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The phenomenon was again noted in by the physicist Frank Benford who tested it on data from 20 different domains and was credited for it.
His data set included the surface areas of rivers, the sizes of US populations, physical The 9 most likely numbers tomolecular weightsentries from a mathematical handbook, numbers contained in an issue of Reader's Digestthe street addresses of the first persons listed in American Men of Science and death rates.
The total number of observations used in the paper was 20, This discovery was later named after Benford making it an example of Stigler's Law. InTed Hill proved the result about mixed distributions mentioned below.
As a rule of thumb, the more orders of magnitude that the data evenly covers, the more accurately Benford's law applies. For instance, one can expect that Benford's law would apply to a list of numbers representing the populations of UK settlements, or representing the values of small insurance claims.
A broad probability distribution of the log of a variable, shown on a log scale. Therefore, the numbers drawn from this distribution will approximately follow Benford's law.
On the other hand, for the right distribution, the ratio of the areas of red and blue is very different from the ratio of the widths of each red and blue bar. Rather, the relative areas of red and blue are determined more by the height of the bars than the widths.
Accordingly, the first digits in this distribution do not satisfy Benford's law at all. On the other hand, a distribution that is mostly or entirely within one order of magnitude e. As the distribution gets narrower, the discrepancies from Benford's law typically increase gradually.
In terms of conventional probability density referenced to a linear scale rather than log scale, i. The reason is that the logarithm of the stock price is undergoing a random walkso over time its probability distribution will get more and more broad and smooth see above.
To be sure of approximate agreement with Benford's Law, the distribution has to be approximately invariant when scaled up by any factor up to 10; a lognormally distributed data set with wide dispersion would have this approximate property.
Unlike multiplicative fluctuations, additive fluctuations do not lead to Benford's law: They lead instead to normal probability distributions again by the central limit theoremwhich do not satisfy Benford's law. For example, the "number of heartbeats that I experience on a given day" can be written as the sum of many random variables e.
By contrast, that hypothetical stock price described above can be written as the product of many random variables i. Multiple probability distributions[ edit ] Formann provided an alternative explanation by directing attention to the interrelation between the distribution of the significant digits and the distribution of the observed variable.
He showed in a simulation study that long right-tailed distributions of a random variable are compatible with the Newcomb-Benford law, and that for distributions of the ratio of two random variables the fit generally improves. However, if one "mixes" numbers from those distributions, for example by taking numbers from newspaper articles, Benford's law reappears.
This can also be proven mathematically: This is not always the case. For example, the height of adult humans almost always starts with a 1 or 2 when measured in meters, and almost always starts with 4, 5, 6, or 7 when measured in feet. But consider a list of lengths that is spread evenly over many orders of magnitude.
For example, a list of lengths mentioned in scientific papers will include the measurements of molecules, bacteria, plants, and galaxies.
If one writes all those lengths in meters, or writes them all in feet, it is reasonable to expect that the distribution of first digits should be the same on the two lists.
In these situations, where the distribution of first digits of a data set is scale invariant or independent of the units that the data are expressed inthe distribution of first digits is always given by Benford's Law. Applying this to all possible measurement scales gives the logarithmic distribution of Benford's law.
Applications[ edit ] Accounting fraud detection[ edit ] InHal Varian suggested that the law could be used to detect possible fraud in lists of socio-economic data submitted in support of public planning decisions. Based on the plausible assumption that people who make up figures tend to distribute their digits fairly uniformly, a simple comparison of first-digit frequency distribution from the data with the expected distribution according to Benford's Law ought to show up any anomalous results.
However, other experts consider Benford's Law essentially useless as a statistical indicator of election fraud in general. The importance of this benchmark for detecting irregularities in prices was first demonstrated in a Europe-wide study  which investigated consumer price digits before and after the euro introduction for price adjustments.
The introduction of the euro inwith its various exchange rates, distorted existing nominal price patterns while at the same time retaining real prices.
While the first digits of nominal prices distributed according to Benford's Law, the study showed a clear deviation from this benchmark for the second and third digits in nominal market prices with a clear trend towards psychological pricing after the nominal shock of the euro introduction. Genome data[ edit ] The number of open reading frames and their relationship to genome size differs between eukaryotes and prokaryotes with the former showing a log-linear relationship and the latter a linear relationship.
Benford's law has been used to test this observation with an excellent fit to the data in both cases. The fabricated results failed to obey Benford's law. Statistical tests[ edit ] Although the chi squared test has been used to test for compliance with Benford's law it has low statistical power when used with small samples.Females are more likely than males to have suicidal thoughts.
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Those who define intelligence as academic aptitude are most likely to criticize: Gardner's concept of multiple intelligences. Vanessa is a very creative sculptress. Aug 08, · Black males are 16 times more likely than white males to be shot and injured in assaults involving guns.
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