The Cellar - Interesting "Laws"

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Interesting "Laws"

Stigler's Law of Eponymy

From Wikipedia:

Quote:

Stigler's law of eponymy is a process proposed by University of Chicago statistics professor Stephen Stigler in his 1980 publication "Stigler’s law of eponymy." In its simplest and strongest form it says: "No scientific discovery is named after its original discoverer." Stigler named the sociologist Robert K. Merton as the discoverer of "Stigler's law", consciously making "Stigler's law" exemplify Stigler's law.

Benford's Law
From Wikipedia:

Quote:

Benford's Law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 about 30% of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than 5% of the time. This distribution of first digits is the same as the widths of gridlines on the logarithmic scale.

This counter-intuitive result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). It tends to be most accurate when values are distributed across multiple orders of magnitude.

The graph (below) shows Benford's law for base 10. There is a generalization of the law to numbers expressed in other bases (for example, base 16), and also a generalization to second digits and later digits.
It is named after physicist Frank Benford, who stated it in 1938, although it had been previously stated by Simon Newcomb in 1881.

http://upload.wikimedia.org/wikipedi...nforda.svg.png

Sod's Law.
It's like ten thousand spoons, when all you need is a knife.

Hahhahaahaa...brother just sent me this today.

Synchronicity!
It must mean something.

Probably being deluged by spoons.

Pareto principle
From Wikipedia:

Quote:

The Pareto principle (also known as the 80–20 rule, the law of the vital few, and the principle of factor sparsity) states that, for many events, roughly 80% of the effects come from 20% of the causes.

Business-management consultant Joseph M. Juran suggested the principle and named it after Italian economist Vilfredo Pareto, who observed in 1906 that 80% of the land in Italy was owned by 20% of the population; he developed the principle by observing that 20% of the pea pods in his garden contained 80% of the peas.

It is a common rule of thumb in business; e.g., "80% of your sales come from 20% of your clients". Mathematically, where something is shared among a sufficiently large set of participants, there must be a number k between 50 and 100 such that "k% is taken by (100 − k)% of the participants". The number k may vary from 50 (in the case of equal distribution, i.e. 100% of the population have equal shares) to nearly 100 (when a tiny number of participants account for almost all of the resource). There is nothing special about the number 80% mathematically, but many real systems have k somewhere around this region of intermediate imbalance in distribution.

Quote:

Due to the scale-invariant nature of the power law relationship, the relationship applies also to subsets of the income range. Even if we take the ten wealthiest individuals in the world, we see that the top three (Warren Buffett, Carlos Slim Helú, and Bill Gates) own as much as the next seven put together.
...
In the systems science discipline, Epstein and Axtell created an agent-based simulation model called SugarScape, from a decentralized modeling approach, based on individual behavior rules defined for each agent in the economy. Wealth distribution and Pareto's 80/20 Principle became emergent in their results, which suggests that the principle is a natural phenomenon.
…
In health care in the United States, it has been found that 20% of patients use 80% of health care resources.

Several criminology studies have found that 80% of crimes are committed by 20% of criminals.

85% of ....

You know the rest. :)

[quote=HungLikeJesus;780792]Benford's Law
From Wikipedia:

Quote:

This counter-intuitive result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). It tends to be most accurate when values are distributed across multiple orders of magnitude.

OK, pls explain to me why this is "counter-intuitive ?
and, what difference would it make if it was distributed across orders of magnitude.

If I am going to count anything, I start with "1". Therefore in a set of any size,
there will always be a larger number of "1's" than of "2's" than of "3's"...etc.
And to point out the obvious, in small sets, there may not even be a "9" or "0"

In the examples of the quote above,
a small town might have street addresses of 100's, 200's... to 700's, but no 800's or 900's. etc.

I guess I'm not getting Benford's idea of the whole thing. :neutral:

As a real-world example, Lamp, let's say that we looked at the number of views for threads in one Cellar forum - Nothingland, for example. Would you expect the first digit to have a uniform distribution, or would you expect it to follow the distribution indicated by Benford's Law?

HLJ, by the same "reasoning" I had above.
The first post (or first view) in the thread must be "1".
and there might or might not be a second ("2")
..If there is a "2" there might or might not be a third ("3")
..If there is a .... and so on up to "N"

That is, two "1's" must occur (1 and 10) before there can be two "9's" as the first digit.

So the probability at any given test of the number of posts is going to be higher for "1's" than any other digit, etc.
Therefore in repeated measurements, the distribution of digits will not be equal.

With respect to the "distribution indicated by Benford's Law", my example might or might not be the same.
But as in most treaties on Statistics, "The derivation is left to the reader" ;)

OK - now someone just has to do the analysis.

Some things - street numbers, numbers of posts in a thread, lend themselves to a natural explanation of the preponderance of lower first digits. This is becaue they are built in a series - you can't have post 3 without post 2, but you can have post 2 with no post 3. So there will be more 2-post threads than 3-post threads.

Interestingly, though, it works just as well with things like river lengths and mountain heights, despite the fact that you CAN have a 3 mile long river without having a 2 mile long river. Weirder still, it holds up just as well no matter what units you measure in. Feet, meters, inches, whatever.

Z, what is the difference between counting posts in a thread,
and measuring height or length of a natural object ?
It's not like counting live and dead cats in a box.

ETA: Above, I said:
...two "1's" must occur (1 and 10) before there can be two "9's" as the first digit."

But in fact,
...eleven "1's" (as the first digit) must occur (1,10,11,12,...and 19)
before there can be two "9's" as the first digit."

Sorry, but I'm just not seeing the significance of Benford's Law.
I must be misinterpreting something or other ???
.

One condition of Benford's Law is that

Quote:

It tends to be most accurate when values are distributed across multiple orders of magnitude

If you make a list of consecutive numbers covering a large range (e.g. from 1 to 999) and count how many start with 1, 2, 3, etc. you'll find that there are an equal quantity starting with each digit (111, in this example). So wouldn't you expect that a list of measurements or values would have this same uniform distribution?

I would.

HLJ and Z, you guy are talking to a dummy here... or a stubborn jackass.
I still don't see the difference.

I can argue that if we were measuring "a single" river,
the probability of leading digits = 1 would be skewed,
because only few rivers are 1 mile or 1,000 miles in length
compared to the number of rivers of 9 or 90 or 900 miles.
But that's a function of our definition of a "river" compared with a brook or stream.

I'll stop now, but I'm hoping someone will continue this discussion.
I'm willing to believe there is significance to this law... I just don't see it yet. :(
.