The Weibull modulus is a dimensionless parameter of the Weibull distribution. It represents the width of a probability density function (PDF) in which a higher modulus is a characteristic of a narrower distribution of values. Use case examples include biological and brittle material failure analysis, where modulus is used to describe the variability of failure strength for materials.

The Weibull distribution, represented as a cumulative distribution function (CDF), is defined by:

${\displaystyle F(x)=1-\exp \left(-\left({\frac {x-x_{u}}{x_{0}}}\right)^{m}\right)}$

in which m is the Weibull modulus. ${\displaystyle x_{0}}$ is a parameter found during the fit of data to the Weibull distribution and represents an input value for which ~67% of the data is encompassed. As m increases, the CDF distribution more closely resembles a step function at ${\displaystyle x_{0}}$, which correlates with a sharper peak in the probability density function (PDF) defined by:

${\displaystyle f(x)=\left({\frac {m}{x_{0}}}\right)\left({\frac {x-x_{u}}{x_{0}}}\right)^{m-1}\exp \left(-\left({\frac {x-x_{u}}{x_{0}}}\right)^{m}\right)}$

Failure analysis often uses this distribution, as a CDF of the probability of failure F of a sample, as a function of applied stress σ, in the form:

${\displaystyle F(\sigma )=1-\exp \left[-\left({\frac {\sigma }{\sigma _{0}}}\right)^{m}\right]}$

Failure stress of the sample, σ, is substituted for the ${\displaystyle x}$ property in the above equation. The initial property ${\displaystyle x_{u}}$ is assumed to be 0, an unstressed, equilibrium state of the material.