A Statistical Distribution Function Of Wide Applicability

A Statistical Distribution Function ofWide ApplicabilityBy WALODDI WEIBULL,l STOCKHOLM, SWEDENThis paper discusses the applicability of statistics to awide field of problems. Examples of simple and complexdistributions are given.·.Isat,lst'VlD2 this conditionF a variable X is attributed to the individuals of a popula ion, the distribution function (df) ofdenoted F(x), may bedefined as the number of all individuals having an X x,dividedthe total number of individuals. This function alsogives the probability P of choosing at random an individualhaving a value of X equal to or less than x, and thus we haveP(X x) F(x)[1]Any distribution iunction may be written in the formF(x) 1e - /I(X) . . . . . . . [2]This seems to be a complication, but the advantage of this formaltransformation depends on the relationship(1p)n e-nrp(x) . . . . . . . . . . . .[3]'The merits of this formula will be demonstrated on a simple.Assume that we have a chain consisting of several links. If wehave found, by testing, the probability of failure P at any load xapplied to a "single" link, and if we want to find the probabilityof failure P n of a chain consisting of n links, we have to base ourdeductions upon the proposition that the chain as a whole hasfailed, if anyone of its parts has failed. Accordingly, the probability of nonfailure of the chain, (1P n), is equal to theprobability of the simultaneous nonfailure of all the links. Thuswe have (1- P ,J (1 - p)n. If then the df of a single link takesthe form Equation [21, we obtainprob m.Pn 1-e-rnp(x) . . . . . . . . . . . . .[4][4] gives the appropriate mathematical eX]:)reI3Slc.nfor theof the weakest link in the chain, or, more generally, for the sizeon failures in solids.The same method of reasoning may beto thegroupwhere the occurrence of an event in any partof anbeto have oocurred in theaand thus we put(x -F(x) 1- exu)mXI. [51The only merit of this df is to be found in the fact that itsimplest mathematical expression of the appropriate form,tion [2J, which satisfies the necessary general conditions.ence has shown that, in many cases, it fits the observationsthan other known distribution functions.The objection h s been stated that this distribution functionhas no theoretical basis. But in so far as the author un lC1er'sts.nds.there are-with very few exceptions-the sameagainst all other df, applied to real populations from naturalbiological fields, at .least in so far as the theoreticalhas anything to do with the population in question. Furthermore, itutterly hopeless to expect a theoretical basis for distributionfunctions of random variables such as strengthof materials or of machine parts or particlethe "Dl rtlCJ S"fly ash,Cyrtoideae, or even adult males, born in the British Isles.It is believed that in such cases the onlyway ofprogressing is to choose a simple function, test itandstick to it as long as none better has been found.aocordancewith this program the df Equation [5], has beennot onlyto populations, for which it was originallyalso topopulations from widely different fields, and, inwithquiteresults. The author has neveropinion that this function is always valid. On the p.nflt.," :l,.'trvery much doubts the sense ofof thebution function, just asis no meaning incorrect strength values of anbut also uponit

a Fouriersmall and the numberlihood of realeasy torealdistributionsIt seems obvious that the components of examples 4 and 5 aredue to real causes. In6 and 7 it is impossible todecide whether the division is a formal one or real one, butfact itself may be a valuable stimulus to a closer examination ofthe observed material.The specific data for the examples follow.The observed values are obtained as routine tests of a Boforssteel, the quality of which was chosen at random for purposes ofdemonstration only. Fig. 1 gives the curve and Table 1 theYIELD STRENGTH OF A BOFORS STEEL(xyield strength in 1.275 kg/mm YIELD STRENGTH OF A BOFORS STEELTABLE 12of Fly- ash.0-0N- 211.S-I--I----.----.-- . - - - - - - - 1.2-1- 1- --- ---f-- 3435363738394042FIG.2SIZE DISTRIBUTION OF FLY ASHFIBER STRENGTH OF INDIAN COTTONThe observed values are taken from R. S. Koshal and A. J.Turner. 33 gives the curve, and Table 3 the values. Theparameters are Xu 0.59 gram, Xo 3.73 grams, m 1.456.If the classes 14 to 16 ared of fare 13 -3 10. ThenX2 11.45 gives a P 0.35.The authors 3 haveabout the lre'QUE nCY

andThe undividedIt istothatmarked N 1one, and that it istoup theinfound that 86 of the individuals betwo parts. By trial itlonged to component No.1, and 14 to component No.2.The parameters are: Component No.1: Xu 3.75 J.l, Xo 63.2J.l, m 2.097. Pooling the classes 2-3, 9-10, and 11-13 givesX2 3.59. The d of fare 73 4, and P 0.47.vo:mp,onlent No.2: Xu122.0 IJ-, Xo 124.1 J.l, m1.479.The number of individuals is too small for the x2-test.FATIGUE LIFE OF AN ST-37 STEELIlog (x- xu).3.5.9-1FIG.3.7.9l1.1FIBER STRENGTH OF INDIAN COTTONTABLELENGTH OF CYRTOIDEAE(xlength in microns),. -ExpectednlThe observed values are t.aken from Muller-Stock. s The frequency curve in Fig. 55 gives no impression of a complex distribution, which, on the other hand, may easily be seen when4 "A StatisticalCores From the East164,of the Size of Cyrtoideae in AlbatrossOcean," ·by W. Weibull, Nature, vol.1047."Derdauernd und unterbrochen wirkender, schwingenderfiolerlJleal1.Splruc::hlLng aufdes Dauerbruchs," by H.l!h :enJfOr8Ch'L na. (March, 1938),5 of this nt 2123456789101112131415162060"01---- ---- -1---- -- -- ----20 1---.- ---.lI----- ---

u- 4.032 I Nu ·'0800m S.95SComponent 275xu· 4.484, Nu ·30S00m log .lues.------Expected va.lues-----.FATIGUE LIFE OF ST-37 079007900790079007900790023It may be pointed out that the frequency curve in Fig. 5 seemsto be the result of a smoothing operation on the cumulativefrequency curve. Accordingly, the sampling errors of the observed values in Table 5 have been eliminated almost entirely(without affecting the function), which explains the really toogood representation of the observed values.The real causes of this splitting up in two components may befound by examining the frequency curve of the yield strength ofthe same material, Fig. 7. It is easy to see that the material,probably not being killed,. is composed of two different kinds.If we suppose that all the specimens with a yield strength of lessthan 25 kg/mm 2 belong to Component No.1, we obtain 14 specimens out of 20, making 70 per cent. Exactly the same proportion has been found by the statistical analysis, as 165/235 70per cent.The reason why this partition is so easily seen in Fig. 7 andnot at all in Fig. 5,of course, upon the much largerscatter in fatigue life than in yield strength.456789101112131415161718ntnl 211971119911199812000nl -----i----t----t-----14"----"---I--L -----11--- --.r---- --- - -i3"----- --;I-- --I--- t ---4.----t----t2 -- --- --4---f---i-- --t--- ----tAppendixThe foregoing statistical methods have beento manyprc

in the ASME Journal ofApplied Mechanics, Transactions ofthe American Society OfMechanical Engineers, September 1951, pages 293-297as described above. Discussion ofhis paper was reported in the ASMEJournal ofApplied Mechanics, Transactions ofthe American Society OfMechanical Engineers, June 1952, pages 233 234 as described on the following pages.