<< A little more on Benford's Law if you please... >>
I checked the effect of various scale factors on the
first 35 prices in amplification of my previous experiment.
Translating dollars to marks (1.77 to the dollar) I got 11
out of 35 beginning with 1. Using British pounds, I got 13
of 35; using a multiple of 2, 9 of 35; of 3, 5 of 35; of 4,
9 of 35; of 5, 11 of 35; of 6, 12 of 35; of 7, 11 of 35; of
8, 11 of 35; of 9, 10 of 35. So we do have a confirmation of
B's law here, I admit.
[out of order:]
....[but] The law definitely does not work, for
example, on the last 4 figures of telephone numbers,
probably because they do not represent quantities....
As the article posted in 597 mentions, Benford's Law does not
apply to numbers that are randomly generated, as the last four digits of phone numbers apparently are. There is certainly no law that says that winning lottery numbers are more likely to begin with 1. But, Benford's Law does not apply to the first of the last four digits -- it applies to the first digit on the left. Fwiw, in NYC, at any rate, the complete phone number is
1+area code+last seven digits.
But there must be some kind of
psychological issue here,...
I agree that there may be a kind of "uniphilia", if you will, at
work in the "selection" of numbers. But I don't think Benford's Law is about numbers that were arbitrarily selected (like "The 1,001
Nights" or "101 Dalmatians"), so much as it is about what happens when numbers are aggregated -- by virtue of the fact that the last digit on the left is one power greater than the digit to its right (see
below).
perhaps something about the way
people measure or scale values is related to what they
perceive....I have a recollection of
the Weber-Fechner law of psychology concerning the response
to a stimulus being logarithmic.
That's an interesting association. Weber and Fechner observed
that a person could manually discern a difference of as little
as, say, 7% between two very light weights as well as two heavy
weights, and various weights in between. This constant ratio is
supposed to hold for other modes of sense perception also, such
as sight and sound. (Of course, at extremes at either end of the
spectrum, the relationship breaks down).
Constant ratios for equally spaced increments imply a logarithmic
distribution of values. For example, an X decibel sound is twice
as loud as an (X-10) decibel sound, and half as loud as an (X+10)
decibel sound. Thus, 16 decibels is twice as loud as 6 decibels,
but half as loud as 26 decibels. Conveniently, each decibel is
about 7% apart (in other words, compounding at 7% per year
doubles the original sum in 10 years).
So, W-F implies that a person who can distinguish the difference
in loudness between a 15 decibel sound and a 16 decibel one can
also distinguish between a 37 decibel sound and a 38 decibel one.
[See: netsrv.casi.sti.nasa.gov]
....Nor does it work for
human heights: most people are between 5 and 7 feet, elves
and giants are rare....
Try centimeters, which almost everyone outside the U.S.'s 5% of
the world's population is using. (I know -- Benford's Law is supposed to be scale invariant. Interestingly, the metric system was
constructed to make a lot of values begin with 1.) But, again,
I don't think it's about single selections or single measurements
-- I think it's most applicable for data that aggregate over
time (see below).
.....I tried
it out for per capita income by (US) state in nominal
dollars using the info in BUSINESS STATISTICS OF THE UNITED
STATES (1996 Edition), Bernan Press. For the 50 states in
1960 the per capita income ranged from $2379 to $4848; not a
single value began with 1. For the 50 states in 1995 the
range was $16683 to $33452, and 13 values began with 1.
What Benford's Law implies is that the annual figure will on average
be recorded in more years between $10,000 and $20,000 than
between $20,000 and $30,000 (assuming steady annualized growth),
because it has to double to get to $20,000, but only increase by
50% to get to $30,000. It will be recorded in even fewer years with
beginning with 3, etc.
CONCLUSION: Whether or not the law applies depends on the
data. I've also tried it out for a random sample of the
atomic weights of chemical elements; in that case it
worked.
It is certainly true that all the "new elements" created in the
laboratory these days have an atomic weight beginning with 1.
It seems to work best if the data are highly variable,
i.e., values are scattered over a wide range. But is there
any reason to suppose that it would apply for the values of
a single quantity over time, like the DJIA?
In fact, a sidebar to the article posted at 597 contains such an
example (but, I was unable to cut and paste the sidebar).
Assuming average annual compounding at 7%, the DJIA will spend 10
years getting from 10,000 to 20,000 -- the same amount of time it
will spend getting from 20,000 to 40,000, and therefore recording
daily closings with totals beginning with 1 approximately as many times as for those beginning with 2 and 3 combined.
I think W-F is easiest to see by viewing a simple multiplication
table and addition table for the digits 1 through 9. It is
immediately apparent that the greatest number of products and
sums begin with 1, then 2, etc., and the fewest with 8 and then
9. This is true no matter what the base. But, since there are
fewer alternative digits, the phenomenon is more pronounced the
lower the base. At the extreme, base 2 (binary notation), all
digits begin with 1.
porc --''''> |
