Mathematical, physical and physiological
background of normal distribution, delta
distribution and log-normal distribution
Chang Lin Zhang
College of Life Science,
Zhejiang University,
Hangzhou, China
Department of Music and Music Education,
Siegen University,
Siegen, Hochstrasse 57,
Siegen D-57076, Germany
E-mail: zhang.biophysik@musik.uni-siegen.de
Abstract: The backgrounds of three statistical distributions of measurement data, namely,
normal distribution, log-normal distribution and delta distribution are discussed here from the
viewpoint of mathematics, physics and physiology. It can proved by means of combination
mathematics that the measurement data which fit into normal distribution come from a
system in ideal chaotic state; whereas, only the data from ideal crystal state fit into delta
distribution. The most important distribution for physiological system, however, is log-normal
distribution which shows that the measurement data come from a system that is in coherent
state, in which every element keeps its independence and at the same time has the ability to
cooperate with all others.
Keywords: normal distribution; log-normal distribution; delta distribution; chaotic state;
coherent state; crystal state.
Reference to this paper should be made as follows: Zhang, C.L. (2008) ‘Mathematical,
physical and physiological background of normal distribution, delta distribution and
log-normal distribution’, Int. J. Modelling, Identification and Control, Vol. 5, No. 3,
pp.200–204.
Biographical notes: Chang Lin Zhang is a Former Professor of Biophysics of Zhejiang
University in China and now Visiting Professor in the University of Siegen in Germany. Since
1991, he has been working in the scientific background of acupuncture, music therapy,
homeopathy, etc. He found that there is a dynamic dissipative structure of electromagnetic field
in living systems, which could be some common background of many holistic medicines and a
new chapter of physiology. Meanwhile, he also develops a mathematical method to
quantitatively evaluate the degree of coherence in living systems.
_______________________________________________________________
1 Introduction: background of normal
distribution
The Gaussian distribution was found by the German
mathematician Karl Friedrich Gauss (1777–1855), the
king of mathematics in the 19th century. The formula and
the curve of Gaussian distribution are as follows
(Figure 1).
Since it has been so widely used in many areas that
people once believed that it is so universal that it may
be called ‘normal’ distribution. It also becomes a base
for biological statistics and medical statistics which
assume that any measurement data fit into normal
distribution.
Since 1970s, the concepts of ‘order’ and ‘disorder’
have been widely studied in science, more and more
scientists have become aware that Gaussian distribution is
not ‘normal’ at all, neither ‘ordinary’ nor ‘universal’, it
does not even exist in real world.
Figure 1 Normal (Gaussian) distribution and one of its
examples (see online version for colours)
It is well known that the basic postulate of Gaussian
distribution is that the measurement data come from a
system in which all elements are independent from one
another. The postulate shows that if the data perfectly fits
into Gaussian distribution then they have come from a
system which is in ‘ideal chaotic state’ such as the concept
of ‘ideal gas’ in thermodynamics and statistical physics.
Mathematical, physical and physiological background 201
Obviously, such an ideal system does not exist in the real
physical systems, let alone in the living systems in biology
and medicine.
Before we discuss two other in the following papers,
statistical distribution, we consider the degree of freedom
in an ideal system from which the measurement data fit
into Gaussian distribution perfectly. Obviously, the degree
of freedom in such a system with N elements is N.
2 Background of delta distribution
In contrast to Gaussian distribution, there is another
statistical distribution in pure mathematics and is
called Delta distribution which describes an ideal
crystal system in which all elements are exactly the
same and strongly bound together like one element.
Therefore, the degree of freedom in such a system with N
elements is always 1, no matter how big the N is. The
formula and the curve of Delta distribution are as follows
(Figure 2).
Figure 2 Delta distribution and one of its examples (see
Obviously, like ideal chaotic system, such an ideal
crystal system also does not exist in the real physical
systems and one can say nothing of biological systems or
medical ones.
3 Background of log-normal distribution
In 1970s some mathematicians (Sachs, 1969) noticed that
the data from many physiological systems such as the
body weight of frog, the height of children, the blood
pressure, etc. do not fit into normal distribution, but into
log-normal distribution. The shape of log-normal
distribution is a little similar to normal distribution.
But normal distribution is symmetrical; whereas, the
log-normal distribution is asymmetrical with a peak shifted
toward left (Figure 3).
The mathematical relationship between normal
distribution and log-normal distribution is quite
simple. If the variable of normal distribution is x and the
variable of log-normal distribution is y, the relationship is
denoted by Formula 1.
Formula 1 The relationship between the variable x in normal
distribution and the variable y in log-normal
distribution
y = ex
Figure 3 Log-normal distribution and one of its examples
(see online version for colours)
Here, the x is variable of normal distribution and the y is
variable of log-normal distribution. Then, the question of
what is the background behind such a simple relationship
comes. Now, let us find the relationship between them with
combination mathematics.
4 Physical model of the three distributions in
a system with three elements
We can describe the three kinds of distributions
aforementioned with a model system with only three
elements and with some unusual arithmetic.
4.1 A system in chaotic state
In a system with three elements and in ideal chaotic state,
the three elements are completely independent from one
another (Figure 4).
Figure 4 A system with three elements which are completely
independent from one another
Note: Such a system is in ideal chaotic state.
In such a system, the degree of freedom is equal to the
number of elements. Therefore, the arithmetic as simple as
Formula 2.
Formula 2 Arithmetic in a system with three elements and the
system is in ideal chaotic state
1+1+1 = 3
Considering again the basic postulate of Gaussian
distribution, the measurement data have come from a
system in which all elements are independent from one
another. The arithmetic in such a system with N elements,
from which the measurement data perfectly fit into
Gaussian distribution, is
Formula 3 Arithmetic in a system with N elements and the
system is in ideal chaotic state
202 C.L. Zhang
4.2 A system in crystal state
In a system with three elements and in ideal crystal state,
the three elements are strongly bounded as one element
(Figure 5).
Figure 5 A system with three elements which are completely
bounded together like one element
Note: Such a system is in ideal crystal state.
In such a system, the degree of freedom is always one.
Therefore, the arithmetic here is a little strange.
Formula 4 Arithmetic in a system with three elements and the
system is in ideal crystal state
1+1+1 = 1
Considering again the basic postulate of Delta distribution,
the arithmetic in such a system with N elements, from
which the measurement data perfectly fit into Delta
distribution, is shown as below.
Formula 5 Arithmetic in a system with N elements and the
system is in ideal crystal state.
4.3 A system in coherent state
In a system with three elements and in ideal coherent state,
the three elements are independent from one another and
at the same time have the ability to cooperate with
all others.
The degree of freedom in a system with three element
and in coherent state can be calculated according to
Figure 6 and with combination mathematics as follows.
Figure 6 A system with three elements which are independent
from one another and at the same time have the
ability to cooperate with all others.
Formula 6 Arithmetic in a system with three elements and the
system is in ideal crystal state.
Or the arithmetic in such a system can be simply written
with following formula:
Formula 7 Arithmetic in a system with three elements and the
system is in ideal coherent state.
In the same principle, same calculation and same
mathematical formula, we can calculate the degree
of freedom of such a system with four elements as
follows.
Formula 8 Arithmetic in a system with four elements and the
system is in ideal coherent state.
Therefore, we can generalise the principle to the system
with N elements.
Formula 9 Arithmetic in a system with N elements and the
system is in ideal coherent state.
If we further generalise the principle to the system with
infinite elements, we would automatically get Formula 1
when the element number N is approaching infinite.
Formula 10 Arithmetic in a system with infinite elements and
the system is in ideal coherent state.
Now, we can see that we already get Formula 1, which is
the relationship between normal distribution and
log-normal distribution, from Formula 9.
In other words, through the calculation of the
possibilities of combinations, namely the cooperation of
all elements we can automatically get log-normal
distribution from normal distribution. Through the
aforesaid procedure, it is easy for us to see the
mathematical and physical backgrounds of normal
distribution, delta distribution and log-normal distribution.
In the viewpoint of physiology, it is also easy to see
why some parameters in physiological systems fit into
log-normal distribution instead of normal distribution. We
may take the relationship between heart and lung as
example. Obviously, it is impossible for the heart and the
lung to work together in the same frequency. However, it is
also dangerous if there is no cooperation and coordination
between the frequency of heart and the frequency of lung.
They are independent from one another and at the same
time have the ability to cooperate with all others.
5 Practical application of the relationship
between the three statistical distribution
Technically, it is impossible to measure the degree of
cooperation between so many organs, tissues and cells.
However, in the viewpoint of electronics, if there is some
weak coupling between two oscillators, a beating
frequency would come out (Figure 7).
Mathematical, physical and physiological background
Figure 7 The beating frequency (f1 − f2) in the frequency
spectrum
It means mathematically, the number of frequencies is
shown in Formula 11.
Formula 11 Calculation of the number of frequencies in a
system with two oscillators which have weak
coupling between them
It is easy to see that Formula 11 is in the same principle of
Formulas 6–10.
In a physiological system, such as a human body, there
are many, almost infinite oscillators such as organs, tissues
and cells which permanently emit electromagnetic waves.
If there are cooperations and coordinations among them, it
means that there are many weak couplings between them.
In such a case, there would be many beating frequencies
coming out. From the viewpoint of spectrum, the ‘white
noise’ would turn to ‘1/f noise’. It means that the energy
would shift from the range of higher frequencies to lower
frequencies (Figure 8).
Figure 8 The frequency spectrums of (a) the ‘white noise’ and
(b) the ‘1/f noise’
It is worth noting that the frequency spectrums from many
physiological systems, including the acoustic frequency
spectrums from classical music and folk music are ‘1/f
noise’ instead of ‘white noise’, since in many cases, the
physiological systems including classical music and folk
music are in coherent state.
The electromagnetic waves which are permanently
emitted from organs, tissues and cells would be reflected
by so many boundaries in a body. Therefore, they would
be put together to form some interference pattern, which is
actually a kind of dynamic dissipative structure of
electromagnetic field and determines the distribution of
electromagnetic field in a human body, namely the energy
distribution in the body.
According to electrodynamics, conductivity (J) is
proportional to electrical field (E), namely,
Formula 12 Relationship between conductivity and electrical field
In the ‘Hilbert space’, any function, including the
statistical distribution function can be regarded a point.
Therefore, we may regard the three ideal distributions
above as three points, and put them on the three tops of a
pyramid.
7 Discussion and conclusion
How to study a body-mind system in a holistic, scientific
and quantitative way for complementary medicines is a big
challenge to modern science and scientists.
Fortunately, many mathematicians in the last two
centuries already developed some important statistical
methods to deal with complicated systems in a holistic
way.
Since the 1970s, more physicists ventured into the
study of complex systems, the study of open systems, the
study of the systems far from equilibrium state, the study
of chaos and the study of dissipative structure.
204 C.L. Zhang
(Haken,1977; Nicolis and Prigogine, 1977; Zhang, 1994,
1996a, 1996b, 1997, 2002, 2003a, 2003b, 2004, 2007;
Kapteina and Zhang, 2008).
Based on the important development of dissipative
structure in physics in 1970s and the mathematics which
have already been developed in the last two centuries, we
discovered in 1990s that there is a dissipative structure of
electromagnetic field in living systems, since all living
systems are open systems in the state far from equilibrium.
This discovery not only gives some scientific
explanation for the ancient and mysterious acupuncture
system, charkas system and homoeopathy but also
establishes a practical, scientific and holistic evaluation
method of the state of a living systems, including the state
of mind-body system.
Finally, it is worth noting that the scientists in
Hamburg University recently found (Zhang, 2007) that
there is another statistical distribution, which they named
‘mirror-log-normal distribution’, which appears when a
person is in good sleep. The scientists recorded
this during their observation in their sleeping lab,
experimentally with excellent reproducibility. It is
obviously an important discovery for studying the state of
body-mind systems with conductivity measurement,
although it becomes a new challenge for mathematicians
and theoretical physicists to find its background.
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