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Applied Mathematics, 2020, 11, 957-984
https://www.scirp.org/journal/am
ISSN Online: 2152-7393
ISSN Print: 2152-7385
DOI:
10.4236/am.2020.1110063 Oct. 19, 2020 957
Applied Mathematics
Scrutinizing Distributions Proves That IQ Is
Inherited and Explains the Fat Tail
Michael Grabinski, Galiya Klinkova
Department of Business and Economics, Neu-Ulm University, Wileystr, Neu-Ulm, Germany
Abstract
The motivation of this paper is to show how to use the information from
given
distributions and to fit distributions in order to confirm models. Our
examples are especially for disciplines slightly away from mathematics. One
minor result is that standard deviation and mean are at most a more or less
good approximation to determine the best Gaussian fit. In our first example
we scrutinize the distribution of the intelligence quotient (IQ). Because it is
an almost perfect Gaussian distribution and correlated to the parents’ IQ,
we
conclude with mathematical arguments that IQ is inherited only which is as-
sumed by mainstream psychologists. Our second example is income distribu-
tions. The number of rich people is much higher than any Gaussian distribu-
tion would allow. We present a new distribution consisting of a Gaussian plus
a modified exponential distribution. It fits the fat tail perfectly. It is also suit-
able to explain the old problem of fat tails in stock returns.
Keywords
Fat Tail, Income Distribution, Chaos, Finance, IQ
1. Introduction
In finance, economics, and many social sciences distributions are important.
However, there are two closely connected puzzling items. Firstly, there is an al-
most dogmatic assumption that there are Gaussian distributions only (with few
exceptions). Secondly, there are partly strange methods to prove that something
must have a Gaussian distribution. The mathematics of distributions is essen-
tially a product of the 19th century, for an overview consider e.g. [1]. Since 100
years one can read in a
textbook
[2] on page 179 “
Everybody believes in the ex-
ponential law
[
i
.
e
. Gaussian distribution]
of errors
:
the experimenters
,
because
they think it can be proved by mathematics
;
and the mathematicians
,
because
How to cite this paper:
Grabinski,
M. and
Klinkova
, G. (2020) Scrutinizing Distribu-
tions Proves That IQ Is Inherited and E
x-
plains the Fat Tail
.
Applied Mathematics
,
11,
957
-984.
https://doi.org/10.4236/am.2020.1110063
Received:
September 4, 2020
Accepted:
October 16, 2020
Published:
October 19, 2020
Copyright © 20
20 by author(s) and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons
Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
M. Grabinski, G. Klinkova
DOI:
10.4236/am.2020.1110063 958
Applied Mathematics
they believe it has been established by observation
”. In non-mathematical
sciences distributions became popular from roughly 1950+.
A typical paper from this time is [3]. Fama observed a “fat tail” in the distri-
bution of stock market prices. This fat tail provoked an avalanche of publications
which are impossible to mention completely, just as an example see [4] [5] [6].
What is most puzzling with it, is that the authors of [3] [4] [5] [6] (and many
more) wrongly assumed a Gaussian distribution in the first place. Ignoring text-
books like [2] they “derived” the Gaussian distribution from a misinterpretation
of the central limit theorem and assumed ergodicity without any justification.
Even worse, once having accepted this derivation as correct, observations of
stock prices and the like showing a fat tail are in contradiction to a Gaussian dis-
tribution. From this one has an experimental proof that some assumptions of
Fama [3] must be wrong. Instead of starting all over, the fat tail is cherished as
one of the greatest discoveries of the 20th century in finance.
Another reason for the wrongly assumed fat-tail in finance is that stock prices
are fluctuating chaotically rather than randomly. The mathematical description
of chaos is pretty old, and a modern summary (and application to physics) can
be found in [7]. The application of chaos to business and economics is much
younger, just as an overview see [8]-[15]. The distinction relevant here between
randomness and chaos has been shown quite recently [16].
The name “fat-tail” originates from physics. However, the wrong doings men-
tioned above are not present there, for a quite recent example see [17].
Dealing with distributions other than Gaussian does rarely cause problems
similar to the one mentioned here. As stated above, especially social sciences as-
sume a Gaussian distribution for almost everything. Only if there is a proof for
another distribution, it is not used. As an example consider the exponential dis-
tribution mostly applicated to describe queues [18] [19] [20] [21]. An already
rarer example is the Poisson distribution [22]. There are also power law distribu-
tions but these are most common in physics within e.g. critical situations like
phase transitions [23]. An application from physics (critical points) for herd be-
havior in financial markets can be found in [24].
So far for a brief summary of the use of distributions especially in
non-mathematical sciences, the purpose of this paper is not to fix the mentioned
problems especially when using a Gaussian distribution wrongly or without justi-
fication. This is hardly possible. Our goal is to explain the use of distributions in
two general situations:
• An experiment or reality shows a strictly known distribution (e.g. Gaussian).
• An experiment or reality shows a strange situation (e.g. Gaussian with fat
tail).
To give a complete answer to the two points would mean writing a textbook.
Of course, such books have existed for roughly 100 years, see e.g. [2]. As stated
above, they are rarely used in fields from economics over finance to psychology.
Therefore we take just two examples which are discussed quite often.
Our first example is the distribution of the intelligence quotient normally re-
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ferred to as IQ. There are lots of data worldwide, and they have one thing in
common: They are perfectly Gaussian (unfortunately without fat tail). From this
it is almost trivial to show in chapter 2 that the IQ is inherited or at least not
changed by conscious action like training. This gives the first mathematical
proof for an old and almost religious argumentation about nature versus nur-
ture.
In chapter 3 we will scrutinize income distributions. We have chosen this
example for two reasons. Firstly, it is an often debated subject especially since
the best-selling book of Piketty [25], though we will not contribute to the politi-
cal or moral debate of it. Secondly, income distributions also show something
like a Gaussian distribution with a fat tail. In subchapter 3.3 we will derive a new
model by using the remarks of Chapman [26] on historical data. It is also (part-
ly) an explanation of the fat tail in finance. Though the general idea is identical,
the effects of speculation make finance more complicated [27].
As a result we get a narrower distribution for the not-very-rich if the su-
per-rich are allowed to have a wider distribution. In other words, without the
super-rich there would be a less equal distribution within the “normal” people.
Fitting the data within subchapter 3.3 is extremely complicated. It shows the
frontiers of numerical mathematics. Therefore we are deferring some of the ma-
thematical derivations to chapter 4.
Chapter 5 gives a summary and ideas for further research.
2. What IQ Distribution Teaches Us
In Figure 1 you can see the probability density for an IQ of German men (wide
blue curve) and women (narrow red curve). Both curves peak at IQ = 100
leading to an average IQ of 100. This is not identical but very similar in most
developed countries. Sometimes the average IQ is higher but sometimes it is
lower. In most third world countries it is even significantly lower. Because
there is less education in these countries, some may argue that this is a proof
for IQ being based on nurture. However, it could also be nature, as most of the
inhabitants of a country live there for generations. So it is neither a proof nor a
disproof.
Even much more universal than the average IQ in the developed world is the
width of the distribution. It is wider for men than women. Again, this is no
proof for nature or nurture. However, setting on nurture would mean that there
is a
universal
difference in the education of boys and girls. Though there are dif-
ferences in education, it would be at least puzzling that this difference is persis-
tent in so many societies.1
1The different width in female and male IQ distribution is quite plausible in terms of classical evol
u-
tion. At least during most of human history
the roles of men and women were clearly defined. Men
had to hunt and gather in order for a family to survive. Women had to take care of raising the kids.
Especially in hunting there may be two successful strategies. Put m
ost of your energy into your
mussels or into your brain. (The brain uses a lot of energy) Raising kids a
low IQ will maybe kill the
kids. On the other side,
a high IQ will not help very much but consumes much energy. Thus stone
age women should have an almost constant average IQ.
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Figure 1. Gaussian distribution of IQ of men (
σ
≈ 16.2) and women (
σ
≈ 13.2).
Though this paper is on mathematics rather than psychology, there is mean-
while an agreement under academic psychologists that IQ is inherited. In a re-
view [28] one can read “…
IQ is fixed throughout your life
:
the only way you’ll
lose it is because of a brain injury
”. One can find only small and rare deviations
from it as stated in [29]. But this is no disproof. Especially, there is no recipe
how to increase IQ, which would be necessary to prove nurture. There are even
hints that it is epigenetic as stated in [30].
Nevertheless, there are
hardliners
(even in academic psychology) sticking to
nurture instead of nature. Because the authors of this paper cannot read minds,
we can only speculate about the reason for it. It looks like
ideology takes over
science
. Of course, an inherited IQ has political consequences. It would be much
harder to argue for a more equal income within countries and especially between
developed countries and the rest of the world where average IQ is partly signifi-
cantly lower. In order to find counter arguments against the mainstream of aca-
demic psychologists it is sometimes said that IQ is a poorly defined measure.
One has to say that there are lots of advances ever since Binet and Simon devel-
oped the first IQ tests in the early 20th century. There are also broader measures
like “fluid intelligence” which include the ability of abstract thinking and
problem solving. Furthermore, many other data like the scores of Americas
standardized entrance exams for university (e.g. GMAT) correlate nicely to the
IQ. So we have extremely many sets of data which are all seeing nature instead of
nurture.
Sometimes it is also stated that one can
learn
how to get a higher score in an
IQ test. It would make the entire concept ridiculous. It is even easy to “prove” it
by taking an IQ test several times. However, this is nothing but
corrupting the
system
. Taking the same or a very similar math exam many times will also im-
prove results.
Up to now we have just summarized about nurture versus nature in IQ. The
motivation for this chapter was to find a clear-cut
mathematical
proof that IQ or
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fluid intelligence is (essentially) inherited. However we are not sure whether it
will convince the above-mentioned
hardliners
. To our own surprise it is ex-
tremely simple.
If IQ increases by training, it should be identical to learning in the sense of
learning curves. Learning curves are used in (industrial) engineering and espe-
cially production management. Maybe first noted in [31], one can find the same
approach in contemporary PhD theses like [32] or even applied to the learning
of terrorists in a paper in Science [33]. Such forms of learning over time
t
show a
decrease in e.g. cost in the form of
t
α
−
(1)
where
α
is a critical positive exponent. Purely numerical it fits most situations
fairly well. Though widely used, it is wrong. Equation (1) is a result of a random
walk [34]. It fits for unconscious learning only. This is like ants “learn” how to
find food. It is also called trial and error. Obviously it does neither apply to
learning how to produce cheaply nor to learning in order to increase IQ. De-
scribing (conscious) learning correctly has been published surprisingly recently
[35]. It has been extended to situations where two sides learn ([11] [36]) like in
terrorist attacks where the aggressor learn how to attack and the protector how
to defend.
Instead of Equation (1), learning takes the following form:
e1
t
τ
−
−
(2)
where
τ
is a typical learning time. If IQ is essentially acquired by “learning,” one
would have the same picture in the IQ distribution. Proportionally to how much
IQ points you have already gained, it will be more and more difficult to get an
additional IQ point. We have a differential equation of the form
IQ IQ∝−
leading to an exponential distribution of IQ:
( )
( )
0
IQ IQ 0
IQ e with IQ IQp
λ
λ
−⋅ −
=⋅≥
(3)
where
λ
is a parameter determining the quality of the education program. A
small
λ
means intense education for everybody and a big
λ
means no education.
IQ0 is the IQ at birth which may have a very narrow Gaussian distribution. The
plot in Figure 2 reveals the difference.
The difference between an inherited distribution of IQ (Figure 1) and a
trained one (Figure 2) should be obvious. From this we have the clear result that
reality (experiment) proves that IQ is essentially inherited. One of the most
striking difference between Figure 1 and Figure 2 is the universal constant IQ0.
There should not be an IQ below it. Unfortunately this is in contrast to all ob-
servations of human IQ.
Some may say that there is both: nurture and nature. Of course there may be a
Gaussian distribution of IQ0 and some IQ achievements due to education. How-
ever the decision whether nature or nurture has (by far) the upper hand is easy.
Does our measured IQ distribution look more like Figure 1 or Figure 2?
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Figure 2. Exponential IQ distribution;
0
IQ 90=
and
0.1
λ
=
would mean
IQ 100
=
.
The above-mentioned hardliners may say that the skills to educate (
i
.
e
.
λ
) is
Gaussian distributed within the parents. But this would still not lead to a Gaus-
sian distribution. Though the distribution for high IQ values would look almost
Gaussian, most strikingly this new distribution would still be very asymmetric
and never show
0
IQ IQ<
values in contrast to observations in the real world.
Of course it is possible to destroy IQ either by physical or mental injury. As an
example of the latter one, one may consider the tragic figure Kaspar Hauser2 liv-
ing in Germany around Nuremberg from about 1812 till 1833. However, there is
never so much abuse to explain symmetry.
If it is possible to increase IQ (massively) by education or training, such me-
thods could be applied to a centralized child education. It should have led to a
massive increase in IQ in systems like the Soviet Union or mainland China.
All the above clearly states that IQ cannot be improved. It is a result of a ran-
dom mix of the genes of the two parents. Due to the central limit theorem such
randomness leads to a Gaussian distribution. It is very hard to imagine that any
other mechanism creates a Gaussian distribution. Actually, mankind is not able
to create (complete) randomness by e.g. computer programs. Astrophysicists are
sometimes in need to scrutinize signals for
exact
randomness. They still have to
rely on natural sources like radioactive decay in order to have a precise reference.
Therefore the Gaussian distribution proves randomness and no conscious ac-
tions.
Some may argue now that the IQ is not inherited but a result of randomness
2He appeared in the Nuremberg area in the age of about 16 and could hardly speak. Obviously
he
had been a mistreated or severely abused child. He learned to speak and told people that he was held
in a dark room by water and bread for almost his entire life though this cannot be true. There were
also rumors that he was aristocratic and betrayed of his inheritance. However,
a genetic analysis of
2002 disproved it once and for all. With learning to speak fluently he invented stories in order to
gain the interest of other people. In some sense
it shows that his IQ was present all the time and its
usage had been recovered.
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having nothing to do with nature or nurture. It would also lead to a perfect
Gaussian distribution. However, there are correlations between the parents’ and
offspring’s IQ. These can only be there if either nature or nurture plays a major
role.
However it is impossible to judge whether this randomness is finished by
conception. Something during the embryonic growth may contribute. At least
for the trait homosexuality in women there seems to be strong evidence for it.
Early childhood may also have an influence on IQ as long as it cannot be influ-
enced consciously. To judge whether such unconscious influences exist is im-
possible to decide because the statistics are identical3.
In breeding animals genetic selection is meanwhile quite common. It is done
by producing many embryos from one pair of parents in a Petry dish. The genes
are scrutinized in order to find e.g. the embryo with the highest potential for a
cow giving lots of milk. In humans such selection should theoretically be possi-
ble too. Though the genes for high IQ are not discovered yet, it will be possible
someday. If done massively (albeit moral and ethical concerns), it would lead
more and more to an exponential distribution of IQ. If done only by the rich
who can afford it, it would lead to a mixture which is a Gaussian with a fat tail. It
would be essentially the same model as we will suggest in subchapter 3.3.
3. Income Distributions
As a start consider the monthly net household income of 2017 in Germany as
given in Figure 3. One may give this income distribution to graduate students
(as we did) to find the best fitting Gaussian of it. Though it is a god math exer-
cise, it does not make very much sense. The distribution is given. The necessary
row data are arbitrarily fine. So why should one find a mathematical function
which is at most a good approximation? As in the last chapter the goal is to learn
something from the form of the distribution. From the exact Gaussian distribu-
tion of IQ one can derive that IQ is inherited only. Find a descend fit of Figure 3,
and it may reveal why it comes to a difference in income.
Though this paper is on mathematics rather than economics, it is an interest-
ing question especially in the age of globalization and Piketty’s book [25] of
popular science and also earlier [38] research. Please note that it is not only de-
bated how narrow or wide an income distribution should be. Looking into the
details, it is even not clear how uneven the distribution is [39]. Most scrutinized
(and envied) are the super-rich as shown in e.g. [39]. However, their total wealth
3There is however at least weak evidence that both factors contribute. In all societies
infidelity is
considered bad. If nurture (even in the womb) is essential for IQ only,
it would be uninteresting who
impregnates whom (except for the pleasure). If it is purely nature, just the maid and not the comp
a-
nion should be selected. If both have some effect, it would be good if maid and companion are ide
n-
tical. So high IQ parents could pass the high IQ more likely to their children while infidelity would
lead to a much narrower IQ distribution. Please note that such trait as fidelity can also be “inh
e-
rited.” It is known as cultural evolution,
and is much less scrutinized than natural selection. It has
been e.g. explicitly proven that in societies were
female genital mutilation exists but not mutilated
women are accepted too, there mutilated women hav
e more kids so that this “culture” unfortunately
spreads [37].
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Figure 3. Distribution of net monthly household income in Germany 2017. Data from
statista.com.
varies very much over the years. Furthermore, the ratio of income from wealth
to work is often overstated.
Here we will consider income only, be it from wealth or work. Within our ac-
curacy goal this distinction is unimportant. We will also always go for net in-
come. Of course the net income depends on the political system and things like
minimum wages and social support. On the other hand, people try to increase
their net income. Again within our accuracy it does not matter very much.
In subchapter 3.1 we will quickly state how values like the one from Figure 3
are classically fitted by a Gaussian distribution. Choosing a Gaussian is consis-
tent with the results from chapter 2. The IQ is strictly Gaussian distributed.
Other characteristics like strength or health are also Gaussian. In a just world
income should therefore be distributed Gaussian. And as long as there is a free
labor market without frictions (unfortunately not existing in the real world), jus-
tice will appear automatically. Redistribution may benefit the poor too little, but
it will not enhance the income of the rich.
We will show that the classical approach using the mean and standard devia-
tion is wrong for principle reasons. A least square fit or better least absolute val-
ue fit [16] gives different numbers. However, both ways cannot explain the exis-
tence of households with a monthly (net) income of €10,000 or more. Though
these are rare they exist in numbers unexplainable by any Gaussian distribution.
And there are quite a few households having such income purely from labor.
Even deviating from a Gaussian in a way that negative incomes are impossible
does not fix the problem.
In subchapter 3.2 we explain why one should not take values from the col-
umns of Figure 3 for a fit. Such ups and downs come from tax rates and social
support up to a certain income, etc. Anyway, it is completely uninteresting to get
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a most precise fitting mathematical function of Figure 3. It will be a polynomial
of nth degree were nothing can be learned from. But even with the results from
subchapter 3.2, households of a monthly income of €10,000 and over should not
exist.
In subchapter 3.3 we will construct a new model by using the historical results
of Chapman [26]. It is essentially a mixture of a Gaussian and exponential dis-
tribution. In doing so one will get quite some households with a monthly income
of € 10,000 and over.
3.1. The Classical Gaussian Fit
Finding a Gaussian describing Figure 3 is usually considered trivial. Just calcu-
late the mean and the standard deviation in order to get
μ
and
σ
of the Gaussian
distribution for the distribution
( )
pE
of the net earnings
( )
( )
2
2
2
1e
2
x
pE
µ
σ
σ
−
−
= ⋅
π
(4)
Please note that we have only an income range in each column of Figure 3.
This is particularly difficult for the first and the last column in Figure 3. They
have (theoretically) average values within the intervals
( )
,900 €−∞
and
[
)
6000 €,∞
, respectively. Of course, the raw data leading to Figure 3 will reveal
the true averages. As an assumption one may take 450 € for the first and 7000 €
for the last column. For all the others it is fair to assume the middle of the re-
spective interval as the average. In doing so one will find
2664.03 €, 1686.12 €
µσ
≈≈
(5)
Though these results are pretty simple to get, a least square fit of Figure 3
with a Gaussian should be the more general choice. It would be the only way for
a fit with an arbitrary distribution. Here one must minimize
( ) ( )
2
2
900 €
6000 €
3792 2276
d d min
40205 40205
Ep E Ep E
∞
−∞
− ++ − →
∫∫
(6)
with respect to
μ
and
σ
contained in
( )
pE
of Equation (4). The (numerical)
solution of the minimization yields:
,2275.75 € 1302.15 €
µσ
≈≈
(7)
As one sees there is quite some difference between the values in Equation (5)
and Equation (7). Please note that this has nothing to do with the assumption of
450 € and 7000 € average of first and last column, respectively. It is easily possi-
ble to show that
any
assumption for the averages of the first and last column of
Figure 3 will not make the values in Equation (5) identical to the one of Equa-
tion (7). In order to get identical values in Equation (5) and Equation (7) one has
to assume that the averages of the first and last column are complex:
333.929 € 1072.79 €i±
and
333.929 € 1787.35 € i
This is of course nonsense.
With the imaginary parts even bigger than the real parts, one cannot speak of a
small deviation due to the numerical calculation.
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There is obviously a mistake in finding
μ
and
σ
in a Gaussian distribution by
using the mean and standard deviation of the given data, even if the raw data
(not clustered) were used. And this mistake can be quite big. We stress it here
because this (wrong) procedure is standard for finding values of
μ
and
σ
in most
non-mathematical sciences. The reason behind it is quite simple. If the given
data are for sure
exactly
Gaussian, it is correct to assume that
μ
equals the mean
and
σ
the standard deviation. However, this is something which will (almost)
never be the case. The standard deviation is a non-linear function of the data.
Although approximately Gaussian distributed data will be nicely fitted by a
Gaussian, the standard deviation of these data is not necessarily an approxima-
tion for
σ
. It can be quite different as this example shows. Though the mean is a
linear function of the data, it will not be identical to
μ
either. This has to do with
the fact that
σ
and
μ
cannot be fitted independently.
Just for completeness we note that the least square fit is an approximation on-
ly as has been shown in [16]. Taking the squares yields positive values but it is
arbitrary. Why not taking the fourth or sixth power? Correctly one has to take
the absolute value:
()( )
900 €
6000 €
3792 2276
d d min
40205 40205
Ep E Ep E
∞
−∞
− ++ − →
∫∫
(8)
This minimization is numerically quite challenging. Maybe that is the reason
why the (wrong) least square fit and not the (correct) least absolute value fit is
normally used. Of course, in many cases least square fit and least absolute value
fit will lead to similar results. However, here it is not the case. Though numeri-
cally tough, it is a well-defined problem with a unique solution. For our values
we will get:
2096 €, 1228 €
µσ
≈≈
(9)
The deviation of Equation (9) from Equation (7) is far from being negligible.
And Equation (9) is even more different from Equation (5) than Equation (7).
Though it is not the main part of this paper, we have two statements especially
for non-mathematically sciences using statistics:
• Taking the standard deviation for
σ
and the mean for
μ
to fit a Gaussian dis-
tribution like in Equation (4) is generally wrong.
• The least square fit is an approximation only. The correct least absolute value
fit will lead to quite different results especially in non-linear fits where the
data vary over orders of magnitude.
Some critics might say that our Gaussian approach is faulty from the begin-
ning. This is because a Gaussian distribution runs from minus to plus infinity.
And negative incomes are impossible. Please note that this is always the case be-
cause nothing runs from minus to plus infinity. The IQ shows a perfect Gaussian
distribution though there is no negative IQ. With income it is not absurd to as-
sume negative values. With e.g. very low IQ and/or very poor health it is not
possible to survive without support from the community which is nothing but a
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negative income. But be it as it may, of course one can start with a Gaussian
running from zero to infinity. Because it needs a new normalization, Equation (4)
will read now
( )
( )
2
2
2
1e
2 1 erf 2
E
qE
µ
σ
µ
σσ
−
−
= ⋅
π⋅ +
(10)
where
()
erf x
denotes the error function defined as
( )
2
0
2
erf d e
xt
xt
−
=π
∫
(11)
within this approach it is also possible to get
μ
and
σ
from the mean and stan-
dard deviation. However, the mean and standard deviation are given by Equa-
tion (25) and Equation (26), respectively. Now we have to solve two coupled
non-linear equations:
( ) ( ) ( )
22
2664.03 € , 1686.12 € ,ms
µσ µσ
=∧=
(12)
The solution of the couple Equation (12) is possible numerically only. As a
result one will get
1946.86 €, 2180.27 €
µσ
≈≈
(13)
Please note that getting
μ
and
σ
this way is incorrect for the same reason as the
result in Equation (5) is wrong.
As stated above the correct way finding
μ
and
σ
is a least square fit. It takes the
form
( ) ( )
2
2
900 €
0 6000 €
3792 2276
d d min
40205 40205
Ep E Ep E
∞
− ++ − →
∫∫
(14)
The solution can be obtained numerically only:
2104.98 €, 1620.90 €
µσ
≈≈
(15)
As explained above and in [16], the least square fit is at most an approxima-
tion. To be precise one has to use a least absolute value fit.
I
.
e
. one has to solve
the following:
( ) ( )
900 €
0 6000 €
3792 2276
d d min
40205 40205
Ep E Ep E
∞
− ++ − →
∫∫
(16)
A numerical solution yields:
1625.60 €, 1707.45 €
µσ
≈≈
(17)
This last result can be considered “exact” within our fit procedure. In the next
subchapter we will learn that the fit procedure does not necessarily give a result
from which one can learn something.
3.2. How to Fit Income Distributions
With Equation (17) we have the best possible Gaussian fit for the distribution of
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Figure 3. The question is whether it is useful? The absolute value fit is a Gaus-
sian distribution looking most similar to the given distribution in Figure 3.
However, there may be some ups and downs in the distribution due to tax rates,
minimum wages or social benefits. These are uninteresting here. We want to
show whether our assumed Gaussian distribution will be in accordance to global
measures such as Gini or median. One may think of many factors useful for this
purpose. But they should also be available in standard data banks. For OECD
countries many data are available for free in OECD. Stat which can be found in
the internet under stats.oecd.org. From this we gathered data for four different
countries. All are based on net monthly income per household in € at German
PPP. (This is our contribution against the $ dominance) The first two columns
in Table 1 (number of households and mean income) are not statistical meas-
ures. They are gauge factors. As we always consider income per household, it
would be smart divide any income by the particular mean income. It would lead
to a mean income of 1 in every country. One would also get rid of any exchange
rates or PPP. However, we avoided this approach in order to have results in real
currency units which might be convenient for many readers especially econo-
mists.
The last two columns in Table 1 (median income and P90/P10)4 are the statis-
tical measures we have chosen. The median is the “middle income.” It is the in-
come to choose if one has only one number instead of the entire distribution.
Some people take the mean as an alternative measure for it. Why this is wrong
can be found in [16]. Additionally or as an alternative, it would be good to con-
sider the Gini coefficient in addition or instead of the median. However, includ-
ing the Gini makes it numerically extremely complicated here. Especially for the
extended model of subchapter 3.3 the authors were unable to perform the ne-
cessary numeric calculations. For the reason behind it please see Chapter 4. The
P90/P10 ratio is a measure for the rich. Were the distribution completely sym-
metric like an ordinary Gaussian of Equation (4), the ratio would be one. We
would love to have something like a P99/P1 ratio. But we did not find such
numbers in free data banks.
One might argue that the standard deviation is also a global measure. So fit-
ting with it like in subchapter 3.1 should not be a bad idea. Firstly, we have to
Table 1. Data from OECD.Stat 2016 taken from stats.oecd.org on 13/02/20, all at GER
PPP.
# of households
mean income
median income
P90/P10
GER
4,0749,525 2148.83 € 1888.50 € 3.7
USA
126,220,000 2957.65 € 2394.29 € 6.1
UK
27,800,000 1915.89 € 1567.63 € 4.2
DK
2,686,035 2213.79 € 2006.73 € 2.9
4Most quantities like mean or median should be well-
known. The P90/P10 ratio is the ratio of the
income which is lower for 90% of the people to the income which is higher for 90% of the people.
More can be found in Chapter 4, Equation (32) and Equation (33).
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note that the standard division is a quite complicated expression as given in Eq-
uation (26) or even more complicated as indicated below Equation (26). Se-
condly, the standard deviation has a very limited meaning here. In measure-
ments like the mass of an elementary particle one will expect
one
value. In sever-
al measurements one will get different results though they should be equal. It
does make sense to build a mean. And one should test whether the measured
data have a Gaussian distribution. (If not, something is systematically wrong)
The Gaussian distribution should be narrow if the measurements are accurate.
As a measure for accuracy one may take the easily obtainable standard deviation.
However, in income distributions such parameter does not make sense at all.
There is a reason why we have an income distribution. It is not an error in mea-
surement. Here we assumed that it has to do with the distribution of skills such
as IQ. The deviations will teach us (in subchapter 3.3) what other effects rather
than skills contribute.
In an extreme socialist country, it may be stated that everybody should have
the same income as ordered by a socialist income committee. In such country it
would be a reasonable idea to measure the real income. The mean should be the
value set by the committee and the standard deviation tells how good the social-
ist ideology has been implemented. This shows another misunderstanding of
statistics in non-mathematical sciences. But as stated in the very beginning of the
introduction, 100 years old books like [2] have never found readers there.
The fitting with median income and P90/P10 from Table 1 must be at least an
absolute value fit in accordance with Equation (16). However, as the quantities
of Table 1 have a different dimension (€ and no dimension) it is formally im-
possible to just minimized a sum of it. Depending on the chosen dimension (e.g.
€, Cent) the result will be strikingly different5. Therefore we have to take the
rel-
ative
deviation. Furthermore, the exact mean is a constraint. Put this together we
have to minimize the following with respect to
μ
and
σ
:
( )
( )
( )
( )
( )
( )
90/10
90/10
, median , P90 P10 min
11
, median , P90 P10
22
np
np
µσ µσ
µσ µσ
−−
+→
++
(18)
with the constraint
( )
, meanm
µσ
=
(19)
( )
,n
µσ
,
( )
90/10
,p
µσ
, and
( )
,m
µσ
must be taken from Equation (31), Equa-
tion (33), and Equation (25), respectively. The values for median, P90/P10, and
mean come from Table 1. Equation (18) and Equation (19) look quite simple.
However, inserting the functions from Equation (31), Equation (33), and Equa-
tion (25), it is already much more complicated. Theoretically one might solve
Equation (19) for
μ
or
σ
and insert it in Equation (18). Unfortunately, there is
neither an analytic solution nor one with arbitrary numerical accuracy. But of
5The problem is not only the dimension. Even with dimensionless quantities
one can be very big
(e.g. around 100) and the other one very small (e.g. around 1). Though both deviations are equally
important, an ordinary absolute value fit will stress the deviation of the big quantity much more.
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course, Equation (19) defines a function
( )
mean,
µσ
or
( )
mean,
σµ
, at least
piecewise. Inserting one of these functions in Equation (18) leads to one variable
in Equation (18). Instead of minimizing one might set Equation (18) to zero.
However, unlike polynomials highly non-linear functions generally need not to
have zeros (even complex ones). And indeed at least with the values from Table
1 there will be no zeros. The classical approach to minimize Equation (18) by
differentiation with respect to
μ
and
σ
and setting to zero is excluded because of
the absolute values. Furthermore, differentiating the functions
( )
,n
µσ
and
( )
90/10
,p
µσ
gives far from trivial results. In addition there is the constraint of
Equation (19).
Nevertheless, Equation (18) and Equation (19) are a well-defined problem
with a solution. The constraint of Equation (19) makes it a one-dimensional
problem in two dimensions. In Figure 4 we have e.g. plotted
( )
( )
( )
( )
( )
90/10
90/10
P90
,
, median P10
11 P90
, median ,
22 P10
p
n
np
µσ
µσ
µσ µσ
−
−+
++
(20)
with the constraint of Equation (19) and the values of Germany from Table 1.
The minimum of the red curve in
μ
-
σ
-space is our desired minimum. Deter-
mining this minimum numerically is not very complicated, once the functions of
Equation (18) and Equation (19) are programmed. Using a software like Mathema-
tica can be very helpful here. The minimum in Figure 4 is at
2106.3705 €
µ
≈
and
992.90175 €
σ
≈
. The expression in Equation (20) takes the value of about
0.119. Instead of using the median one could have used the Gini
( )
,g
µσ
of
Equation (36). Though this will consume more CPU time, the result is surpri-
singly similar6. The minimum in a correspondingly changed Figure 4 will be at
2106.3698 €
µ
≈
and
992.90455 €
σ
≈
. The expression in Equation (20) (with
( )
,g
µσ
instead of
( )
,n
µσ
) takes the value of about 0.163.
The similarity makes two things likely. Firstly, our fit procedure is not just
luck. Secondly, the income distribution is essentially Gaussian as long as global
measures like median or Gini are concerned. It is also intriguing to compare the
result of the fit procedure of subchapter 3.17 with our results here. It will show
how wrong the approach of subchapter 3.1 is. The first column of Table 2 gives
the “exact” data from OECD.Stat. The calculated data from the fit here (fit 3.2)
shows a quite good match. Because of the constraint in Equation (19), the mean
is of course identical. The calculated data based on the results of the previous chap-
ter (Equation (17)) deviate dramatically in the Gini and P90/P10 ratio. Taking
them at face value, Germany’s income distribution is as unequal as in the USA.
The results for all four countries are summarized in Table 3. Please note that
6Mathematically this is explained easily. In both version
( )
90/10 ,p
µσ
was exactly 3.7,
which implies
a fixed value for
μ
and
σ
because of Equation (19).
7We are aware that the data from
Figure 3
are from 2017 while data from 2016 are used here. U
n-
fortunately, we have for no data from 2016 for the Graphics in
Figure 3. We also don’t have data
from 2017 for the quantities of
Table 1
. But the difference seems to be unimportant.
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Figure 4. Plot of Equation (18) with constraint from Equation (19) with values for Ger-
many.
Table 2. Comparison between classical fit (3.1) and new procedure (3.2) for GER.
OECD.Stat
fit 3.2
fit 3.1
mean
2148.83 € 2148.83 € 2147.55 €
median
1888.50 € 2127.46 € 1993.36 €
P90/P10
3.7 3.7000 8.1030
Gini
0.294 0.2496 0.3508
Table 3. Summary of the four countries considered here.
μ
σ
Gini this fit
Gini OECD.Stat
GER
2106.37 € 992.902 € 0.250 0.294
USA
2631.14 € 1918.25 € 0.315 0.391
UK
1852.56 € 967.013 € 0.267 0.351
DK
2202.08 € 848.398 € 0.213 0.261
μ
and
σ
are just fit parameters. They do not have the meaning of mean and
standard deviation like in the Gaussian distribution of Equation (4). Due to the
constraint of Equation (19) we have in effect only one fit parameter. With one
parameter only, the Gaussian model used in this subchapter describes reality
nicely. This is especially surprising as an income distribution is not a result of
one or a few constants as it is often the case in e.g. physics. Here many million
people interact, and all have a free will. Obviously we have a quite just world. As
most skills (especially IQ) are Gaussian distributed, the income is in accordance
with it.
Having a closure look at Table 3 one can see that the Gini from our fit is al-
ways smaller than the observed one. The income in our modelled world is more
equally distributed than reality shows. As stated in the introduction already, with
a pure Gaussian distribution very wealthy people cannot exist. We have some-
thing like a fat tail. To solve this puzzle is the main point of the following sub-
chapter.
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3.3. The Extended Model
In the previous subchapter we have shown how to fit an income distribution
with one effective fit parameter. The results are quite fine. However, they deviate
for the rich. With the numbers of Table 1 and Table 3 it is easy to calculate the
number of households (#
HH
) with a net income of 10,000 € or more per month
by integrating
( )
qE
in Equation (10) from 10,000 € to infinity. One will get the
following:
8
10 14
# 3.86 10 # 8437
# 5.14 10 # 5 2
,
. 3 10,
GER USA
UK DK
HH HH
HH HH
−
−−
=×=
=×=×
(21)
A net household income of 10,000 € is for sure not common in neither of the
four countries as the median is roughly four times lower in each of the four.
However, it does exist8, and for sure it is possible without any income from
wealth. In contrast to it, Equation (21) teaches that such households should not
exist in three out of four countries considered. And even in the USA the number
is incredibly tiny. It should be bigger by at least a factor of 103.
Though the necessary skills to create income are fairly well Gaussian distri-
buted, at least higher incomes are much more likely than any Gaussian distribu-
tion would predict. One thing is the income of a
leader
. This is a person who has
subordinates. And part of the value created by these subordinates will contribute
to the income of the leader. Generally this is just because only the leader has the
skills enabling the subordinates to create so much value. It is an explanation why
leaders may have hourly wages several times the wages of their subordinates.
However, these leadership skills will also show a Gaussian distribution. It will
not lead to the observed
fat tail
in income distributions from work. It is also im-
possible that the households with lower income are betrayed by their bosses. It
would be possible in totalitarian states but for sure not in the four OECD coun-
tries considered here. Democracy and a working labor market will always lead to
justice. If a boss pays too little, the most skilled workers will leave making the
company less profitable. It is the same as with ordinary goods. The market de-
termines the price.
That people will allow for a redistribution, be it by tax or even free giving is
not impossible even in market economies. Using an extended Edgeworth cube
[40] it can be shown that it does make sense to consider “social peace” as a good
people are willing to trade. Though this is a reasonable and likely effect, it would
lead to an even narrower distribution especially for lower incomes and some-
thing like a “slim” tail.
However, without a working labor market everything is possible. Considering
median incomes it is hard to imagine that the labor market is not working in this
area. This is probably even accepted by trade unions and the like. They demand
minimum wages, child support, etc. just because the free market is creating a too
8Unfortunately the authors have no data on households with an income of over 10,
000 € per month.
However, it should be reachable with two academics working full time in above average
positions in
all four countries.
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broad spread of incomes. A labor market works if there are many similar posi-
tions and many potential people able to fill the position. For people having sev-
eral times the median income, there are less and less potential positions. Even
if they imagine that they are creating much more value for their bosses, it will
be much more difficult to find an alternative. One chance will be a spinoff. But
it is rarely realistic. Rightly, there is also no lobby for people having several
times the median income. They may suffer from injustice but not from finan-
cial hardship.
Now we have shown why a labor market may partly not work. It is another
question who takes why an advantage of it. The answer to this question is only at
first glance obvious. Unlike e.g. chimpanzees, humans are not homines
oeconomici. They are not altruist either. They go for more money in order to
become richer than their peers.9 It is the case especially within rich people. This
is neither new nor is it just a gut feeling. Recently historic data from the second
half of the 19th century has been analyzed in detail [26]. Such historic data have
the advantage of being “natural.” They are not biased by modern social policies.
The essential result from [26] is, that the rich are even willing to give to the
poorer ones as long as the rich remain distinct in income.
Putting this into our Gaussian distribution of Equation (10) would mean a
sigma growing with income. Though we do not say that this ansatz is not worth
pursuing, it has two disadvantages. Firstly, it comes a little bit unmotivated. Se-
condly, it bears technical problems. If
( )
E
σσ
=
in Equation (10), the leading
power of
( )
E
σ
must be less than linear. In other words,
( )
a
EE
σ
∝
with
1
a<
in order to make normalization possible. With such low powers the effect
is pretty tiny (besides making the math complicated especially for
12 1a<<
).
This technicality can be fixed by introducing an income cutoff. Having a maxi-
mum possible income in the world is even realistic. Our income distribution like
in Equation (10) is in that sense unrealistic because it will give a (very small)
probability that someone has ten times the world income. On the other hand,
setting a cutoff value seems arbitrary. It looks like an unmotivated fit parameter.
Therefore we did not pursue this ansatz.
Our model used goes back to the effect that richer people tend to be leaders
getting their money from advising subordinates. Getting up the hierarchy the
number of people will be less and less. This alone would lead to an exponential
distribution like in Equation (3). To make the number of assumptions as small
as possible, we say that everybody tries to get money through subordinates.
However, the will to do it
and
the possibility is proportional to income. This
leads to an
2
E
term in front of the exponential distribution:
2
e
E
E
λ
−
⋅
We have to add this modified exponential distribution to our Gaussian one of
Equation (10). After normalization we will have
9Of course there are many
more models. Though our purpose is to show the mathematics behind
modeling, the example presented here supports this assumption.
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( )
( )
2
2
2
23
1ee
4
2 2 1 erf 2
EE
E
qE
µ
σλ
λ
µ
σσ
−
−−
= ⋅ +⋅
π⋅ +
(22)
Formally we have now an identical optimization as given in Equation (18) and
Equation (19). Instead of
( )
qE
from Equation (10) we have to use Equation
(22) now. Our problem is the following:
( )
( )
( )
( )
( )
( )
90/10
90/10
, , median , , P90 P10 min
11
, , median , , P90 P10
22
np
np
µσ λ µσ λ
µσ λ µσ λ
−−
+→
++
(23)
with the constraint
2
2
1
Γ,2
22
2mean
33
1 erf 2
r
µ
σ
µ
λµ
σ
−−
= +⋅
+
(24)
( )
,,n
µσ λ
and
( )
90/10
,,p
µσ λ
are given in Equation (39) and Equation (40),
respectively. This minimization problem is well defined. The constraint is even
simpler as in Equation (19). However, the highly non-linear functions must be
determined numerically which consumes quite some CPU-time and RAM. The
reason behind it is stated in the next chapter. There we also explain why it is
virtually impossible to use the Gini
( )
,,g
µσ λ
in Equation (23).
Making a 3D-Plot of Equation (23) (with
λ
substituted via Equation (24))
shows the areas of local minima. One should not just calculate points and con-
nect them. Gradients should be considered too. This will make sure that there is
really and minimum. It will increase the number of points to be calculated by
perhaps a factor of ten times ten. But it is necessary because the minimum will
be typically at a non-analytic point due to the absolute values in Equation (23). A
software like Mathematica is very helpful here. As the problem lies in the inverse
functions, Mathematica analyses the original functions and tries to get at least
piece-wise analytic inverse functions. This is of course not always possible. So
one has to choose by hand which interval should be considered. Even this way it
costs quite some CPU-time. And it is neither straight forward nor can it be au-
tomated. Having identified the area with the smallest local minimum it has
proven practical to find its value iteratively by guessing the value for
σ
and then
making a one-dimensional plot over
μ
which will yield a minimum at a certain
value of
μ
. With this value of
μ
one can plot over
σ
, and so forth until sufficient
accuracy has been reached.
In Table 4 we summarized the data for our four countries. We are confident
the digits shown there are accurate. But we are far from being able to produce
results with arbitrary accuracy. Even this accuracy is pretty tedious and at least
the authors see no way to automize it. The most interesting column in Table 4 is
the last one. Now the number of households with a net monthly income of 10,000 €
or more is at least about correct though most likely still understated. As indicated
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Table 4. Data for the coefficients and the number of housholds with ≥104 € per month.
μ
σ
λ
# of households
GER
1719 € 539 € 859 € 1.44∙104
USA
1772 € 928.2 € 1361 € 1.43∙106
UK
1341 € 416 € 830 € 6.96∙103
DK
1955 € 340 € 824 € 6.27∙102
Figure 5. Gaussian and exponential part of the income distribution for Germany.
in footnote 7, we do not know the exact number of these households. Else it
would be smart to use it as a quantity to be fitted directly.
In Figure 5 we have plotted the Gaussian (blue, narrow) and exponential
(beige, wide) of the income distribution for Germany separately. Both parts peak
at around the median income10. The total distribution is the sum of both curves.
The exponential part clearly enlarges the peak at the median income. The num-
ber of households getting about the median income also increases. In addition
one gets a “fat tail.”
4. Derivation of Some Equations
In subchapter 3.1 we defined in Equation (10) a “Gaussian” distribution
q
which
runs from zero to infinity. Of course it is straight forward to calculate the mean
( )
,m
µσ
and variance (=square of standard deviation)
()
2
,s
µσ
:
( ) ( )
0
,dm EE q E
µσ
∞
= ⋅
∫
( ) ( )
( )
( )
2
2
0
,d ,
s EE m qE
µσ µσ
∞
=−⋅
∫
These integrals are tedious but straight forward to solve. A lengthy calculation
yields
10Formally the Gaussian part (blue curve) peaks at
μ
and the exponential part (beige curve) at 2
λ
.
This makes the USA in Table 4 a particular case.
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( )
2
2
1
2,
22
,
1 erf 2
r
m
µ
µσ
µσ µ
σ
−Γ −
=
+
(25)
( )
( )( )
()
()
2
2
2
22
3
22 22
22
2
22
22 22
12
, e 1 erf 2
1 erf 2
2 erf 2 erf
22
11
4Γ,,
22
22
5 erf 2 er
2
rr
s
−
= +
π
+
+ ++ − ++ +
+ − − Γ−
− ++ + +
µ
σ
µ
µ σ µσ σ
µ
σ
µµ
µσ σµµσ µσ
σσ
µµ
µσσ
µ
µσ µσ σ
2
2 22
22
2 22
f2
1 11
4, , ,
2 22
2 22
r rr
− Γ− + Γ− Γ−
µ
σ
µ µµ
µµ
σ σσ
(26)
The error function erf has been defined in Equation (11) already.
r
Γ
is the re-
gularized incomplete gamma function with
( ) ( )
( )
,
,
r
ax
ax a
Γ
Γ=
Γ
(27)
where
( )
,axΓ
and
( )
aΓ
are incomplete and “normal” gamma function, re-
spectively with
( ) ( ) ( )
( )
11
0
1
1
, deand de !
k
at at
k
x
a x tt a tt kka
∞∞
∞
−− −−
=
−
Γ = Γ= + +
∑
∫∫
(28)
Please note the sum in Equation (28). Normally, the gamma function is dis-
played by an integral only. But this only works for positive arguments. In the en-
tire paper the first argument of
r
Γ
is −1/2. So we have
32
1111
,Γ, de
22
22
t
r
x
x x tt
∞−−
Γ−=− −=−
ππ
∫
(29)
Equation (29) does not lead to much simplification in the numerical calcula-
tions.
To be consistent with an absolute value fit one should consequently write
( ) ( ) ( )
0
,d ,s EE m q E
µσ µσ
∞
=−⋅
∫
This integral is easily solved by splitting it in one running from 0 to
( )
,m
µσ
and one running from
( )
,m
µσ
to
∞
. Though the solution is straight forward,
it is much more complicated than Equation (26) and has about double its length.
In subchapter 3.2 we used the median, which we will denote
n
here (because
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m
is already used for the mean). For any distribution
( )
qE
normalized to one
the median is
( ) ( )
1
1with
2
Qx
n Q qx
x
−
∂
= =
∂
(30)
As always, the exponent −1 denotes the reverse function with
()
( )
1
f fx x
−
=
.
Applying this to the
( )
qE
of Equation (10) (Gaussian from zero to infinity)
leads to
( )
11
, 2 erfc 1 erfc
22
n
µ
µσ µ σ σ
−
=+⋅ −
(31)
where erfc is the complementary error function with
( ) ( )
erfc 1 erfxx= −
.
The P90/P10 (abbreviated as
90/10
p
) ratio one gets for a general (normalized)
distribution
( )
qE
( )
P10 1
11
d P10
10 10
Ep E Q
−
−∞
=⇒=
∫
and
( )
1
P90
11
d P90
10 10
Ep E Q
∞−
=⇒=−
∫
to
1
90/10 1
1
10
1
10
Q
p
Q
−
−
−
=
(32)
where
Q
is defined as in Equation (30). Applied to the
( )
qE
of Equation (10)
(Gaussian from zero to infinity) leads to
( )
1
90/10
1
11
2 erfc erf
10 10 2
,99
2 erfc erf
10 10 2
p
µ
µσ σ
µσ µ
µσ σ
−
−
+⋅ +
=
+⋅ +
(33)
Here erf and erfc are defined as in Equation (11) and Equation (31), respectively.
Though not used here, a few words about the Gini coefficient
g
. It is defined
for distributions
( )
qE
running from zero to infinity only.
( )
( )
( )
10
00
d
1 2d d
Ef
q
gf q
ηη η
ηη η
∞
⋅
= − ⋅
∫
∫∫
(34)
with
( )
E Ef=
as inverse function of
( ) ( )
0
d
E
fE q
ηη
=∫
(35)
Taking the
( )
qE
of Equation (10) leads to after a tedious but straight forward
calculation
M. Grabinski, G. Klinkova
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Applied Mathematics
( )
2
2
1
2
2
2
2 erf 2erfc 1 erf 2
2
2
2 e erf erfc 2
22
1 erf 2
,1 1
1 erf , 2
22
2
r
g
µ
σ
µ
σσ
σ
µµ
µσ µ µ
σσ
σ
µσ µµ
µσ
σ
−
−
⋅ +−
+ +⋅ + −
π
π+
= −
+ Γ− −
(36)
r
Γ
is defined as in Equation (27) and Equation (28). Though this expression
looks pretty clumsy, it consists of functions which can be evaluated with arbi-
trary accuracy. This is in contrast to the Gini coefficient we would need in sub-
chapter 3.3.
Calculating the mean
( )
,,m
µσ λ
for the distribution of Equation (22) is
simple because an integral over a sum is the sum of the integrals.
( )
2
2
11
1,
22
23
,, 2
1 erf 2
r
m
µ
µσ
µσ λ λ
µ
σ
−Γ−
= +
+
(37)
The constraint from Equation (19)
( )
, , meanm
µσ λ
=
can be solved for
λ
.
2
2
1,2
22
2mean
33
1 erf 2
r
µ
σ
µ
λµ
σ
Γ− −
= +⋅
+
(38)
with Equation (38) the additional parameter
λ
can be eliminated.
To calculate the median
( )
,,n
µσ λ
for the distribution of Equation (22) is
formally like in Equation (30).
( )
( )
1
22
2
1
2
erfc
e2
with 1 2 2
44 2erfc 2
x
nQ
x
Qx x x
−
−
=
−
=− ++ −
−
λ
µ
σ
λλ µ
λ
σ
(39)
Unfortunately there is no closed inverse function of
( )
Qx
. Building an inverse
function is numerically simple. But the amount of data is very big within our
problem. Of course, we can insert
λ
from Equation (38) into
( )
Qx
of Equation
(39). This leaves us with two parameters
μ
and
σ
which are supposed to be de-
termined eventually. In the sense of a Mont Carlo simulation one may assume
103 different values for each parameter. So we have 106 different functions
( )
Qx
.
In order to build an inverse function, we may have to assume 103 different values
for
x
. It leaves us with 109 values which must be calculated. As each calculation
contains integrals, this will consume quite some computing power.
Building the P90/P10 ratio for the distribution from Equation (22) causes the
same problem. Formally
( )
90/10 ,,p
µσ λ
is given like in Equation (32)
M. Grabinski, G. Klinkova
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( )
( )
( )
1
90/10 1
22
2
1
10
,, 1
10
wit
erfc
e2
1 22
44 2er
h
fc 2
x
Q
p
Q
x
Qx x x
λ
µσ λ
µ
σ
λλ µ
λ
σ
−
−
−
−
=
−
=− ++ −
−
(40)
The parameter
λ
can be eliminated with Equation (38). Again, we will need the
inverse function of
( )
Qx
given in Equation (40). In order to find it we also
have to calculate 109 data points. (However, it is mostly an identical calculation)
Just for completeness we will also show how to calculate the Gini coefficient
for the distribution given in Equation (22). The Gini
( )
,,g
µσ λ
is formally
given in Equation (34). The necessary function
( )
fE
in Equation (35) is easily
determined to be
( )
( )
22
2
erfc
e2
1 22
44 2erfc 2
E
E
fE E E
λ
µ
σ
λλ µ
λ
σ
−
−
=− ++−
−
(41)
For obvious reasons
f
and
Q
are identical. Putting it together we have
()( ) ()
( )
1
00
2
,, 1 d d
,,
Ef
g fq
m
µ σ λ ηη η
µσ λ
=−⋅
∫∫
(42)
( )
,,m
µσ λ
is given in Equation (37).
λ
can be eliminated with Equation (38).
( )
Ef
is the reverse function from
( )
fE
given in Equation (41), and
( )
q
η
is defined in Equation (22). Inserting all this will make Equation (42) look much
more complicated. But this is not the real problem. Getting the function
( )
Ef
one needs to make 109 calculations as stated above. Furthermore we have to take
two integrals in Equation (42). Even going for a not too high accuracy, we may
divide each integration interval into 100 pieces. This leads us with 109+2+2 data-
sets. Even storing these 1013 data is critical. Therefore, a fit with the Gini instead
or in addition to the median is impossible. Making some simplifications was not
possible; at least the authors did not find any way. Finding the inverse function
( )
Ef
of
()
fE
may involve far less than the mentioned 109 data. A smart
software like Mathematica is able to find gradients in
( )
fE
and also proves
continuity. With it, it is (mostly) able to construct an
( )
Ef
in a much simpler
way with the required accuracy. However, this
( )
Ef
is useless for the two
(numerical) integrations in Equation (42).
5. Conclusions and Further Research
We have shown how to use distributions, and what conclusions can be drawn.
We have taken two examples and used mathematics which is well-known for
over 100 years. This alone would disqualify our work as a journal publication.
M. Grabinski, G. Klinkova
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But we have chosen two particular examples: IQ distribution and income distri-
bution. These examples belong to psychology and neighboring fields, and eco-
nomics and finance and the like. These disciplines have in common that they use
distributions frequently though they are not too close to mathematics. Over
simplified two statements are prevalent there:
• Every distribution is a Gaussian distribution.
• The mean and the standard deviation determine
μ
and
σ
, respectively.
At most a Χ2 test is applied in order to prove Gaussian behavior. Applying the
central limit theorem wrongly sometimes produces Gaussian distributions which
are by no means justified. The statements make life easy, but they are wrong and
may lead to false conclusions. To show examples for it was the main motivation
to write this paper.
Our first example attacks the first statement. IQ is distributed almost perfectly
Gaussian. At first glance that seems to confirm the statement. However, if dis-
tributions are always Gaussian, a Gaussian IQ distribution is a tautology. A
Gaussian distribution appears only if something happens by chance. It is very
difficult to produce such distribution otherwise, as everybody might know once
trying to fake lab data with the tools of the early 1980 ties. The accidently mixing
of genes generates a certain IQ. In chapter 2 we concluded from this that IQ
must be inherited or at least not being created by conscious actions. Though this
is assumed by the vast majority of academic psychologist, we have presented
mathematical proof for it.
As our proof is clear-cut, it is hard to imagine any further research. However,
two things may be worth scrutinizing. One is the width of the IQ distribution. It
differs for men and women. But even the total IQ distribution does most likely
not have the same width in every country. The big problem is finding sources of
data for it. Even for the average IQ in different countries there are no complete
reliable data. In developing countries, it is particularly dim. Data for the width
are not available, at least not for the authors. However, many factors might con-
tribute to IQ and especially its distribution. Suspects are the frequency of mar-
rying cousins, fidelity, religion, and many more.
In this paper we came also across cultural inheritance [37]. As at least some
traits are inherited genetically, some are inherited culturally. It is much less scru-
tinized. The distribution of such traits might shed some light on the mechanism.
Our second examples are income distributions. Income is based on skills such
as IQ. Many of them show a Gaussian distribution. Therefore it is a reasonable
assumption that income shows a Gaussian distribution. Subchapters 3.1 and es-
pecially 3.2 confirm it in most points. Puzzling is a “fat tail”. It means that there
are far too many rich than any Gaussian distribution can predict. Before solving
this problem, we showed in subchapter 3.1 that even if assuming a Gaussian dis-
tribution from minus to plus infinity, fitting
μ
by the mean and especially
σ
by
the standard deviation can be tremendously wrong. In subchapter 3.2 we as-
sumed a Gaussian distribution with positives incomes only. Such “half” Gaus-
sian is a much more realistic assumption in many cases ranging from infection
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rates to certain measurement errors. There the standard deviation has nothing to
do with
σ
. In Equation (26) we have displayed a lengthy formula for the standard
deviation in such distribution. It is a (complicated) function of
μ
and
σ
. Though
we got in subchapter 3.2 a decent fit with our half Gaussian, it is still far from
possible to explain the fat tail.
In subchapter 3.3 we introduced in Equation (22) a distribution consisting of
a Gaussian part and a (modified) exponential distribution. We were motivated
to it by the evaluation of historical data by Chapman [26]. It led to a tremend-
ously better description of the fat tails. The predicted number of households
with a monthly income of €10,000 or above increased by a factor ranging from
1016 (DK) to 103 (USA) to much more realistic values. With monthly net in-
comes of €10,000 and especially above, the income from (inherited) wealth will
be more and more important. Therefore it does not make sense to scrutinize
very high incomes in detail within our model.
As a further research within income distributions one could extend the pro-
cedure of subchapter 3.3 to other countries and by using other measures to fit.
As the (numerical) mathematics is very complicated, the authors will not pre-
sume actions in this direction. We also see no way to simplify or automate the
numeric calculations, though it would be very welcome. Maybe tools of
big data
can help.
It would be worthwhile investigating in other areas where the results of [26]
play a major (quantitative) role. Sloppily paraphrasing the results of [26] reads:
“You are rich if you have more money than your neighbor”. That envy influ-
ences behavior is well-known. We would like to look for quantitative effects
other than income distributions. We give a warning to use envy as a variable
which can be done e.g. in game theory. Assuming a value such as 3.7 for envy is
possible but these envy values are not elements of a field (in a mathematical
sense). Though calculations such as addition or multiplications are technically
possible, the results are ludicrous [10].
A quite obvious extension of our results is fat tails in finance. The general
mechanism must be the same. The income from stocks is nothing more than the
sum of the values created by workers. Furthermore, even companies are neither
homines oeconomici nor “machinae oeconomici”. Companies are always led by
humans. Therefore it comes as no surprise that many bosses want to make their
company more profitable than a rival company, even if the total profit of both
companies reduces this way. Quite a few (proud) stock holders will accept it.
The reason why we have not taken the fat tail from finance as an example has
many sources. There are fundamental errors in the work of Fama [3]. To under-
stand them one has to understand quite many newer publications such as [9] to
[11] and [27]. This would have led away from the point we want to make. Com-
menting briefly, the following can be said. Fama assumes that there is a real fair
value for e.g. stocks and the market prices fluctuates around it due to imperfect
information. This would lead to a perfect Gaussian distribution and a fat tail
would be a
big
surprise. But even the fellow Nobel laurate of 2013 (Robert Shiller)
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disagrees with it. (Giving a shared Nobel prize to two contradicting theories in
one year remains a conundrum to the authors) As explicitly shown in [9], the
price of a stock does not fluctuate around an underlying value of the company
considered. On average it overstates the company value several times. Up to now
we are speaking of the fluctuation in time of a single stock. Assuming ergodicity
the fluctuation should be the same over a portfolio of stocks. But assuming ergo-
dicity is not justified even if Fama’s approach was correct. The underlying true
price is not something like an energy minimum. It changes in time differently
for each stock. Furthermore the fluctuations in price are chaotically rather than
by chance as has been shown in [11] and [27]. Chaotic fluctuations look like
chance, but they are distinct [16] though this may or may not play a major role
here. Chaotic fluctuations are deterministic instead random though they look
random. Therefore ergodicity does not need to hold. The origin of these fluctua-
tions is not an adjustment to true market prices but speculation. Though with
some flaws the effect has been analyzed quantitatively in [41] by using Fourier
analysis.
Acknowledgements
This publication is dedicated to the late Thomas Dierks. A distinguished scien-
tist, and high school classmate and comrade of M.G.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this
paper.
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