnova raspodele populacije hemijskih
elemenata (atoma) u geološkom telu pomoću 6 teorema i 3 zakona raspo-
dele. Matematička osnovica ovih teorema i zakona raspodele geološke
slučajne promenljive prevazilazi obim ovog rada, koji predstavlja refe-
rat izložen na 27. Međunarodnom geološkom kongresu u Moskvi dana
6. avgusta 1984. godine, te je zato izostavljena.
Dvojna priroda probe kao jedinične veličine uslovljava postojanje
dve geološke populacije izučavanog hemijskog elementa u geološkom
telu. Ovo saznanje predstavlja otkriće, a ono suštinski menja metodo-
logiju statističkih ispitivanja u geologiji (geohemiji). Napori autora su
usmereni na razradu ove nove metodologije statističkih izučavanja raspo-
dele hemijskih elemenata u geološkim telima.
256 Veljko Ornai]ev
From the geochemical point of view, atoms of chemical elements are the
basic building elements of geological bodies of the Earth's crust. The concept
of an atom is identical to the concept of a chemical element (Vernadskiy,
1983). The geological population of a chemical element in a geological body
is, in its essence, the population of atoms of a certain kind. This population is
infinitely large, continuous and uncountable even in relatively small geological
bodies or their parts (samples).
Geological population of atoms of a chemical element is a natural pheno-
menon of statistical character and it obeys the law of large numbers. The
atoms react, thus producing minerals which constitute geological bodies. The
distribution of the geological population of chemical elements is indirectly
connected to the distribution of minerals in a geological body.
Redistribution of the population of atoms in minerals has the character of
separation in the sense that considerable differences in concenitrations of che-
mical elements are made on the level of elementary volumes (these volumes
correspond, in dimension, to mineral grains in rocks).
The basic geological population of atoms of a chemical element in a geo-
logical body cannot be investigated directly. Therefore sampling methods have
been developed as a means of practical investigation of geological bodies.
Sampling is an experiment which reveals concentrations of elements in a samp-
le; these concentrations are statistically reliable events (the content of an
element is always realized, taking into account also the conditional "zero",
below the threshold of the analytical method).
A sample being the material representative of a geological body constitutes
a unit value, while the set of samples represents the basic geological popu-
lation of the element, i. e., it is the statistical sample of a geological body. The
characteristics of a geological population in a statistical sample are attributed
to the basic geological population of an element ini a geological body.
The distribution of a chemical element in a geological body is defined by
means of a statistical sample in 6 theorems and 3 distribution laws. These theo-
rems and laws will be presented and commented in this paper. The theoretical
basis of these theorems and laws lies beyond the scope of this paper; it has been
presented in previous publications (O m a 1 j e v , 1977, 1978, 1982a).
Theorem 1: A sample, being the material representative of a geological
body, has two characteristics: 1) the unit mass of the sample material (q), and
2) the unit mass of the investigated chemical element (X), shown by concen-
tration (x) as a measure of the contents of its atoms; this represents the geo-
logical information.
The sample material consists of the atoms of all present elements, therefore
the sum of atomic masses makes the unit mass of the material of a sample (q).
The unit mass of an investigated element (X) consists of its atoms and it is
a part of the mass contained within the sample matter. The unit mass of
a sample (q) represents the "wrapping" for the unit mass of the investigated
chemical element (X). This fact is of extreme importance, and it is the basis
of the new methodology of statistical investigation of the geological population
distribution. The unit mass (of samples) represents the "quantum" in statistical
investigations; it can be accepted or rejected.
Theorems and laws defining the distribution of the geological random variable_257
i. e. this is unit concentration, and it is numerically equal to the unit mass of
an investigated chemical element (X).
In this way, the unit concentration (x) unites the two characteristics of the
unit volume of the sample (v): 1) the unit mass of a sample (q), and 2) the unit
mass of an investigated chemical element (X); these is information about
a geological population according to characteristics of (q) and (X).
A set of samples of which consists the statistical sample is also a geological
population. This geological population is related to the sample's characteristics;
i. e. the statistical sample contains two geological populations: 1) the popula-
tion of samples as unit masses of material (q), and 2) the population of unit
masses of an investigated chemical element (X).
The statistical sample as a geological population is of discrete (corpuscular)
character. The number of data is finite and countable. On the other hand, the
basic geological population in a geological body is infinitely large, continuous
and uncountable. In order to overcome the discrete nature of a statistical
sample, which has to be fulfilled in order to represent the basic geological po-
pulation in a geological body, an infinitely large number of data is required.
Depending on the thereshold of sensitivity of the statistical investigation,
it is possible to define a conditionally large number of data in a statistical
sample; this infinitely large number is defined when the relative frequency
of the unit mass of a sample (q) becomes, conditionally, equal to zero. For
example:
Theorem 2: Unit masses of sample material (q) and unit masses of an in-
vestigated chemical element (X) are functions of concentration (x); they are
represented by symbols f(x) and fx(x).
Graphically, the illustration of these functions can be a dual polygon, a dual
histogram, or a dual diagram (fig. 1, 2, 4 and 5). Concentration (x), as an
independent geological variable, is plotted on the abscissa, while the dependent
variables, f(x) and fx(x), are plotted on the ordinate.
17 - Geologija 28/29
The concentration of a chemical element (x) in a sample, being the measure
of the chemical element's contents in the mass of the sample, is represented as
follows:
258
Veljko Omaljev
Theorems and laws defining the distribution of the geological random variable_259
Fig. 1. The normal (symmetrical) distribution of geological random variable
a) Dual histogram of distribution of functions f(x) (pillars) and fx(x) (full pillars)
b) Density curves of distribution of functions f(x) and fx(x)
c) Commulative curves of distribution of functions F(x) and Fx(x)
SI. 1. Normalna (simetrična) raspodela geološke slučajne promenljive
a) Dvojni histrogram raspodele funkcija f(x) (stubovi) i fx(x) (puni stubici)
b) Krive gustine raspodele funkcija f(x) i fx(x)
c) Kumulativne krive raspodele funkcija F(x) i Fx(x)
The dual nature of polygons, histograms and diagrams is derived from the
fact that two geological populations exist within a statistical sample. The pillars
of dual histograms (fig. 1, 2, 4 and 5) represent the intensities of the function
f(x) and they contain pillars which represent the function fx(x). However, it
is only a graphical representation, since both widths are the same on the ab-
scissa. These functions are:
f(x) —   The relative frequency distribution of the sample s material unit masses
(q) versus the independent geological variable (x).
f.\(x) — The relative frequency distribution of the investigated chemical ele-
ment's unit masses in the sample's material (X) versus the independent
geological variable (x).
The quotient of concentrations of chemical elements in a sample (Na/K,
Th/U) has also the property of an independent geological variable.
If more concentrations of chemical elements are investigated in a sample
(multicomponent sample analysis), then all investigated elements (components)
have their own geological populations. Unit masses of each of the elements
behave like functions (dependent geological variables) of concentrations of
each of the investigated chemical elements.
In a two-component system there are 6 functions (unit mass of a sample q,
and each of the two investigated elements X and Y is a function of two con-
centrations X and y), while in three-component systems there are 12 functions,
etc. In multi-component systems the number of functions grows rapidly, and
therefore the investigation of these systems becomes extremely complicated.
Theorem 3: The statistical scale is a standardization measure of the inde-
pendent geological variable (x); it defines the position and size of each of the
classes and isolines of concentration (geochemical equidistance) of an investi-
gated chemical element on the abscissa of the histogram (diagram) and on
the geochemical map.
The relief of the geochemical field, according to the new definition given
by the author (Omaljev, 1982 b, 1983), is basically similar to the relief
of Earth's surface. The graphical methods of topography and of g&ochemical
field representation have to be, basically, the same; both surfaces, are presented
by isolines of potential on geochemical maps: 1) elevation for topography, and
2) concentration intensities of an investigated chemical element for the relief
of the geochemical. Of course, the equidistance standardization has to be done
according to the same principles.
Geochemical equidistance is a number which shows the interval between
adjacent concentration isolines; its standardization is called the »Statistical
Scale«.
Theorems and laws defining the distribution of the geological random variable 261
Fig. 2. The lognormal distribution of geological random variable
a) Dual histogram of distribution of functions f(x) (pillars) and fx(x) (full pillars)
b) Density curves of distribution of functions f(x) and fx(x)
c) Cummulative curves of distribution of functions F(x) and Fx(x)
SI. 2. Lognormalna raspodela geološke slučajne promenljive
a) Dvojni histogram raspodele funkcija f(x) (stubovi) i fx(x) (puni stubici)
b) Krive gustina raspodele funkcija f(x) i fx(x)
c) Kumulativne krive raspodele funkcija F(x) i Fx(x)
The linear statistical scale is the ratio of unity to the geochemical equidi-
stance, and it is presented in the usualy way (1 : 1, 1 : 2, 1 : 5, 1 : 10, 1 : 100, etc.).
Geochemical equidistance is the interval of concentration variation of each
of the classes on the abscissa of dual histograms. It is essential that the value
"zero", for concentration, is placed at the coordinate origin.
The logarithmic statistical scale is defined in a similar way: it is the ratio
of unity to the geochemical equidistance in logarithmic relations. The loga-
rithmic geochemical equidistance is represented by equal segments on the
logarithmic abscissa (log x — fig. 2). As we move the decimal point, the whole
relative numbers of logarithms correspond to decades of natural numbers; that
is what we call "logarithmic decades". Logarithmic decades are equal segments
on the logarithmic abscissa (log x). Logarithmic geochemical equidistance must
contain the whole logarithmic decade or its whole number quotient.
Logarithmic statistical scale is denoted by "log 1 : 1" (the whole logarithmic
decade), "log 1 :5 (1/5 of the logarithmic decade), "log 1 : 10" (logarithmic
decade divied in 10 parts — fig. 2), etc.
Theorem 4: The distribution of the geological random variable is defined
by three parameters: 1) mathematical expectation (//) of the population (ari-
thmetical mean x of the statistical sample), 2) median (M), and 3) median of
the investigated chemical element (Mx). The medians divide geological popu-
lation iji two halves (the depleted one and the enriched one) according to the
mass of material (q) and according to the mass of an investigated chemical
element (X).
Theorem 5: Mathematical expectation (//) of the population (arithmetical
mean x of the statistical sample) is the common point (the intersection point)
of the density curves f(x) and fx(x) and it is the point of equalization of unit
relative frequencies {XjH X = q/^ q = 1/N) of these functions.
The mean contents, of a chemical element in a geological body is the mean
value E(x) of the geological random variable, and it represents mean concen-
tration of the atoms of an investigated chemical element. This value is unk-
nown, and there is no method for accurate determination of mathematical
expectation. For this reason arithmetical mean is introduced as an estimator
of the mathematical expectation, being the mean contents.
Mathematical expectation (arithmetical mean), graphically taken, represents
the intersection point (fig. 1, 2, 4 and 5) of the density curves f(x) and fx(x);
this points corresponds to the equalization point of relative frequencies of unit
masses (q) and (X) of these functions (Х^ 2" = q/2'q = 1/N). The value of
mathematical expectation (arithmetical mean) is equal to the ratio of areas
under the density curves fx(x) and f(x):
262 Veljko Omaljev
Median (M) is a numerical value of the concentration of the investigated
chemical element of the central member of the statistical sample population.
It represents the grouping central of a geological population of the statistical
sample, but it does not have the character of the mean value E(x).
The population of unit masses of an investigated chemical element (X),
in a statistical sample, has its own median of an investigated element (Mx).
It is the middle member of the set which divides the population in two halves
so that each of them contains, approximately, one half of the atoms, i. e. of the
mass of an investigated chemical element.
Function f.\(x) and median Mx are new in statistics (Omaljev, 1977,
1978, 1982a).
Moduses (Mo and Mox) are the most frequent members of the set of samples
and they are positioned at the maximus of density curves f(x) and fx(x). Com-
plex geological populations consist of several partial populations which have
their own moduses. Modus is not the characteristics parameter of the distri-
bution, like the mean contents (x) and medians (M and Mx), as defined by
Theorem 4.
Theorem 6: The position of a geological population on the abscissa of the
independent geological variable is defined by the interval of concentration
of a chemical element (хшах — x„,in, being the element of a geochemical field)
in the samples of the statistical sample. Outside of this interval the geological
random variable is not defined.
The form of the density curve f(x) is strictly dependent on the character
of the distribution; of the geological random variable; the determination of the
distribution type is based on this fact. Five density curve types have been
described (fig. 3) so far (B ogatskiy, 1963). This author is, however, of
the opinion that there are only three types of the geological random variable
distribution (Omaljev,  1978, 1982a):
1) The symmetrical distribution
2) The left-asymmetrical distribution
3) The right-asymmetrical distribution
The type of the geological random variable distribution is mathematically
defined by the relation (position) of parameters which represent the grouping
centre of the statistical sample's data. On the basis of Theorem 4 these are:
1) mathematical expectation of the population (arithmetical mean of the stati-
stical sample), 2) median (M), and 3) median of an investigated chemical ele-
ment (Mx).
The laws of the geological random variable distribution mathematically
define the distribution type.
First law: The position of the mathematical expectation of the population
(arithmetical mean of the statistical sample) in symmetrical distribution, ap-
proximatelly coincides on the abscissa with the median (M):
Theorems and laws defining the distribution of the geological random variable
263
Fig. 3. The types of density curves of distribution of geological random variable
1 Normal (symmetrical) distribution, 2 Left-asymmetrical distribution, 3 Right-asym-
metrical distribution, 4 Left-asymmetrical hyperbolic distribution, 5 Right-asym-
metrical hyperbolic distribution
SI. 3. Tipovi krivih gustina raspodele geološke slučajne promenljive
1 Normalna (simetrična) raspodela, 2 levoasimetrična raspodela, 3 desnoasimetrična
raspodela, 4 levoasimetrična hiperbolična raspodela, 5 desnoasimetrična hiperbolična
raspodela
X ~ М< Мх
Symmetry of the density curve f(x) is defined with respect to the median
(M) as a result of Theorem 4. The normal distribution (fig. 1) is the special
case of absolute coincidence of mathematical espectation (arithmetical mean)
and median (x = M); it represents the central ("zero") position within the
spectrum of various density curve forms caused by the left (positive) or right
(negative) asymmetry, or, in other words, by the different degrees of flatness.
Second law: The position of the mathematical expectation of the population
(arithmetical mean of the statistical sample) on the abscissa, in the left-asym-
metrical distribution (positive assymetry), is between the two medians:
M < X < Mx
The left (positive) asymmetry (fig. 4) might be of various shapes, even
hyperbolic in the extreme case. By application of the logarithmic statistical
scale, the elongatiori of the curve to the right side is eliminated (logarithmic
abscissa "compresses" the curve to the left side), and therefore the second law is
also called the "logarithmic law".
264
Veljko Omaljev
Theorems and laws defining the distribution of the geological random variable_265
Numerous authors refer to the logarithmic law as to the "lognormal law"
of distribution (Ahr ens, 1963; Rod ionov, 1961; C a r 1 i e r , 1964; and
many others). However, the lognormal density curve f(x) is only the special
case when it becomes symmetrical with respect to its median M (fig. 2).
The same effect is gained if we apply the exponential relations on the ab-
scissa (x"^), for m not equal to one (m = 1 is the linear relationship), and there-
fore the second law can be generalized as the "exponential law" of distribution..
Logarithm is only a special form of exponential relations, so the logarithmic
law is a special case of the exponential law of geological random variable di-
stribution (the second law).
This problem has not been considered with full attention in literature, and
that is the reason for different interpretations of analysis of the left-asym-
metrical distribution of a geological random variable.
Third law: The position of the mathematical expectation of the population
(arithmetical mean of the statistical sample) on the abscissa, in right-asym-
metrical distribution (negative asymmetry), is to the left of medians:
X < M < Mx
Right (negative) asymmetry (fig. 5) may take various forms, up to the
hyperbolic density curve f(x). For this type of distribution a procedure for the
elimination of elongation to the left side has not been found yet.
Some density curve forms, in the case of left-asymmetrical and right-
asymmetrical distribution types, are symmetrical with respect to the normal
distribution median. It is the result of the law of large numbers in the course
of generation of basic geological populations. It also means that there is unity
between the processes of accumulation of chemical elements in nature. On the
basis of central limit theorem's assumptions all distributions show a tendency
towards the symmetrical (normal) distribution, and this is the reason why it
has the central position in the spectrum of different density curves forms
(fig. 3).
The characteristics of the geological random variable distribution are mathe-
matically defined by the distribution parameters (Theorem 4), while their mu-
tual relations define the distribution type. Distribution parameters are unique
for all types of density curves, while the distribution laws unify the treatment
of these parameters. The application of exponential and logarithmic relations
on the abscissa is used only to symmetrize the density curves f(x) with respect
to their medians. Bearing this in mind it is not possible to use the geometrical
or exponential mean to calculate the mean value E(x). According to Theorem 5,
the mathematical expectation (arithmetical mean) is the only estimate for the
mean value E(x)._
Fig. 4. The ideal left-asymmetrical distribution (positive asymmetry) of geological
random variable
a) Dual histogram of distribution of functions f(x) (pillars) and (full pillars)
b) Density curves of distribution of functions f(x) and fx(x)
c) Cummulative curves of distribution of functions F(x) and Fx(x)
SI. 4. Idealna levoasimetrična raspodela (pozitivna asimetrija)  geološke slučajne
promenljive
a) Dvojni histogram raspodele funkcija f(x) (stubovi) i fx(x) (puni stubići)
b) Krive gustine raspodele funkcija f(x) i fx(x)
c) Kumulativne krive raspodele funkcija F(x) i Fx(x)
266
Veljko Omaljev
Teoreme i zakoni koji definišu raspodelu geološke slučajne promenljive 267
Teoreme i zakoni koji definišu raspodelu geološke slučajne promenljive
Izvod
Osnovni elementi građe geoloških tela zemljine kore, sa aspekta geohemije,
su atomi hemijskih elemenata. Geološka populacija elemenata (atoma) u geolo-
škom telu je beskonačno velika, neprekidna i neprebrojiva. Ova osnovna po-
pulacija nije dostupna direktnom opažanju i zato su razvijene metode oproba-
vanja. Proba predstavlja jediničnu veličinu, a skup proba čini statistički uzorak
koji predstavlja osnovnu populaciju.
Raspodela elemenata (atoma) u geološkim telima je definisana sa 6 teorema
i 3 zakona raspodele. Teoretska osnova ovih teorema je ranije izložena u publi-
kacijama (Omaljev, 1977, 1978, 1982a).
Teorema 1: Proba kao materijalni predstavnik geološkog tela ima dva obe-
ležja: 1) jediničnu masu materijala probe (q) i 2) jediničnu masu ispitivanog
hemijskog elementa (X) iskazanu koncentracijom (х) kao merom zastupljenosti
njegovih atoma, što predstavlja geološku informaciju.
Koncentracija elementa se izražava odnosom х = X'q, a za jediničnu masu
materijala probe ravnu jedinici (q = 1) dobijamo da je jedinična koncentracija
brojno jednaka jediničnoj masi hemijskog elementa u probi (х = X).
Kao posledica teoreme 1 u statičkom uzorku postoje dve geološke populacije:
1) populacija proba kao jediničnih masa materijala (q) i 2) populacija jediničnih
masa ispitivanog elementa (X) u probi.
Statistički uzorak, kao populacija, ima diskretan karakter; broj podataka
je konačan i prebrojiv. Zavisno od praga osetljivosti, moguće je definisatl
uslovno beskonačno veliki broj podataka u statističkom uzorku; a to je broj
podataka kada je relativna frekvencija jedinične mase probe uslovno jednaka
nuli (q/2'q = l/N = 0%).
Teorema 2: Jedinične mase materijala probe (q) i jedinične mase ispitivanog
hemijskog elementa (X) su funkcije koncentracije (х); što predstavljamo sim-
bolima f(x) i fx(x).
f(x) —  Raspodela frekvencija jediničnih masa materijala probe (q) po neza-
visnoj geološkoj promenljivoj (х).
fx(x) — Raspodela frekvencija jediničnih masa ispitivanog hemijskog elementa
(X) u masi probe po nezavisnoj geološkoj promenljivoj (х).
Grafička ilustracija ovih funkcija može biti dvojni poligon, histogram ili
dijagram. Na apscisu se nanose vrednosti nezavisne geološke promenljive (х),
a na ordinatu vrednosti funkcija f(x) (stubovi) i fx(x) (puni stubići) kao za-
visne geološke promenljive (njihove frekvencije).
Fig. 5. The ideal right-asymmetrical distribution (negative asymmetry) of geological
random variable
a) Dual histogram  of distribution of functions f(x) (pillars) and fx(x) (full pillars)
b) Density curves of distribution of functions f(x) and fx(x)
c) Cummulative curves of distribution of functions F(xj and"Fx(x)
SI. 5. Idealna desnoasimetrična raspodela (negativna asimetrija) geološke slučajne
promenljive
a) Dvojni histogram raspodele funkcija f(x) (stubovi) i fx(x) (puni stubići)
b) Krive gustine raspodele funkcija f(x) i fv(x)
c) Kumulativne krive raspodele funkcija F(x) i Fx(x)
268_ Veljko Omaljev
Srednji sadržaj izučavanog hemijskog elementa u geološkom telu E(x) je
srednja vrednost geološke slučajne promenljive (matematičko očekivanje //).
Ova veličina je uvek nepoznata, a u praksi se umesto nje uzima aritmetička
sredina izučavanog hemijskog elementa u statističkom uzorku (reprezentu geo-
loškog tela).
Teorema 6: Položaj geološke populacije na apscisnoj osi geološke slučajne
promenljive (х) je određen intervalom varijacija koncentracija hemijskog ele-
menta (x,„ax — x„ii„ kao elementom geohemijskog polja) u probama statističkog
uzorka. Van intervala variacija geološka slučajna promenljiva nije definisana.
Tip raspodele geološke slučajne promenljive, odnosno forma krive gustine
f(x) je matematički određen položajem parametara raspodele (po teoremi 4) na
apscisnoj osi. Zakonima raspodele definisan je tip raspodele geološke slučajne
promenljive, i to:
Prvi zakon: Položaj matematičkog očekivanja populacije (aritmetičke sredine
statističkog uzorka) na apscisnoj osi kod simetrične raspodele približno se
poklapa sa medijanom (M):
x~ M < Мх
Normalna raspodela je specijalan slučaj simetrične raspodele, kada se ma-
tematičko očekivanje (aritmetička sredina) potpuno poklapa (po brojnoj vred-
nosti) sa medijanom (х = M).
Odnos koncentracija Kemijskih elemenata u probi (Na'K, Th/U) takođe ima
osobine nezavisne geološke promenljive.
Teorema 3: Statistička razmera je mera standardizacije geološke nezavisne
promenljive (х), koja određuje položaj i veličinu svake klase i izolinije kon-
centracije (geohemijsku ekvidistancu) ispitivanog hemijskog elementa na apscisi
histograma (dijagrama) i geohemijskoj karti.
Geohemijska ekvidistanca je interval koncentracije između susednih izolini-
ja. Statistička razmera je odnos jedinice prema geohemijskoj ekvidistanci, a ona
može biti linearna (npr. 1 : 1, 1 : 10, 1 : 100 //g'g) i logaritamska (npr. log 1 : 10,
log 1 :20//g/g).
Teorema 4: Raspodela geološke slučajne promenljive je određena sa tri
parametra: 1) matematičkim očekivanjem (fi) populacije (aritmetičkom sredi-
nom X statističkog uzorka), 2) medijanom (М) i 3) medijanom ispitivanog hemij-
skog elementa (Мх). Medijane dele geološku populaciju na dve polovine (osiro-
mašenu i obogaćenu) po masi materijala (q) i po masi ispitivanog elementa (X)
u probi.
Teorema 5: Matematičko očekivanje (//) populacije (aritmetička sredina х
statističkog uzorka) je zajednička tačka (tačka preseka) krivih gustina f(x)
i fx(x) i predstavlja tačku izjednačenja jediničnih relativnih frekvencija
(Xs/i: X = q/2 q = l/N) ovih funkcija.
Theorems and laws defining the distribution of the geological random variable_269
References
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Drugi zakon: Položaj matematičkog očekivanija populacije (aritmetičke sre-
dine statističkog uzorka) na apscisnoj osi kod levoasimetrične (pozitivna asi-
metrija) raspodele je između dve medijane:
M < X < Мх
Primenom logaritamskih odnosa na apscisi (log х) može se eliminisati leva
asimetrija krive gustine f(x) i zato se drugi zakon još naziva »logaritamski
zakon«.
Treći zakon: Položaj matematičkog očekivanja populacije (aritmetičke sredi-
ne statističkog uzorka) na apscisnoj osi kod desnoasimetrične (negativna asi-
metrija) raspodele je levo od medijana:
X < М< Мх
Za desnoasimetrične raspodele nije pronađen postupak eliminacije asimetrije
krive gustine f(x).
Parametri raspodele (po teoremi 4) su jedinstveni za sve tipove krivih
gustina, a zakoni raspodele odbacuju mogućnost različitog tretmana pomenutih
parametara. Primenom logaritamskih (i eksponencijalnih) odnosa na apscisi
nezavisne geološke promenljive (х) služi samo za formalizaciju u smislu posti-
zanja relativne simetrije krive gustine f(x) u odnosu na svoju medijanu (М),
a nikako ne daju mogućnost primene geometrijske ili eksponencijalne sredine
za izračunavanje srednje vrednosti E(x). Matematičko očekivanje (aritmetička
sredina) je jedina vrednost za srednju vrednost E(x) geološke slučajne pro-
menljive, što je definisano teoremom 5.