# A Simple Estimation Method of Weibull Modulus and Verification with Strength Data
## 2 Approximate Methods of Weibull Modulus Estimation  
The most common index of data scatter assumes the normal (or Gaussian) distribution and determines the average \(<\sigma>\) and its standard deviation, S. Another is the Weibull distribution, as shown in Equation (1). These parameters are related. Equation (2) above connects the \(<\sigma>\) value to the Weibull parameters, \(\sigma_{\rm o}\) and m. Similarly, standard deviation, S, is given by [3; 11; 13; 14]:  
\[{\rm S}=\sigma_{\rm o}\ [\Gamma(1+2/{\rm m})-(\Gamma(1+1/{\rm m}))^{2}]^{0.5}, \tag{4}\]  
where \(\Gamma({\rm x})\) is again the gamma function. See [14] for the steps of its derivation. Note that \(\Gamma({\rm x})\) is readily available in Microsoft Excel(r) (version 16.16.8, Microsoft, Redmond, WA, USA, 2016). When Equations (2) and (4) are combined, the coefficient of variation, CV, is defined as [3; 11; 13; 14]:  
\[{\rm CV}={\rm S}/{<\sigma>}=[\Gamma(1+2/{\rm m})-(\Gamma(1+1/{\rm m}))^{2}]^{0.5 }/\Gamma(1+1/{\rm m}). \tag{5}\]CV is a function only of m, and is bounded by 1/m and (\(\pi\)/\(\sqrt{6}\))/m = 1.283/m [13]. In expressing m in terms of the inverse of CV, we have two bounds:  
\[\text{m}=1/\text{CV}=\textless\alpha\textgreater/\text{S}\qquad\text{and}\qquad \text{m}=1.283/\text{CV}=1.283\textless\alpha\textgreater/\text{S}. \tag{6}\]  
This implies that m can be approximated by \(<\)\(\alpha\textgreater/\text{S}\) or the inverse of CV. In 1972, [PERSON] [11] first derived this relationship between m and \(<\)\(\alpha\textgreater/\text{S}\) (or 1/CV), and also gave approximate expressions as:  
\[\text{m}=1.2<\)\(\alpha\textgreater/\text{S}, \tag{7}\]  
and:  
\[\text{m}=(<\)\(\alpha\textgreater/\text{S})^{1.064},\,(1.1<\text{m}<60). \tag{8}\]  
Equation (7), or the [PERSON] relation, has 15% error at m = 2, going down to a 6% error at m = 100. Equation (8) has 1% error for \(1.1<\text{m}<60\). When two constants are used, Equation (5) can be represented as:  
\[\text{m}=1.0461\textless(<\alpha\textgreater/\text{S})^{1.049},\,(\text{R}^{2} =0.9997), \tag{9}\]  
achieving an excellent match. In practice, this can be deemed exact. When numerical values are calculated for CV using Equation (5) by supplying a series of m values, one can then exchange the two sequences, making CV a variable. Thus, Equations (7), (8), and (9) can be compared to Equation (5) and it is found that the last two indeed represent the values of m well. In contrast, Equation (7) has about 7% error. A regression analysis of CV versus m from Equation (5) (with an m value of 1.1 to 127) produces a linear equation with a constant of 1.271 in lieu of 1.2 in Equation (7), yielding a high regression coefficient of R\({}^{2}\) = 0.9999. This is a slight improvement over Equation (9). Thus, [PERSON]'s Equation (7) can be converted to an essentially exact representation using:  
\[\text{m}=1.271<\)\(\alpha\textgreater/\text{S}. \tag{10}\]  
In 1981, [PERSON] et al. [16] derived [PERSON] Equation (7) as an approximation for m \(>\) 2. However, their method is questionable, since the actual constant (\(\sqrt{\text{K}}\)) from their equation (A29) varies from 1.05 at m = 2 to 1.25 at m = 20, reaching the upper limit of 1.283 at m = 60. They selected the constant to be \(\sqrt{1.44}\) = 1.20, which is only valid at m = \(9\pm 1\). Thus, their value appears to be an arbitrary choice, which happened to coincide with [PERSON]'s constant. The [PERSON] equation was incorporated into ASTM C1499 [26]. By 1986, it was apparently well-known, as [PERSON] [27] used it to confirm the m values he obtained using the corresponding CV values. In 1989, [PERSON] [28] presented the same [PERSON] relation, but without citing prior works [11; 16; 26].