# Subatomic particle mass and Benford's Law

Nine out of 15 subatomic particle masses begin with the digit one. Benford's Law would imply 30% with one as the leading digit. Is there any significance to this frequency distribution, such as perhaps it implies 15 undiscovered subatomic particles?

• Benford's law is certainly not a law like conservation of energy is a law. Although physical constants as a group tend to obey it there is no guarantee that masses of particles need to. The idea that more undiscovered particles can be implied from a statistical observation applying to most sets of data is unfounded. – Reid Erdwien Feb 24 '14 at 22:33
• Interesting idea but a terrible abuse of statistics. – Brandon Enright Feb 24 '14 at 22:48
• "Nine out of 15 subatomic particle masses begin with the digit one." In which set of units? And how does the number vary in other common sets? – dmckee Feb 24 '14 at 23:00
• And I presume you are counting six leptons (three of which have as yet ill-known masses), six quarks (two of which are only roughly known masses) and the three distinct heavy boson masses ($W^\pm$, $Z^0$, and $H$). Is that correct? But if so, what values are you using for the neutrinos? – dmckee Feb 24 '14 at 23:03
• dmckee and Emilio both asked about units. An interesting aspect of Benford's law is that it actually turns out that unit don't matter. You can change units as much as you want, and on average the law still holds. – industry7 Jun 30 '17 at 16:55

Benford's law is pretty cool. It states that, for many sets of data, a leading digit of n has a probability of $Pr(n) = log_{10}(1+1(n))$ Plugging in our n values we find that we can expect low values of n to have a higher probability of being our leading digit.

The most (initially) boggling thing is that our $Pr(1) = .301$ stays independent of units. If we take our set of physical constants and use a length of light-years, time of years and atomic mass units we can expect that our leading digit would still be one ~30% of the time.

Why is this? Well, In a sense because there is less "room" between 8 and 9 than there is between 1 and 2. Think about it in terms of money. There are more people hanging around 1-2 million than 8-9 million. Why is this? Well, because once you are at 8-9 million that extra million to get to 10 million isn't as hard. I know many mathy/statistical readers will be unsatisfied, but I don't know a proof.

One of the coolest applications of Benford's law is that it has been used to detect fraud. If a company or individual is lying about there incomes/outcomes and thinks "In order to not draw suspicion I need to deliver randomish numbers" Benford's law may catch them. But it comes back to implications. The IRS can't arrest you for tax fraud because your tax numbers don't follow Benford's law but it can sometimes imply that something strange is going on. I hesitate to to even say that because it implies OP might be up to something which I don't think it true.

Benford's law is an interesting point, but I think the main problem is that we can't assume so much from such a small amount of data, or indeed, rely on a purely statistical method to tell us about the nature of the world. For those curious, based on a simple calculation $\mathrm{binompdf}(15,.3,9) = .01159$ Meaning there is about a 1% chance that we have $9/15$ would randomly have a leading digit of one. Ultimately though that's the way the world appears to us.

Note: As with everything in statistics sample size matters greatly, in the history section of wikipedia Benford used about 100 physical constants when first showing they followed Benford's law.

A really good explanation link about Benford's law:

http://datagenetics.com/blog/march52012/index.html

Wikipedia has some good applications written down

http://en.wikipedia.org/wiki/Benfords_law#Applications

For the Benfords' law to apply, one needs a set of unrelated numbers that span many many orders of magnitude. The "natural" comparison scale for these very different numbers turns out to be the logarithmic scale. If the numbers are really unrelated, then you can use Pascal's principle of indifference to say that the probability to have a given number in the interval of measurement (expressed in logarithmic scale) is uniform (in the log scale). Now, it turns out that in log scale the size of the interval going from the first digit as 1 to the first digit as 2 is about 30% of the size necessary to increase the value to the next order of magnitude...hence Benford's law.

In the case of subatomic particles, I reckon looking only at the masses is pretty limited. However if you put all the parameter values (charge, mass, Lande coefficient etc...) of all the known subatomic particles, atoms and some molecules, you might/should get Benford's.