**Behavioral finance FAQ / Glossary (Distribution)**

This is a separate page of theDsection of the Glossary

Dates of related message(s) in the

Behavioral-Finance group(*):

Year/month, d: developed / discussed,

i: incidental

(statistical) Distribution curve anomalies

03/6i -04/2i,7i + see clusters,

fat tails, kurtosis, asymmetry,

extreme, quant, probability,

random... + bfdef3

Ordering data into ranks and files for the parade,

and taking a picture thatshows

if one head (or tail?) sticks out.

Can looking at that picture help predictwhere all those heads and tails are going?

The quest for order in a sea of data.

A statistical distribution is made by

sorting(related to physical or economic events, or to

numerical data

populations, for example ages, income...) according toincreasing.

or decreasing values

The distribution shows how many times the same values are found.

In other words, a data sistribution indicates the

frequency

of each value, or group of close values.Those frequencies are usually converted into percentages.

For example, the percentage of people who are 0-4 year old.

Same thing for 5-9, for 10-14, and so on.Not to forget Methuselah in the 995-999 class. A typical, albeit

mythical "rare event" (see that phrase).

From distribution to probabilities

Assuming that future dice will behave

like good old pastdice.

Statistical distributions, that give the frequency of past values, are

widely used as one of the main tool to determineprobabilities(see

that word).

In a set of 100 future events, how frequent the same one will

be found ,

Never ? Once ? 10 times ? 90 times ? Always ?

The purpose is to help makeprevisions and decisions, for example

financial decisions.But their use might be

complicated when data are distributed in an(anomaly, singularity..).

irregular fashion

Therefore the first thing to analyze is whether

that "data distribution"* either is totally erratic and uncommon,

* or fits a well-known usual pattern (distribution

law).

This analysis is a three step process:

1) The data are gathered, processed and put in order on a scale

of numbers (usually in the form of a chart).For example, applied to a statistic of people heights,

the analysis compares how many people are about as high as

the average, how many are smaller, and how many are higher.

Those past frequencies are used as probabilities.

A garment shop can thus structure its inventories to match

the sales probability for everysize, taken from

official professional statistics.

OK, but it cannot replace fully (see numeracy bias) its

neurons, its experience and some knowledge of what is

happening in town (an arrival of oversized giants?) and

of its competitors inventories.Mind you, this should not be taken as an advice to sneak

into their premises at night ;-)

The next step is to see

2)how , under what pattern, those data

are "distributed"

Are they

random? (the next occurrence cannot be

predicted from the last one or last ones)

What is their

range?What is the

shape(statisticians would say the mode) of

the curve ?Is it regular (see below "typical distributions": bell curve,

L curve...What is the

mean(the central value for which there are as

many smaller ones than higher ones)?

3)Then this distribution is compared with classical sets / patterns:

(Gauss, Pareto...: see below).Standard distribution curves, distribution functions and

distribution laws

In many cases, and when well applied, this gives useful

information toinferprobabilities.It is used for stochastic calculations (see stochastic), for

previsions and decisions.

When a distribution is distorted compared to those classical, we can call that a distribution

laws"anomaly".

Also, when a distribution has

no similarity whatever with, we have a distribution

any law"singularity".Kinds of statistical distributions

Photograph or movie?The

statistical distribution of events, is either:A

static distribution.It shows how things are at a precise time.It can be applied in many fields.

A typical example is the situation of a population as the one stated

above.Another one is

comparing financial ratios(profit margin, debt /

equity...)between various businesses so as to determine management

"benchmarks".

Or atime-distribution.

It shows how things evolve during a period.This is the distribution of data found in historical data

series.Time distributions are of

intensive use in financial markets, as

applied to price, return, volatility data.

Visualization of distributions

Join the dots!Statistical distributions are

often shown visually on charts(as shown

below) in which:

The

horizontal scale

gives thevalues, i.e market

prices or whatever

numerical data.It shows for example the range

of a stock's daily returns in a

market.The scale will be graduated, say,

from minus 19-20 % to plus 19

-20%, as everything can happen.The

vertical scaleshowshow manytimes(number

of days in the above example)

each value occurs.Unless the distribution is fully non-

linear (irregular broken lines), the

graph obtained by joining the dots

shows a distribution curve.

Typical distributions: A) The normal law

Classical bell ding dongsDistributions fits most of the times in

typical curves.A highly common one, shown below,is

The Gauss / Laplace "normal law",

also called "bell curve".It is one of the main laws that applies to various kinds of

random.

events

Randomnessapplies to events that cannot be predicted

individually but which occurrences follow someprecise, as seen

frequency that matches a "law of large numbers"

when looking at the broad picture.

In the normal law (see the graph below),

* Most data concentrate on or

near the mean(also called the

"central limit"), forming the top of the bell,* They follow a

symmetricrepartition around it,* They end in slim "distribution tails" on both sides.

If the data obey the normal law, their dispersion around the mean

can be calculated as the"standard deviation".This is mathematically explained in the "volatility" article.

But to give a simpler approach, if:

* the mean is 50

*

68 % of the dataare between 46 and 54,* then the standard deviation is 4, on both sides, as equal

to (54 - 50) and to (50 - 46).Obviously, if the standard deviation of the wave sizes is 2

centimeters, you are in a farmyard looking at the ducks'

wakes in the pond, if it is 5 meters, you are at the seaside

observing a surfer's paradise.

Gaussian distributionXXXXXXXXXXXX

XXX

X

X

Horizontal scale = values(i.e. shoe sizes)

Vertical scale = frequencies (number of times each value occur)Also, by using (legitimate) mathematical tricks, some asymmetric

distributions might be presented as symmetric ones.It is the case of the "log-normal" law, which takes into account

the tendency of economic values to rise exponentially in many

time-distributions.

A price change from 100 to 120, and one from 200 to 240,

expressan identical20% growth rate.

Typical distributions: B) The power law

ThePareto "power law";or "law of extremes", is also called

the"L curve"(or in some cases inverted L curve), or the(20% of events / cases make 80% of the volumes or

"20/80 law"

variations at play).

In that distribution, there is an

asymmetricconcentration of data

near oneextreme value(the vertical branch of the L).This occurs for example, in time-distributions, when the data

evolutions start snowballing towards extremes.

It is considered that the Pareto law

applied to economic incomescalled

represents an optimum,"Pareto efficiency", in which:* A "flattening" of the curve would certainly bring less inequality,

but would also lower the income of the whole population.

* An increase in the asymmetry would also bring lower general

results to the population.

Distribution anomalies in

finance and economics

Dents and cracks in the bell.

Use probabilities but with a lot of precautions

Many classical economists considered that the bell curve and other

classical (also!) curves matched closely thetime-distribution of.

prices, returns, risk / volatilitiesThis is far from being fully the case.

Economic and financial realities differ rather often, and sometimes

widely, from what this idealized paradigm predicts.

It is said in such cases that there are

distribution anomalies

or "defects".For example,

The bell is often slightly flattened at the top, while showing

at its low extremities two

"fat tails"(see that phrase)

Often also, it isleptokurtic(see that word) with ahigher

peakaround the mean, less thick sides but fatter tails.In other cases the curve is decisively

asymmetric (skew):

see that word.

It shows often also several dataclusters(see that word),

looking like camel humps.Those deviations from the "normal" law (or the log-normal law) can

have two origins. They signal:

1)Eitherbehavioral biases.

This glossary describes extensively many of them.

2)Orstructural market imperfections(poor liquidity).

=> There is here a danger when the normal law is applied

inprediction modelsinvolving for example

financial markets:They might

underestimate the real risks, by ignoring those distribution

anomaliesThere is also the danger of skirting the needed effort to make plans

for someextreme scenarios.The human mind can be abler than a stochastic law to foresee them.

Never forget also that

situations might change.Because of that divergence, old statistics might not apply to

the new state of affairs.

On the other hand,to ignore or neglect statistical distributioncan: see base rate fallacy

be dramatic

Quantitative analysis tools

(

Kurtosisconcentration / dispersionof data) is a mathematical

measurement

ofhow far from the bell curvea statistical series is.There are mathematical models (see Garch, heteroskedasticity) that

take into account theinstability ("non-constant variance")in

historical data series

Distribution / Accumulation phaseSee accumulation / distribution +

congestion / (price) cluster,

percolation + investors type

Amassing fresh goodies inside the castle

and throwing stale ones at the barbarians.Accumulation and distribution in financial market are periods in which

prices seem to stabilize at the end of long and massive bull or bear period,which

can precede a trend inversion.

At the end of a bull

market,

"big hands"

(investors who own a large

quantity of assets) are the

first to sell

(distribution phase).They take advantage that small

investors keep on buying.Prices tend to stop rising, as if

blocked by an invisible ceiling,and to get more volatile.

After that - unless the big

hands were wrong - the trend

might become bearish.

At the end of a bear

market, before a

recovery, the opposite

phase,by big

accumulation

hands, often happensA kind of invisible floor

seems to block the fall,although it might not

prevent a sudden last day

of panic

(see "capitulation")

It might then be followed

by the recovery and a

bullish trend.

Those phenomena can be classified as forms of

percolations(see

that word) that lead totipping points/ breakthroughs in a new

direction.

Can such a market phase bedetected ?

Big hands weighing on the charts.

Accumulation create statisticalclusters(see that word) in time-distribution

graphs (*) of prices and volatilities.The market prices are stuck (clustered) in some horizontal

time-zone of the graph that look like a ceiling or a floor.It might signal (but not always), as seen above that a change of trend

is near.However interesting those phenomena are, they can be misleading, or at

least their effects might be hard to spot visually or mathematically on the

market.

Prices can either rebound on a ceiling or a floor, or percolate through

them (see percolation).Thus the behavior - as well as the

relative market power

(financial strength) - of each type of investors (see

"investor types" and "agent based models") arenot so easily(and of course predictable).

measurableThe effects might be

more complex and diluted in timethan an immediate

trend reversal.(*) Distribution in the statistical sense (see distribution curve), and

distribution phases as occasional market phenomenon described here,

are non related homonyms, even if time distribution series might help

to spot such a market phase.

(*)To find those messages: reach thatBF groupand, once there,

1) click "messages", 2) enter your query in "search archives".

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