A statistical distribution function of wide applicability

Tags: distribution function, values, distribution functions, STOCKHOLM, SWEDEN, Mechanical Engineering, procedure, Applicability, problems, statistics, individuals, JOURNAL OF APPLIED MECHANICS, Statistical Distribution Function, ASME, normal distribution, ASME Journal of Applied Mechanics, American Society Of Mechanical Engineers, applied mechanics, J. Greenberg, experimental data
Content: A statistical distribution Function of Wide Applicability
By WALODDI WEIBULL,l STOCKHOLM, SWEDEN
This paper discusses the applicability of statistics to a wide field of problems. Examples of simple and complex distributions are given. ·. F a variable X is attributed to the individuals of a popula~ion, I· the distribution function (df) of denoted F(x), may be defined as the number of all individuals having an X ~ x, divided the total number of individuals. This function also gives the probability P of choosing at random an individual having a value of X equal to or less than x, and thus we have
P(X ~ x) = F(x)
[1]
Any distribution iunction may be written in the form
F(x) = 1 e-This seems to be a complication, but the advantage of this formal transformation depends on THE RELATIONSHIP
(1
p)n = e-nrp(x) . ........···..··. [3]
'The merits of this formula will be demonstrated on a simple
prob~m.
.
Assume that we have a chain consisting of several links. If we
have found, by testing, the probability of failure P at any load x
applied to a "single" link, and if we want to find the probability
of failure P n of a chain consisting of n links, we have to base our
deductions upon the proposition that the chain as a whole has
failed, if anyone of its parts has failed. Accordingly, the proba-
bility of nonfailure of the chain, (1 P n), is equal to the
probability of the simultaneous nonfailure of all the links. Thus
we have (1- P,J = (1 - p)n. If then the df of a single link takes
the form Equation [21, we obtain
= P n
1 - e-rnp(x) . ...........··. [4]
[4] gives the appropriate mathematical eX]:)reI3Slc.n
for the
of the weakest link in the chain, or, more gen-
erally, for the size
on failures in solids.
The same method of reasoning may be
to the
group
where the occurrence of an event in any part
of an
be to have oocurred in the
a
sat,lst'VlD2 this condition
and thus we put
(x - xu)m
F(x) = 1- e
XI
. [51
The only merit of this df is to be found in the fact that it
simplest mathematical expression of the appropriate form,
tion [2J, which satisfies the necessary general conditions.
ence has shown that, in many cases, it fits the observations
than other known distribution functions.
The objection h~s been stated that this distribution function
has no theoretical basis. But in so far as the author unlC1er'sts.nds.
there are-with very few exceptions-the same
against All Other df, applied to real populations from natural
biological fields, at .least in so far as the theoretical has any-
thing to do with the population in question. Furthermore, it
utterly hopeless to expect a theoretical basis for distribution
functions of random variables such as strength
of ma-
terials or of machine parts or particle
the "Dl~rtlCJ~~S"
fly ash,Cyrtoideae, or even adult males, born in the British Isles.
It is believed that in such cases the only
way of
progressing is to choose a simple function, test it
and
stick to it as long as none better has been found. aocordance
with this program the df Equation [5], has been
not only
to populations, for which it was originally
also to
populations from widely different fields, and, in
with
quite
results. The author has never
opinion that this function is always valid. On the p.nflt.,"~:l,.'tr
very much doubts the sense of
of the
bution function, just as
is no meaning in
correct strength values of an
but also upon
it
a Fourier
small and the number
lihood of real
easy to
real
distributions
It seems obvious that the components of examples 4 and 5 are
due to real causes. In
6 and 7 it is impossible to
decide whether the division is a formal one or real one, but
fact itself may be a valuable stimulus to a closer examination of
the observed material.
The specific data for the examples follow.
yield strength OF A BOFORS STEEL The observed values are obtained as routine tests of a Bofors steel, the quality of which was chosen at random for purposes of demonstration only. Fig. 1 gives the curve and Table 1 the
TABLE 1 YIELD STRENGTH OF A BOFORS STEEL
(x yield strength in 1.275 kg/mm 2)
Observed
Normal
values
distribution
n
n
n
1
32
2
33
3
34
4
35
5
36
6
37
7
38
8
39
9
40
10
42
2 3 4 5 6 7 8 9 10 11 12 13 14 .4-0 .0-0 of Fly- ash N- 211 .S-I--I----.----.--~~.-+---+---1 .2-1-~1-+---+---f--+---'-----'----t 0'1 .2 .8-2-.2 FIG. 2 size distribution OF FLY ASH
FIBER STRENGTH OF INDIAN COTTON
The observed values are taken from R. S. Koshal and A. J.
Turner. 3
3 gives the curve, and Table 3 the values. The
parameters are Xu = 0.59 gram, Xo = 3.73 grams, m = 1.456.
If the classes 14 to 16 are
d of fare 13 -3 = 10. Then
X2 == 11.45 gives a P = 0.35.
The authors3 have
about the lre'QUE~nCY
I log (x- xu)
l
.9-1
.3
.5
.7
.9
1.1
FIG. 3 FIBER STRENGTH OF INDIAN COTTON
TABLE LENGTH OF CYRTOIDEAE (x length in microns) ,...__-Expected nl 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
and
The undivided
marked N 1
It is
to that
one, and that it is
to up the
in
two parts. By trial it found that 86 of the individuals be-
longed to component No.1, and 14 to component No.2.
The parameters are: Component No.1: Xu = 3.75 J.l, Xo = 63.2
J.l, m = 2.097. Pooling the classes 2-3, 9-10, and 11-13 gives
X2 = 3.59. The d of fare 7 3 = 4, and P = 0.47.
vo:mp,onlent No.2: Xu 122.0 IJ-, Xo = 124.1 J.l, m 1.479. The number of individuals is too small for the x2-test.
FATIGUE LIFE OF AN ST-37 STEEL The 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 when
4 "A Statistical
of the Size of Cyrtoideae in Albatross
Cores From the East
Ocean," ·by W. Weibull, Nature, vol.
164,
1047.
5 "Der
dauernd und unterbrochen wirkender, schwingender
fiolerlJleal1.Splruc::hlLng auf
des Dauerbruchs," by H.
l!h~:enJfOr8Ch'L~na. (March, 1938),
5 of this paper.
Observed values nt+2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
60 "01----+----+-1----+--+--+---20 1---.-+---.....lI-----+---~
296
57
58
59
60
61
62
63
64
65
66
67
Component'
68
xu- 4.032 I Nu ·'0800
69 70
m · S.95S
71
72
Component 2
73 74
xu· 4.484, Nu ·30S00
75 76
m · 1.215
77
log (x-xu)
.o-t
·...-1
-8-'
-8-'
FIG. 6 FATIGUE LIFE OF ST-37 STEEL
It may be pointed out that the frequency curve in Fig. 5 seems
to be the result of a smoothing operation on the cumulative
frequency curve. Accordingly, the sampling errors of the ob-
served values in Table 5 have been eliminated almost entirely
(without affecting the function), which explains the really too
good representation of the observed values.
The real causes of this splitting up in two components may be
found by examining the frequency curve of the yield strength of
the 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 less
than 25 kg/mm2 belong to Component No.1, we obtain 14 speci-
mens out of 20, making 70 per cent. Exactly the same propor-
tion has been found by the statistical analysis, as 165/235 = 70
per cent.
The reason why this partition is so easily seen in Fig. 7 and
not at all in Fig. 5,
of course, upon the much larger
scatter in fatigue life than in yield strength.
Appendix
7
of
...------Expected va.lues-----..
Observed va.lues
nl
nt
nl+2
nl+1
32
2
130
3
400
4
1011
5
2145
32 130 400 1011 2145
32 135 374 998 2185
6
3832
7
5718
3832 5718
3835 5718
8
7140
486
7626
7648
9
7761
1525
9286
9286
10
7890
2510
10400
10416
11
7900
3229
11129
11153
12
7900
3671
11571
11580
13
7900
3908
11808
11801
14
7900
4022
11922
11911
15
7900
4071
11971
11968
16
7900
4091
11991
11992
17
7900
4098
11998
11998
18
7900
4100
12000
12000
6..-----..,..-------r----.,..----.-----.-----..----
5.....-.--t----."O'C-.....-.-----ii------i----t----t-----1
4"----"---I--L+-----11---~--...r----+---+-~-i 3 "-----+--;I--~--I---~+__t_---4.----t----t
2~--++---+--4---f---i--+--t---+----t
The foregoing statistical methods have been
to many
prc)blenJLS outside the field of Applied Mechanics. It may
be of interest to have examples of this
and for this reason,
the
two given with the
only:
23 24
28 29
7 FIG.
FREQUENCY CURVE OF YIELD STRENGTH OF ST-37 STEEL
of
yield
STATURES FOR ADULT
The
values
This distribution
of
BORN IN THE BRITISH
WEIBULL-A STATISTICAL DISTRIBUTION FUNCTION OF WIDE APPLICABILITY
297
If the classes 17-18 are pooled, the value of X' - 4.50, and the doff 9 - 31/, - 51/ , give a P - 0.56. It may be of interest to compare this result with those of Charlier and Crama-. Charlier says that, at the first look, the agreement with the Normal Distribution seems very satisfactory, but that a closer examination shows a small negative skewness and a small posi- tive kurtosis. CraIOOr has calculated the values of X' on the hypotheses of
normal distribution and asymptotic expansions from it. The result was as follows: Normal distribution X'" 196.5 doll 13 P < 0.001 First approximation X' - 34.3 d of f 12 P < 0.001 Second approximation X' - 14.9 d of f 11 P - 0.19 The agreement is satisfactory in the third case only, requiring four terms Df the series. This operation is certainly of a purely formal chara.cter.
Wallodi Weibull published "A Statistical Distribution Function of Wide Applicability" in the ASME Journal of Applied Mechanics, Transactions of the American society of Mechanical Engineers, September 1951, pages 293-297 as described above. Discussion of his paper was reported in the ASME Journal of Applied Mechanics, Transactions of the American Society Of Mechanical Engineers, June 1952, pages 233234 as described on the following pages.
DISCUSSION
233
A Statistical Distribution Function of Wide Applicability'
T. C. Tst:.1: The author should be congratulated (or having de- vised a distribution function of truly wide applicability, 8.8 evidenced by the seven examples presented in his paper. Since the writer is currentl)' concerned ~itb the problems of particle-size distribution in aerosols, he Lot int.erested in the p0ssible utilization of the author's method to reduce tbe necessary Imouni. of experimental l\"Ork. In this connection he would like to ask the following quesLjons.
1 In applying the author's distribution function it is necessary to determine the parameters x., Xo, and m. If the distribution function is a true representation of the observed data, then any three seta of the values of P and x wouJd be sufficient to evaluate these three parameters. In the author's examples he did not specify how his parRmctcrs were obtained. Would he care to diecU88 this point brieJly?
2 When the funotion is: applied to an UOknOVl'D distribution. bow many observed data arc necessary to yield the pArtuDeters reliably? Considers.ble practical value would be added to the author's function if ita application oould result in a saving of ex· perimental vtork.
3 The relations shown in I'igs. I, 2, and 3 in the paper, appear
to represent the equation
P-l-e
(r-ro). n
ratber than
Could that be Ii, misprint? The values for log (x - z .. ) in Fig. 2 do not correspond to the given values of x nnd x" in the second exwnpJe (l!ize di.stribut.ion of fly ash), the discrepancy being appreeiable when x is small. Could there be some numerical errors? If so, would the author kindl:,' flhow fl corrected figure?
tribution to the literature on distribution functions. The range of fields t-reated in his enmples is also impressive. However, the reason for introducing the minimum value I", and ignoring the maximum :r", is not entirely clear. Proba.bly it r(llates to the original applications, ",-hich may have been the Cystoidea. or the yield strengths snd fatigue-lire data of steels. NoW", ror such a case as Fig. 2 of the paper, one would expect tbe maximum particle to be more tangible, and also more significant practically, thun the minimum. Incorporation 01 both a ma.:ximum and a minimum "alue of x will bring EqUAtion [5J into the form F(x) = l - e -'(::r...~-~)' which will ngllin redu('(> to Equation 15 j as x ... beoomes infiJlite, and to the Rosin-Rammler type of ('Quation 4 as x" vanishes. Of course one may start with kny di$t.ribution function where the arb'Urnent has infinite rlWge. alld convert it to onc where tbe range is finite. This has been illustrated in the case of the lognormal distribution b;y Van Uven' and more recently by Mugele and Evans.' The latter reference also gives a Critical Review of the Rosin-Rammler and other distribution functions. A word of warning also should be added in regard to t.he e:~ amples in the paper: They coutain some arithmetical and dimeQsional errors. However, when these are corrected, the examples illustrate e.X"celJently the generalstatementa of the text. F. A. MCCLINTOCK.l The distribution function suggested by the aut.hor is atf.ra.ctive because oC its eimplicity, the ea!le ~th which it can be applied to studying the size effect, and its implication of a lower limit to a distribution. In order to apply the di'ltribution impartially, however, some systematic means of fitting it to expcriment.a.l data should be used. For a simple distribution the following procedure appears useful. The para.meters, x". x.., and m can be chosen so that the first three moments of the distribution function coincide wi.th thoac of· the data. The nth moment of the theoretical distribution is first calculated from t.he cdmulative distribution
..... (lJ
DifI'erentiation gives the frequency dilliribution
f _ dF _ "!. (x_-x.).-. exp [_(x_-x.)·J.... 121
h
~
~
~
The nth moment ahout .:t. is
""' - /.' (x-x.)'fdx - /.' (x-x.),"!. (x-x.)--.
z"
XJJ :..
X"
cxp [ - (X :.
X. dz .... [31
On changing the va.riable of integration to
R. A. MUGELE.J The author's trealment is definitely a (:on- · "Extended Limit. design criteria for Continuous Medill.... by D. C. Drucker. W. Pmger. and n. J. Greenberg. Quarlerlll 01 ApplUd .\fatAernaliu. vol. 9. no. 4, JAnUAry. 1952, pp. 381-389. I By Waloddi Weibull. published in the Sepr,.ember, 19b1, iseue of the JO'tl'RNAL OF ApJ'LlED MECHANICS. Trans. ASME, vol. 73, pp. 203-297. 'Aeeociate Pro(e8llOr or EnpneeriD& Research, The PennlJ)'lvania 8tat.e College, State CoUese. Pa. Mem. ASME. , Oakland. Ca.lif.
· "Feiohcit uod Btruktur des KohlenstAuba," by P. Rosin a.nd E . HAmmier, Zeitsehrilt de. Vereine. thuucher IngenU:urY, "01. 71, 1927, pp.I-7. ." Ske" Frequency Curvcs," by M. J. Van Uven, Proc. KOIl.Ak.ad. v. Weteoa, vol. 19. 1917. p. 670. · "Drople:.SiM Distribution io SpraYlI," by R. A. :\{ugele and H. D. Evana, IndlUtriol and Knqinurinq Ch~Wrv. vol. 43,1951, PP. 1317- 1324. f Assistant Prol'eMOr or :'dechanical Engineerina:, Muaachulletta Institute of Techcology, Cambrida:e. Mus. Jun ASME.
JOURNAL OF APPLIED MECHANICS
JUNE, 1952
this becomes
fo'" 1'0,.' = xo"
J1..I exp(-71)dJ1:
I4]
This integral can be expressed in terms,of the Gamma function
+ 1J.,,' = x/' r (1 nlm)
15J
The second and third moments about the mpan are
+ + 1J.: =- x.,2[f(J 21m) - rt(I 11m)]
+ and J1.s ... xoS[f(I 31m) - 3r(I + 21m)r(l + 11m)
+ + 2r~ (1 l/m)]
16J
From these a mea..'lUf(> of the skewness ('an bp obtained
0'3:= J1.J/pt'/z.
. .... 171
Since ~ is a function of m only, the value of m can be chosen 80 t.hat the values of Јl3 for the theoretical distribution and the experimental data. coincide. Then since the second moment about the mean, that is, the square of the standard deviation, of the experimental data is known, the relation
+ + ~,(x.· - r(1 2(mJ - P(I I(m)
181
can be solved for Xo' Finally, the relation
+ ~,'fx. - (x-x.J(x, - r(1 I(m)
[91
can be solved for x.., since the mean of the experimental data is known. Plom of the quantities ~, Pot/x.,!, and 1l1'lxo are given in Figs. 1 and 2 of this discussion.
-LO
0
'0
'.0
20
5
T7
E 2 '0 5
"': & ~ l"'-
['F::: f--
~ t---..
.85 .90 .95 1.0 E:J
i<.
Fln.l
,
E · , 001
I i 0.02
02
0.' '0
ACTHOR'S CLOSURE The author appreciat-es the comments made by the dh,cussers. The proposal of Professor Tsu to take any three sets of the values P a.nd :J; is quite correct but does not use the data efficiently. This method may be improved by taking the set from a smoothed curve. Up to the past year the author's usual method has been to plot the data as shown in the paper llnd to choose the value :r.. to give the best straight line. In this way it is easy to decide if the distribution is simple or complex, but the procc<1ure is not entirely free of subjectiveness. Aboul tl. year dogO the author decided that it would be better to x st.art by standardizing t.he variable x, Le., by putting z = (x- x)lu, where is t.he mean and u the standard deviation :Uld elimi- nating two of the parameters, for instance, .r" and .tn. The distribut.ion funct.ion then takes the form P -1-exp!-I,v!.-(2a)-"-'(a) + ,,-(a)I';a! where a = 11m. A curve paper {or different values of a, also including .the stand· ardizcd Gaussian distribution, may be prepared. By plotting the points (P, z) on this paper, it is easy to decide wbetber the dis· tribution is simple or complex and to estimate, with a. good approximation, the value of 0'. As to the third question, the parentheses are an awkward misprint. The values for Jog (x - x,,) in Fig. 2 do not correspond to the given value x. =1.5 X 201-1 but to x... -1 X 20.u. It should b~ mentioned that the x-values are mid-point values and should correctly have been increased by 1/2. Thue the value x" = 30 ~ is the correct olle. The introduction of a maximum value x... proJ)08ed by Mr. Mu· gele is a valuable extension of the function. It was not found necessary to introduce this new parameter in the field of strength of materials, probably because the theoretical strength may be perhaps a. hundred times higher than the technical strength. Rut in other fields conditions may be quite different. The method proposed by Professor McClintock to use the first three momentB is quite good if the distributioD is simple and the population not. too small. The author has been a.ware of thi~ possibility of computing the parameters and has mentioned it(with some different notation for thc gamma function) in an earlier paper.s Actually, however, he has never app1ied this method . but admits that it may sometimes have its advantages. As to the question of a systematic procedure when tbe distribution is complex, the author is sorry to admit that so far he has found no better method than to cut and try. This is, of course, not very 83tisfactory, but a simple electronic computing machine, recently eompleted, facilitates the otherwi,<:;e tedious cornpuw.· tions.
FlO. 2
The writer would like to ask what procedure, preferably systematic, should be followed in the case of a "complex" distribution. An extension of the foregoing procedure looks impractical, and yet the writer .....ould like to try applying the distl'ibution in other cases. For example, it would be interesting to see whether the other data on the ST-37 steel reported by ~-tul1er..stock would result in the same division of the population as found from Figs. 6 and 7 of the pa.per.
8 "The Phenomenon of Rupture in Solids," by Waloddi Wei bull. IVA Handling, No. 153, p. 23. 1 By E. H. Lee and B. \'L Shaffer, published in the December, J951. issue of the JOURNAl. OF' ApPLn:n M!:CIIANlC8. Trans. ASME, vol. 13. pp. 405--413. ! AS.'>istant Profe&5or of Mechanical engineering, Unh'ersity of IIIinoi6, Urbana, 111. I Professor of Mechanical Engineering, University of ll1iTlo8. Mem. ASME.

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