{smcl} {* 12dec2007}{...} {cmd:help mata mm_benford()} {hline} {title:Title} {p 4 14 2} {bf:mm_benford() -- Benford's Law (distribution of digits)} {title:Syntax} {p 8 23 2} {it:real vector}{space 1} {cmd:mm_benford(}{it:digits} [{cmd:,} {it:position}{cmd:,} {it:base}]{cmd:)} {pstd} where {p 7 16 2} {it:digits}: {it:real vector} providing the digits for which the probabilities be computed; the digits must be in {c -(}0,...,{it:base}-1{c )-} {p 5 16 2} {it:position}: {it:real scalar} specifying the position of the digits {p 9 16 2} {it:base}: {it:real scalar} specifying the base of the numeric system {title:Description} {pstd} {cmd:mm_benford()} returns the probabilities of the specified {it:digits} according to Benford's Law (Newcomb 1881, Benford 1938, Hill 1998). The formula for the 1st (leading) digit in the base 10 number system is: P(d) = log10(1 + 1/d) {pstd} with d in {c -(}1,...,9{c )-}. {pstd}The general formula for the p-th digit in the base b number system is b^(p-1) - 1 {hline 3} 1 \ {c TLC} 1 {c TRC} P(d, p, b) = {hline 5} ln{c |}1 + {hline 5} {c |} ln(b) / {c BLC} k*b+d {c BRC} {hline 3} k = b^(p-2) {pstd} with d in {c -(}0,...,b-1{c )-} (see, e.g., {browse "http://www.mathpages.com/home/kmath302/kmath302.htm"}). {pstd}The limit of P(d) with rising p is 1/b. Computation becomes exceedingly intensive with rising p since the number of operations is in the order of b^p - b^(p-1). {title:Remarks} {pstd} Examples {pstd} Fist digit Benford distribution: {com}: (1::9), mm_benford(1::9) {res} {txt} 1 2 {c TLC}{hline 29}{c TRC} 1 {c |} {res} 1 .3010299957{txt} {c |} 2 {c |} {res} 2 .1760912591{txt} {c |} 3 {c |} {res} 3 .1249387366{txt} {c |} 4 {c |} {res} 4 .096910013{txt} {c |} 5 {c |} {res} 5 .079181246{txt} {c |} 6 {c |} {res} 6 .0669467896{txt} {c |} 7 {c |} {res} 7 .057991947{txt} {c |} 8 {c |} {res} 8 .0511525224{txt} {c |} 9 {c |} {res} 9 .0457574906{txt} {c |} {c BLC}{hline 29}{c BRC}{txt} {pstd} Second digit Benford distribution: {com}: (0::9), mm_benford(0::9,2) {res} {txt} 1 2 {c TLC}{hline 29}{c TRC} 1 {c |} {res} 0 .1196792686{txt} {c |} 2 {c |} {res} 1 .1138901034{txt} {c |} 3 {c |} {res} 2 .108821499{txt} {c |} 4 {c |} {res} 3 .1043295602{txt} {c |} 5 {c |} {res} 4 .1003082023{txt} {c |} 6 {c |} {res} 5 .0966772358{txt} {c |} 7 {c |} {res} 6 .0933747358{txt} {c |} 8 {c |} {res} 7 .0903519893{txt} {c |} 9 {c |} {res} 8 .0875700536{txt} {c |} 10 {c |} {res} 9 .0849973521{txt} {c |} {c BLC}{hline 29}{c BRC}{txt} {pstd} Second digit Benford distribution in base 6: {com}: (0::5), mm_benford(0::5,2,6) {res} {txt} 1 2 {c TLC}{hline 29}{c TRC} 1 {c |} {res} 0 .2019648465{txt} {c |} 2 {c |} {res} 1 .1841218513{txt} {c |} 3 {c |} {res} 2 .1697091954{txt} {c |} 4 {c |} {res} 3 .1577441976{txt} {c |} 5 {c |} {res} 4 .1476009617{txt} {c |} 6 {c |} {res} 5 .1388589474{txt} {c |} {c BLC}{hline 29}{c BRC}{txt} {title:Conformability} {cmd:mm_benford(}{it:digits}{cmd:,} {it:position}{cmd:,} {it:base}{cmd:)}: {it:digits}: {it:r x} 1 or 1 {it:x c} {it:position}: 1 {it:x} 1 {it:base}: 1 {it:x} 1 {it:result}: {it:r x} 1 or 1 {it:x c}. {title:Diagnostics} {p 4 4 2}{cmd:mm_benford(0,1,}{it:base}{cmd:)} evaluates to missing. {title:Source code} {p 4 4 2} {help moremata_source##mm_benford:mm_benford.mata} {title:References} {phang} Benford, Frank (1938). The law of anomalous numbers. Proceedings of the American Philosophical Society 78: 551-572. {phang} Hill, Theodore P. (1998). The first digit phenomenon. American Scientist 86: 358. {phang} Newcomb, Simon (1881). Note on the frequency of use of the different digits in natural numbers. American Journal of Mathematics 4: 39-40. {p_end} {title:Author} {p 4 4 2} Ben Jann, University of Bern, jann@soz.unibe.ch {title:Also see} {p 4 13 2} Online: help for {bf:{help m4_statistical:[M-4] statistical}}, {bf:{help moremata}} {p_end}