# A Statistical Distribution Function of Wide Applicability

Waloddi Weibull

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Waloddi Weibull. A Statistical Distribution Function of Wide Applicability. Journal of Applied Mechanics, 1951. hal-03112318

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
The correctness of fit has been checked by applying the chi-square method

Of those populations, Nos. 1-3 are distributed in good agreement with the df Equation [5], whereas the four remaining populations have to be split up into two components, before such an agreement is obtained. The first type will be called a "simple" and the second type a "complex" distribution.

The fundamental question now arises, whether this splitting-up is a purely formal operation, or whether it might unveil some hidden real causes. It may be said that any distribution may be represented by a sum of a sufficiently great number of simple distributions, just as any periodical function may be developed in a Fourier series. However, if the number of the components be small and the number of observations sufficiently large, the likelihood of real causes seems to increase. In any case, it is very easy to produce real complex distributions by syntheses.

It seems obvious that the components of examples 4 and 5 are due to real causes. In examples 6 and 7 it is impossible to decide whether the division is a formal one or a real one, but the 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><caption>TABLE 1 YIELD STRENGTH OF A BOFORS STEEL</caption><caption>(x = yield strength in 1.275 kg/mm<fnref n="2" />)</caption><thead><tr><th></th><th>x</th><th>Expected values n</th><th>Observed values n</th><th>Normal distribution</th></tr></thead><tbody><tr><td>1</td><td>32</td><td>10</td><td>10</td><td>8</td></tr><tr><td>2</td><td>33</td><td>36</td><td>33</td><td>28</td></tr><tr><td>3</td><td>34</td><td>84</td><td>81</td><td>71</td></tr><tr><td>4</td><td>35</td><td>150</td><td>161</td><td>141</td></tr><tr><td>5</td><td>36</td><td>224</td><td>224</td><td>225</td></tr><tr><td>6</td><td>37</td><td>291</td><td>289</td><td>301</td></tr><tr><td>7</td><td>38</td><td>340</td><td>336</td><td>351</td></tr><tr><td>8</td><td>39</td><td>389</td><td>389</td><td>376</td></tr><tr><td>9</td><td>40</td><td>383</td><td>383</td><td>380</td></tr><tr><td>10</td><td>42</td><td>389</td><td>389</td><td>388</td></tr></tbody></table>

<figure><img src="image_2.png" /><figcaption>FIG. 1 YIELD STRENGTH OF A BOFORS STEEL</figcaption></figure>

values, observed and calculated. The parameters are $x_u = 38.57$ kg/mm<fnref n="2" />, $x_0 = 7.74$ kg/mm<fnref n="2" />, $m = 2.934$. Without pooling, the degrees of freedom (d of f) are 9—3 = 6. Then $\chi^2 = 5.40$ gives $P = 0.49$. The agreement is thus very satisfactory.

As a comparison, the values expected on the hypothesis of a normal distribution have been computed and are given in the last column of Table 1. If the classes 9-10 are pooled, the d of f are 8—2 = 6. Then a $\chi^2 = 18.17$ gives a $P = 0.008$, which is not satisfactory at all.

### SIZE DISTRIBUTION OF FLY ASH

The observed values are taken from J. M. Dalla Valle's work.<fnref n="2" /> Fig. 2 gives the curve and Table 2 the values. The parameters are $x_u = 30\mu$, $x_0 = 128\mu$, $m = 2.288$. Without pooling, the d of f are 12—3 = 9. Then $\chi^2 = 8.44$ gives a $P = 0.49$. If the classes 2-3 and 13-14 are pooled, the d of f are 7 and $\chi^2 = 8.44$ gives a $P = 0.29$.

<table><caption>TABLE 2 SIZE DISTRIBUTION OF FLY ASH</caption><caption>(x = particle diameter in 20 microns)</caption><thead><tr><th>x</th><th>Expected values n</th><th>Observed values n</th></tr></thead><tbody><tr><td>2</td><td>3</td><td>3</td></tr><tr><td>3</td><td>14</td><td>14</td></tr><tr><td>4</td><td>34</td><td>34</td></tr><tr><td>5</td><td>62</td><td>56</td></tr><tr><td>6</td><td>92</td><td>85</td></tr><tr><td>7</td><td>122</td><td>126</td></tr><tr><td>8</td><td>150</td><td>150</td></tr><tr><td>9</td><td>172</td><td>175</td></tr><tr><td>10</td><td>188</td><td>188</td></tr><tr><td>11</td><td>199</td><td>197</td></tr><tr><td>12</td><td>205</td><td>202</td></tr><tr><td>13</td><td>209</td><td>208</td></tr><tr><td>14</td><td>211</td><td>211</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>FIG. 2 SIZE DISTRIBUTION OF FLY ASH</figcaption></figure>

### FIBER STRENGTH OF INDIAN COTTON

The observed values are taken from R. S. Koshal and A. J. Turner.<fnref n="3" /> Fig. 3 gives the curve, and Table 3 the values. The parameters are $x_u = 0.59$ gram, $x_0 = 3.73$ grams, $m = 1.456$. If the classes 14 to 16 are pooled, the d of f are 13—3 = 10. Then $\chi^2 = 11.45$ gives a $P = 0.35$.

The authors<fnref n="3" /> have pointed out that the most striking feature about the frequency curve is its asymmetry, showing a well-marked predominance of weak fibers. It was found—they say—that the observation curve would be well fitted by a theoretical curve of Pearson's Type 1, having the following equation

$$y = 599.3 \left( 1 + \frac{x}{18.777} \right)^{0.876716} \left( 1 - \frac{x}{29.1947} \right)^{13.631234}$$

In this equation $y$ represents the frequency of any strength $x$, expressed in grams.

The values computed from this not very handy equation are shown in the last column of Table 3. The d of f are 13—5 = 8 (as there are 5 parameters). Then $\chi^2 = 14.43$ gives a $P = 0.07$.

<footnote n="2">² "Micromeritics," by J. M. Dalla Valle, Pitman Publishing Corporation, New York. N. Y., 1948, p. 57, Fig. 2.</footnote>

<footnote n="3">³ "Studies in the Sampling of Cotton for the Determination of Fiber Properties," by R. S. Koshal and A. J. Turner, Journal of the Textile Institute Transactions, vol. 21, 1930, pp. 325-370.</footnote>
<table><caption>TABLE 3 FIBER STRENGTH OF INDIAN COTTON</caption><caption>(x = tensile strength in grams)</caption><thead><tr><th>x</th><th>Expected values<br>n</th><th>Observed values<br>n̄</th><th>Pearson Type 1<br>n̄</th></tr></thead><tbody><tr><td>1</td><td>118</td><td>177</td><td>127</td></tr><tr><td>2</td><td>645</td><td>667</td><td>639</td></tr><tr><td>3</td><td>1232</td><td>1219</td><td>1255</td></tr><tr><td>4</td><td>1751</td><td>1729</td><td>1777</td></tr><tr><td>5</td><td>2161</td><td>2153</td><td>2184</td></tr><tr><td>6</td><td>2461</td><td>2465</td><td>2480</td></tr><tr><td>7</td><td>2667</td><td>2664</td><td>2683</td></tr><tr><td>8</td><td>2802</td><td>2813</td><td>2816</td></tr><tr><td>9</td><td>2886</td><td>2887</td><td>2899</td></tr><tr><td>10</td><td>2937</td><td>2933</td><td>2949</td></tr><tr><td>11</td><td>2966</td><td>2962</td><td>2978</td></tr><tr><td>12</td><td>2982</td><td>2985</td><td>2994</td></tr><tr><td>13</td><td>2991</td><td>2991</td><td>3003</td></tr><tr><td>14</td><td>2996</td><td>2995</td><td>3007</td></tr><tr><td>15</td><td>2999</td><td>2999</td><td>3009</td></tr><tr><td>16</td><td>3000</td><td>3000</td><td>3010</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>FIG. 3 FIBER STRENGTH OF INDIAN COTTON</figcaption></figure>

<table><caption>TABLE 4 LENGTH OF CYRTOIDEAE</caption><caption>(x = length in microns)</caption><thead><tr><th></th><th></th><th>Expected values</th><th>Observed values</th></tr><tr><th>x</th><th>n₁</th><th>n₂</th><th>n₁+n₂</th></tr></thead><tbody><tr><td>1</td><td>10</td><td>1</td><td>0</td></tr><tr><td>2</td><td>20</td><td>1</td><td>5</td></tr><tr><td>3</td><td>30</td><td>13</td><td>12</td></tr><tr><td>4</td><td>40</td><td>23</td><td>24</td></tr><tr><td>5</td><td>50</td><td>35</td><td>38</td></tr><tr><td>6</td><td>60</td><td>47</td><td>43</td></tr><tr><td>7</td><td>70</td><td>58</td><td>58</td></tr><tr><td>8</td><td>80</td><td>67</td><td>69</td></tr><tr><td>9</td><td>90</td><td>74</td><td>70</td></tr><tr><td>10</td><td>100</td><td>79</td><td>80</td></tr><tr><td>11</td><td>110</td><td>82</td><td>82</td></tr><tr><td>12</td><td>120</td><td>85</td><td>84</td></tr><tr><td>13</td><td>130</td><td>86</td><td>86</td></tr><tr><td>14</td><td>110</td><td>4</td><td>90</td></tr><tr><td>15</td><td>150</td><td>7</td><td>93</td></tr><tr><td>16</td><td>160</td><td>86</td><td>95</td></tr><tr><td>17</td><td>170</td><td>86</td><td>97</td></tr><tr><td>18</td><td>180</td><td>86</td><td>98</td></tr><tr><td>19</td><td>190</td><td>86</td><td>99</td></tr><tr><td>20</td><td>200</td><td>86</td><td>100</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>FIG. 4 LENGTH OF CYRTOIDEAE</figcaption></figure>

In spite of the greater number of parameters, the fit of this distribution function is not as close as that of the first one.

## LENGTH OF CYRTOIDEAE

This is the first example of a complex distribution. The observed values have been obtained from investigations by Dr. Gustaf Arrhenius, on submarine cores from the Swedish Deep-Sea Expedition With *Albatross*. The measurements were made by Dr. C. Jungk, taking samples from each 10 cm of the core, corresponding to an age interval of about 100,000 years. Some fifty populations have been analyzed statistically. About 20 per cent of the populations showed a simple distribution, as exemplified in a previous paper.<fnref n="4" /> The remaining samples showed a two-component distribution.

Fig. 4 gives the curves and Table 4 the values of one of the complex populations. The undivided sample gives the curve marked $N_1 + N_2$. It is easy to see that the distribution is a complex one, and that it is necessary to split up the population in two parts. By trial it was found that 86 of the individuals belonged to component No. 1, and 14 to component No. 2.

The parameters are: Component No. 1: $x_u = 3.75 \mu$, $x_s = 63.2 \mu$, $m = 2.097$. Pooling the classes 2-3, 9-10, and 11-13 gives $\chi^2 = 3.59$. The d of fare 7 — 3 = 4, and $P = 0.47$.

Component No. 2: $x_u = 122.0 \mu$, $x_s = 124.1 \mu$, $m = 1.479$. The number of individuals is too small for the $\chi^2$-test.

## FATIGUE LIFE OF AN ST-37 STEEL

The observed values are taken from Müller-Stock.<fnref n="5" /> The frequency curve in Fig. 5⁵ gives no impression of a complex distribution, which, on the other hand, may easily be seen when

<table><caption>TABLE 5 FATIGUE LIFE OF ST-37 (Rotating-beam test at $\pm 32$ kg/mm²)</caption><thead><tr><th></th><th>N</th><th colspan="3">Expected values</th><th>Observed values</th></tr><tr><th></th><th>10<sup>2</sup></th><th>n<sub>1</sub></th><th>n<sub>2</sub></th><th>n<sub>1</sub>+<sub>2</sub></th><th>n<sub>1</sub>+<sub>2</sub></th></tr></thead><tbody><tr><td>1</td><td>17.5</td><td>4.6</td><td>...</td><td>4.6</td><td>4.6</td></tr><tr><td>2</td><td>22.5</td><td>47.4</td><td>...</td><td>47.4</td><td>47.4</td></tr><tr><td>3</td><td>27.5</td><td>125.1</td><td>...</td><td>125.1</td><td>125.1</td></tr><tr><td>4</td><td>32.5</td><td>161.2</td><td>8.1</td><td>169.3</td><td>169.2</td></tr><tr><td>5</td><td>37.5</td><td>164.9</td><td>28.0</td><td>192.9</td><td>192.7</td></tr><tr><td>6</td><td>42.5</td><td>165.0</td><td>41.8</td><td>206.9</td><td>207.3</td></tr><tr><td>7</td><td>47.5</td><td>165.0</td><td>51.0</td><td>216.0</td><td>215.9</td></tr><tr><td>8</td><td>52.5</td><td>165.0</td><td>57.0</td><td>222.2</td><td>222.2</td></tr><tr><td>9</td><td>57.5</td><td>165.0</td><td>61.0</td><td>226.0</td><td>226.9</td></tr><tr><td>10</td><td>62.5</td><td>165.0</td><td>63.7</td><td>228.7</td><td>228.7</td></tr><tr><td>11</td><td>67.5</td><td>165.0</td><td>65.6</td><td>230.6</td><td>230.5</td></tr><tr><td>12</td><td>72.5</td><td>165.0</td><td>66.9</td><td>231.9</td><td>231.9</td></tr><tr><td>13</td><td>77.5</td><td>165.0</td><td>67.9</td><td>232.9</td><td>232.9</td></tr><tr><td>14</td><td>82.5</td><td>165.0</td><td>68.6</td><td>233.6</td><td>233.5</td></tr><tr><td>15</td><td>87.5</td><td>165.0</td><td>69.1</td><td>234.1</td><td>233.9</td></tr><tr><td>16</td><td>92.5</td><td>165.0</td><td>70.0</td><td>235.0</td><td>235.0</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>FIG. 5 FREQUENCY CURVE OF FATIGUE LIFE OF ST-37 STEEL (Number of specimens versus number of stress cycles.)</figcaption></figure>

<footnote n="4">⁴ "A Statistical Analysis of the Size of Cyrtoidae in Albatross Cores From the East Pacific Ocean," by W. Weibull, *Nature*, vol. 164, 1949, p. 1047.</footnote>

<footnote n="5">⁵ "Der Einfluss dauernd und unterbrochen wirkender, schwingender überbeanspruchung auf die Entwicklung des Dauerbruchs," by H. Müller-Stock, *Mittelungen Kohle- und Eisenforschung*, (March, 1938), by measurements from his Fig. 13; reproduced in Fig. 5 of this paper.</footnote>
using the plots in Fig. 6. The parameters are: Component No. 1: $x_u = 4.032$, $m = 5.956$; Component No. 2: $x_u = 4.484$, $m = 1.215$. Table 5 shows the close agreement between the observed and the calculated values.

<figure><img src="image_7.png" /><figcaption>FIG 6 FATIGUE LIFE OF ST-37 STEEL</figcaption></figure>

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 observed 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 specimens out of 20, making 70 per cent. Exactly the same proportion 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, depends, of course, upon the much larger scatter in fatigue life than in yield strength.

## Appendix

The foregoing statistical methods have been applied to many problems outside the field of applied mechanics. It may perhaps be of interest to have examples of this kind, and for this reason, the following two are given with the tables only:

### STATURES FOR ADULT MALES BORN IN THE BRITISH ISLES

The observed values are taken from Yule and Kendall.<fnref n="6" /> This distribution is classified by the authors as being approximately of the symmetrical type, and there is no mention of its being composed of two parts.

By trial it was found that the population had to be split up into two parts: $N_1 = 6200$ and $N_2 = 2385$. The parameters are as follows:

Component No. 1: $x_u = 50.0$ in., $x_0 = 16.2$ in., $m = 9.6865$. If the classes 1-2 and 14-15 are pooled, we get $\chi^2 = 11.80$. As we have 7 parameters altogether, (one of them is the partition of the population), we take $3^{1/2}$ to each of the components. The $d$ of $f$ are then $12 - 3^{1/2} = 8^{1/2}$, which gives a $P = 0.20$.

Component No. 2: $x_u = 67.4$ in., $x_0 = 2.3$ in., $m = 1.4662$.

<caption>TABLE 6 STATURES FOR ADULT MALES BORN IN THE BRITISH ISLES</caption>

(x = height in inches)

<table><caption>TABLE 7 PHASEOLUS VULGARIS
(Breadth of beams = 0.25 x + 6.70 mm)</caption><thead><tr><th rowspan="2">x</th><th colspan="3">Expected values</th><th rowspan="2">Observed<br>values<br>n<sub>1</sub> + 2</th></tr><tr><th>n<sub>1</sub></th><th>n<sub>2</sub></th><th>n<sub>1</sub> + 2</th></tr></thead><tbody><tr><td>57</td><td>2</td><td>...</td><td>2</td><td>2</td></tr><tr><td>58</td><td>6</td><td>...</td><td>6</td><td>6</td></tr><tr><td>59</td><td>20</td><td>...</td><td>20</td><td>20</td></tr><tr><td>60</td><td>56</td><td>...</td><td>56</td><td>61</td></tr><tr><td>61</td><td>143</td><td>...</td><td>143</td><td>144</td></tr><tr><td>62</td><td>333</td><td>...</td><td>333</td><td>313</td></tr><tr><td>63</td><td>702</td><td>...</td><td>702</td><td>707</td></tr><tr><td>64</td><td>1350</td><td>...</td><td>1350</td><td>1376</td></tr><tr><td>65</td><td>2351</td><td>...</td><td>2351</td><td>2366</td></tr><tr><td>66</td><td>3641</td><td>...</td><td>3641</td><td>3589</td></tr><tr><td>67</td><td>4917</td><td>...</td><td>4917</td><td>4918</td></tr><tr><td>68</td><td>5787</td><td>339</td><td>6126</td><td>6148</td></tr><tr><td>69</td><td>6134</td><td>1079</td><td>7213</td><td>7211</td></tr><tr><td>70</td><td>6197</td><td>1671</td><td>7868</td><td>7857</td></tr><tr><td>71</td><td>6200</td><td>2039</td><td>8239</td><td>8249</td></tr><tr><td>72</td><td>6200</td><td>2233</td><td>8433</td><td>8451</td></tr><tr><td>73</td><td>6200</td><td>2324</td><td>8524</td><td>8530</td></tr><tr><td>74</td><td>6200</td><td>2363</td><td>8563</td><td>8562</td></tr><tr><td>75</td><td>6200</td><td>2378</td><td>8578</td><td>8578</td></tr><tr><td>76</td><td>6200</td><td>2383</td><td>8583</td><td>8583</td></tr><tr><td>77</td><td>6200</td><td>2385</td><td>8585</td><td>8585</td></tr></tbody></table>

<table><thead><tr><th rowspan="2">x</th><th colspan="3">Expected values</th><th rowspan="2">Observed<br>values<br>n<sub>1</sub> + 2</th></tr><tr><th>n<sub>1</sub></th><th>n<sub>2</sub></th><th>n<sub>1</sub> + 2</th></tr></thead><tbody><tr><td>1</td><td>32</td><td>...</td><td>32</td><td>32</td></tr><tr><td>2</td><td>130</td><td>...</td><td>130</td><td>135</td></tr><tr><td>3</td><td>400</td><td>...</td><td>400</td><td>374</td></tr><tr><td>4</td><td>1011</td><td>...</td><td>1011</td><td>998</td></tr><tr><td>5</td><td>2145</td><td>...</td><td>2145</td><td>2185</td></tr><tr><td>6</td><td>3832</td><td>...</td><td>3832</td><td>3835</td></tr><tr><td>7</td><td>5718</td><td>...</td><td>5718</td><td>5718</td></tr><tr><td>8</td><td>7140</td><td>486</td><td>7626</td><td>7648</td></tr><tr><td>9</td><td>7761</td><td>1525</td><td>9286</td><td>9286</td></tr><tr><td>10</td><td>7890</td><td>2310</td><td>10400</td><td>10416</td></tr><tr><td>11</td><td>7900</td><td>3229</td><td>11129</td><td>11153</td></tr><tr><td>12</td><td>7900</td><td>3671</td><td>11571</td><td>11580</td></tr><tr><td>13</td><td>7900</td><td>3908</td><td>11808</td><td>11801</td></tr><tr><td>14</td><td>7900</td><td>4022</td><td>11922</td><td>11911</td></tr><tr><td>15</td><td>7900</td><td>4071</td><td>11971</td><td>11968</td></tr><tr><td>16</td><td>7900</td><td>4091</td><td>11991</td><td>11992</td></tr><tr><td>17</td><td>7900</td><td>4098</td><td>11998</td><td>11998</td></tr><tr><td>18</td><td>7900</td><td>4100</td><td>12000</td><td>12000</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>FIG. 7 FREQUENCY CURVE OF YIELD STRENGTH OF ST-37 STEEL
(Number of specimens versus yield strength in kg/mm².)</figcaption></figure>

If the classes 20-21 are pooled, $\chi^2 = 5.11$. The $d$ of $f$ are 8 - $3^{1/2} = 4^{1/2}$, which gives a $P = 0.35$.

### BREADTH OF BEANS OF PHASEOLUS VULGARIS

This is a classical example, quoted from Charlier<fnref n="7" /> to exemplify the expansion in Edgeworth's series.

If the population is divided into two parts, $N_1 = 7900$ and $N_2 = 4100$, each of them may be very well fitted to a simple distribution function with the following parameters:

Component No. 1: $x_u = -3.0$ (= 5.95 mm), $m = 6.2805$. Without pooling we have the $\chi^2 = 7.70$, and the $d$ of $f$ 10 - $3^{1/2} = 6^{1/2}$, giving a $P = 0.29$.

Component No. 2: $x_u = +7.2$ (= 8.50 mm), $m = 1.6098$.

<footnote n="6">⁶"An Introduction to the Theory of Statistics," by G. U. Yule and M. G. Kendall, eleventh edition. J. B. Lippincott Company, Philadelphia. Pa., 1937. pp. 94 and 111.</footnote>

<footnote n="7">⁷"Die Grundlagen der Mathematischen Statistik," by C. V. L. Charlier, second edition, 1920, p. 73, quoted by Harald Cramér in his book: "Mathematical Methods of Statistics," Princeton University Press, Princeton, N. J., 1945, p. 440.</footnote>
If the classes 17-18 are pooled, the value of $\chi^2 = 4.50$, and the $d$ of $f = 9 - \frac{3}{2} = \frac{5}{2}$ give a $P = 0.56$.

It may be of interest to compare this result with those of
Charlier and Cramér.

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.

Cramér has calculated the values of $\chi^2$ on the hypotheses of

normal distribution and asymptotic expansions from it. The
result was as follows:

Normal distribution $\chi^2 \approx 196.5$ d of $\mathfrak{f}$ 13 $P < 0.001$

First approximation $\chi^2 \approx 34.3$ d of $\mathfrak{f}$ 12 $P < 0.001$

Second approximation $\chi^2 \approx 14.9$ d of $\mathfrak{f}$ 11 $P \approx 0.19$

The agreement is satisfactory in the third case only, requiring
four terms of the series. This operation is certainly of a purely
formal character.